Deterministic Algorithm for Non-monotone Submodular Maximization under Matroid and Knapsack Constraints

Deterministic Algorithm for Non-monotone Submo dular Maximization under Matroid and Knapsac k Constrain ts ∗ Shengminjie Chen 1,2 , Yiw ei Gao 1,2 , Kaifeng Lin 1,2 , Xiaoming Sun 1,2 , and Jialin Zhang 1,2 1 State K ey L ab of Pr o c essors, Institute of Computing T e chnolo gy, Chinese A c ademy of Scienc es, Beijing 100190, China 2 Scho ol of Computer Scienc e and T e chnolo gy, University of Chinese A c ademy of Scienc es, Beijing 100049, China Marc h 17, 2026 Abstract Submo dular maximization constitutes a prominent research topic in com binatorial op- timization and theoretical computer science, with extensiv e applications across diverse do- mains. While substan tial adv ancements hav e b een ac hiev ed in appro ximation algorithms for submo dular maximization, the ma jorit y of algorithms yielding high approximation guaran tees are randomized. In this work, we inv estigate deterministic appro ximation algorithms for maximizing non-monotone submo dular functions subject to matroid and knapsac k constraints. F or the tw o distinct constraint settings, we prop ose nov el determin- istic algorithms grounded in an extended m ultilinear extension framework. F or matroid constrain ts, our algorithm ac hieves an appro ximation ratio of (0 . 385 − ε ) , while for knap- sac k constrain ts, the prop osed algorithm attains an appro ximation ratio of (0 . 367 − ε ) . Both algorithms run in p oly ( n ) v alue queries, where n is the size of the ground set, and impro v e up on the state-of-the-art deterministic appro ximation ratios of (0 . 367 − ε ) for matroid constrain ts and 0 . 25 for knapsac k constrain ts. 1 In tro duction Submo dularit y , capturing the principle of diminishing returns, naturally arises in man y functions of theoretical in terest and provides a p ow erful abstraction for man y subset selection problems suc h as game theory [20, 21, 19], maximal so cial w elfare [26, 35, 32], inﬂuence maxi- mization [36, 16, 31], facilit y lo cation [2, 1], data summarization [40, 41, 45], etc. Maximizing submo dular functions sub ject to v arious constrain ts, due to the N P -hard prop ert y , is a funda- men tal problem that has b een extensively studied in b oth op erations researc h and theoretical ∗ Alphab etical Order. Corresp onding Author: Shengminjie Chen, csmj@ict.ac.cn 1 computer science, pla ying a central role in combinatorial optimization and approximation al- gorithms. F ormally , for a ground set N , the submo dular function f : 2 N → R deﬁned on N satisﬁes the following prop erty: for an y subset A ⊆ B ⊆ N and any element u ∈ N \ B , f ( A ∪ { u } ) − f ( A ) ≥ f ( B ∪ { u } ) − f ( B ) . Equiv alen tly , a function f is submo dular if and only if for an y sets S, T ⊆ N , f ( S ) + f ( T ) ≥ f ( S ∪ T ) + f ( S ∩ T ) . The systematic study of submo dular maximization dates bac k to 1978, when Nemhauser et al. [38] abstracted the concept of submo dularity from v arious combinatorial problems and established the 1 − 1 /e approx imation ratio for monotone functions via a greedy algorithm. Since then, numerous com binatorial algorithms hav e b een dev elop ed, particularly for cardinality constrain ts [42, 37, 11], knapsack constraints [44, 34, 3, 25, 33], matroid constrain ts [27, 10, 30], and the unconstrained case [12, 39]. A signiﬁcant adv ancemen t came with the introduction of the m ultilinear extension framework by V ondrák [46]. This framew ork extends a discrete submo dular function to a con tinuous m ultilinear function F M E : [0 , 1] N → R , F M E ( x ) = E [ f ( R ( x ))] = X S ⊆N f ( S ) Y i ∈ S x i Y j / ∈ S (1 − x j ) , where R ( x ) is a random subset of N in which each element i ∈ N is included indep endently with probabilit y x i . By applying a F rank-W olfe-lik e contin uous optimization algorithm, a frac- tional solution is obtained, whic h is then con v erted into a discrete solution via a rounding pro cedure suc h as Pipage Rounding [18] and Swap Rounding [13]. This approach ac hiev es an appro ximation ratio 1 − 1 /e for monotone functions under matroid constraints. Crucially , the optimization comp onen t of this framework generally requires only that the constraint b e a down-closed conv ex p olytop e. Giv en that lossless rounding tec hniques exist for cardinal- it y and matroid constraints, and that knapsac k constraints admit nearly lossless rounding via con tention resolution sc hemes [14], the multilinear extension framework supp orts a v ersatile algorithmic design that is not tied to sp eciﬁc constraint types. Subsequent researc h has con- tin uously adv anced the approximation ratio for non-monotone submo dular maximization, from the long-standing 1 /e ( ≈ 0 . 367) [46] to (0 . 372 − ε ) [22], (0 . 385 − ε ) [6], and (0 . 401 − ε ) [7]. Closing the remaining gap b etw een these re sults and the theoretical upp er bound of 0 . 478 [28] remains a signiﬁcant op en challenge in the ﬁeld. Computing the v alue of the m ultilin- ear extension exactly typically requires exp onen tially many function ev aluations, and random sampling is widely regarded as the only p olynomial-query approach for estimating its v alue. Ho wev er, suc h randomness can b e undesirable in applications where deterministic outcomes are required, suc h as safet y-critical or repro ducibilit y-sensitive en vironmen ts. Randomized al- gorithms provide guaran tees in exp ectation but not necessarily in the worst case, making them less predictable in certain scenarios. Designing deterministic algorithms for submo dular maximization, whic h returns the same solution on every run and guarantees an appro ximation ratio in the worst case rather than only in exp ectation, has emerged as an imp ortan t and active research direction in recent y ears. Broadly sp eaking, deterministic algorithms for submo dular maximization follow t wo 2 main paradigms: (i) designing algorithms that are deterministic by construction, and (ii) de- randomizing existing randomized algorithms while preserving their approximation guarantees. Under the ﬁrst paradigm, man y classical algorithms for monotone submo dular maximization are inherently deterministic. Representativ e examples include greedy-lik e algorithms, threshold- based metho ds, and discrete lo cal search algorithms. The standard greedy algorithm ac hiev es a (1 − 1 /e ) -approximation under a cardinality constrain t [38] and a 1 / 2 -approximation under matroid constraints [27], while Sviridenko prop osed a (1 − 1 /e ) -appro ximation under a knap- sac k constraint [44]. In addition, threshold decreasing algorithms, which iteratively lo wer a marginal-gain threshold and select elemen ts exceeding the current threshold, pro vide a deter- ministic and oracle-eﬃcien t alternativ e to the standard greedy approach, and achiev e near- optimal approximation guarantees with signiﬁcan tly improv ed running time [4]. Buc h binder et al. [10] sho w ed that for monotone submo dular maximization under a matroid constrain t, the split-and-gr ow algorithm enables deterministic algorithms to surpass the 1 / 2 barrier, achieving a (1 / 2 + ε ) -appro ximation. F or non-monotone objectives, deterministic lo cal searc h algorithms w ere systematically studied by F eige et al. [23], establishing constant-factor guaran tees for un- constrained submo dular maximization. Bey ond these general framew orks, several algorithms ha ve b een explicitly designed to achiev e strong deterministic guaran tees. Notably , the twin greedy algorithm prop osed b y Han et al. [29] achiev es a 1 / 4 -appro ximation for non-monotone submo dular maximization under matroid constrain ts, whic h w as later extended to knapsac k constrain ts by Sun et al. [43] while preserving the same approximation ratio. T o the b est of our kno wledge, this remains the strongest kno wn deterministic guaran tee for non-monotone submo dular maximization under knapsac k constrain ts. The second paradigm fo cuses on der andomizing pr eviously r andomize d algorithms , and has witnessed substantial progress in recen t y ears. Early examples include the random greedy algorithm for cardinality constraints [11] and the double greedy algorithm for unconstrained non-monotone maximization [12], both of whic h were later systematically derandomized by Buc hbinder and F eldman [5]. More recen tly , Buch binder and F eldman [8] derandomized the non-oblivious lo cal searc h algorithm, obtaining a deterministic (1 − 1 /e − ε ) -appro ximation for monotone submo dular maximization under matroid constraints. In addition, Chen et al. [17] derandomized the measured contin uous greedy algorithm [6], achieving approximation ratios of 0 . 385 − ε under cardinalit y constrain ts and 0 . 305 − ε under matroid constraints. V ery recen tly , Buc hbinder et al. [9] introduced the Extende d Multiline ar Extension (EME), a nov el derandom- ization framework that maintains a distribution with constant-size support and enables exact ev aluation without random sampling. Using this framew ork, they further obtained a deter- ministic (1 /e − ε ) -approximation for non-monotone submo dular maximization under matroid constrain ts. While the Extended Multilinear Extension framework oﬀers a promising av en ue for deran- domizing submo dular maximization algorithms, exploiting this framework in broader settings presen ts substantial technical challenges. At a high lev el, the p o wer of EME critically relies on maintaining a distribution whose fractional supp ort v alue remains constant throughout the algorithm. This structural requiremen t fundamentally alters the nature of the optimization pro cess: unlik e classical multilinear extension metho ds, the optimization under EME can no longer b e carried out via purely con tin uous pro cedures. Instead, each up date step must care- fully combine contin uous adjustmen ts with embedded com binatorial subroutines that preserv e 3 the b ounded supp ort prop ert y . Designing suc h h ybrid steps is non trivial, as standard contin u- ous optimization tec hniques are no longer applicable. Moreov er, the existing derandomization approac h for EME is tightly coupled with the exchange prop erties of matroids, and the re- sulting algorithmic framework relies hea vily on matroid-sp eciﬁc com binatorial structures. As a consequence, these tec hniques do not readily extend to other algorithms and constraint fami- lies. F or more general combinatorial constrain ts, additional diﬃculties arise: one m ust design new rounding pro cedures that are compatible with the EME framework while sim ultaneously guaran teeing that the supp ort size remains constant. Balancing these requirements p oses a cen tral obstacle to extending EME-based derandomization b ey ond the matroid setting. These c hallenges naturally raise the follo wing question: Can we lever age the EME fr amework to der andomize c ontinuous optimization algorithms and achieve impr ove d appr oximation r atios under diﬀer ent c ombinatorial c onstr aints? Our Contribution. In our w ork, w e consider the problem of maximizing a non-monotone submo dular function sub ject to constraints of the form max { f ( S ) : S ∈ C } where C denotes the constraint of the submo dular maximization problem, that is, the collection of feasible sets: • Matroid constrain ts. A matr oid M = ( N , I ) is a set system, where I ⊆ 2 N is a family of subsets of the ground set N satisfying the following prop erties: (i) ∅ ∈ I ; (ii) If A ⊆ B and B ∈ I , then A ∈ I ; (iii) If A, B ∈ I with | A | < | B | , there exists an elemen t u ∈ B \ A suc h that A + u ∈ I . The members of I are called indep endent sets , and the maximal indep enden t sets are called b ases . All bases of a matroid hav e the same cardinalit y , which is referred to as the r ank of M . F or a subset S ⊆ N (not necessarily indep enden t), its r ank is deﬁned as the size of its maxim um indep endent subset, i.e., rank( S ) = max T ⊆ S, T ∈I | T | . The sp an of a set S ⊆ N is deﬁned as the set of elements u ∈ N such that r = rank( S ∪ { u } ) = rank ( S ) . • Knapsac k constrain ts. Each element u ∈ N is asso ciated with a non-negativ e cost w ( u ) , and a set S ⊆ N is feasible if P u ∈ S w ( u ) ≤ B , where B > 0 denotes the knapsac k budget. F or simplicit y , W e denote the weigh t of a set S by w ( S ) = P u ∈ S w ( u ) . W e ﬁrstly address these c hallenges b y derandomizing the aided contin uous greedy algorithm under matroid constraints. Our approach exploits a discrete stationary p oint to aid the opti- mization tra jectory and p erforms a piecewise optimization o v er the EME, thereb y maintaining a constant-size supp ort throughout the pro cess. Ho wev er, extending the conv ergence analysis to the EME setting is non trivial. Existing conv ergence analyses of the aid measured con tinuous greedy algorithm rely on the Lov ász extension as a low er b ound of the multilinear extension [6] or on the directional conca vity of the multilinear extension of a submo dular function [15]. These prop erties do not carry ov er to the EME, whic h implies us adopt a new analysis pro cess. When combined with the rounding sc heme of Buch binder et al. [9], our framework yields a de- terministic (0 . 385 − ε ) -approximation. This strictly impro v es up on the previously b est-kno wn deterministic guarantee of (1 /e − ε ) under matroid constraints, whic h was also achiev ed in [9]. The formal statement is as follows: 4 T able 1: Comparison of algorithms for non-monotone submo dular maximization under matroid and knapsac k constrain ts. The table includes state-of-the-art results, our deterministic algo- rithms, and algorithms w e derandomize (mark ed with † ). Constrain ts Appro ximation Ratio Query Complexity Type Reference 0 . 385 − ε O ε ( n 11 ) [6] † 0 . 401 − ε p oly( n, 1 /ε ) Random [7] 0 . 305 − ε O ε ( nk ) [17] 0 . 367 − ε O ε ( n 5 ) [9] Matroid 0 . 385 − ε O ε ( n 5 ) Deterministic Ours 0 . 367 − ε O ε ( n ε − 2 ) [24] † 0 . 401 − ε O ε ( n ε − 2 ) Random [7] 0 . 25 O ( n 4 ) [43] Knapsac k 0 . 367 − ε O ε ( n ε − 2 ) Deterministic Ours Theorem 1. Given a non-ne gative submo dular function f : 2 N → R ≥ 0 and a matr oid M = ( N , I ) , Algorithm 1 is a deterministic algorithm that uses O ε ( n 5 ) queries, r eturns a set S satisfying the matr oid c onstr aint M = ( N , I ) , and achieves f ( S ) ≥ (0 . 385 − O ( ε )) f (OPT) . F or knapsack constraints, the lac k of matroid exc hange prop erties necessitates a diﬀerent and more delicate approach. W e b egin by enumerating a set of O (1 /ε 2 ) elements, which allows us to assume without loss of generality that all remaining elements are smal l enough. Under this small-element assumption, we dev elop an optimization algorithm o ver the EME that is guided by element densities. This density-based optimization admits a prov able approximation guaran tee on the EME solution and can b e implemen ted while maintaining a constant-size supp ort. W e then apply a rounding pro cedure inspired b y Pipage Rounding, which rep eatedly selects t wo non-integral elements in the support of the EME and mov es along a suitable conv ex direc- tion to integralize one of them. Unlike the matroid setting, this pro cess ma y leav e at most one elemen t in a fractional state. Ho w ever, since this element is guaranteed to b e small, feasibilit y can b e ensured b y executing the optimization step with a reduced knapsack capacit y of (1 − ε ) , thereb y reserving suﬃcient slac k to accommo date the ﬁnal rounding. Ov erall, this framew ork yields a deterministic (1 /e − ε ) -appro ximation for submo dular maximization under a knapsac k constrain t, substan tially impro ving upon the 1 / 4 approximation ratio of t win greedy methods. Theorem 2. Given a non-ne gative submo dular function f : 2 N → R ≥ 0 and a weight function w : N → R ≥ 0 , A lgorithm 5 is a deterministic algorithm that uses O ε ( n ε − 2 ) queries, r eturns a set S satisfying the knapsack c onstr aint P u ∈ S w ( u ) ≤ B , and achieves f ( S ) ≥ (1 /e − O ( ε )) f (OPT) . Organization. Our paper is organized as follo ws: in Section 2, w e in tro duce basic notations and review the deﬁnition and fundamen tal prop erties of the extended multilinear extension. In Section 3, w e present our algorithm for the matroid constrain t: Section 3.1 describ es the dis- crete lo cal searc h pro cedure, Section 3.2 explains how a lo cal search stationary p oint guides the Deterministic aided Contin uous Greedy algorithm, and Section 3.3 analyzes the p erformance 5 of the ov erall algorithm. In Section 4, we present our algorithm for the knapsack constraint: Section 4.1 b egins with the enumeration of elemen ts, Section 4.2 introduces a deterministic con tinuous greedy algorithm for the knapsac k constrain t, Section 4.3 presents our rounding pro cedure, and Section 4.4 analyzes the p erformance of the entire algorithm. Finally , in Sec- tion 5, we summarize our work and discuss possible directions for future researc h. 2 Preliminary In this section, w e ﬁrst introduce some basic deﬁnitions and essential lemmas adopted in our algorithm design. Without ambiguit y , we denote n = |N | and use OPT to represent the optimal feasible solution. W e do not explicitly consider how the submo dular function f is computed; instead, we assume access to a value or acle that returns f ( S ) for any set S ⊆ N in O (1) time. In addition, for problems under matroid constrain ts, we assume access to a memb ership (or indep endenc e) or acle that determines whether a given set is indep endent in O (1) time. The total num b er of calls to the v alue oracle and the mem b ership oracle made b y an algorithm is referred to as its query c omplexity . P olytop e of Constrain ts. Algorithms based on the con tin uous optimization to ol are typi- cally executed o ver a feasible region, whose form dep ends on the underlying constrain ts and is referred to as the p olytop e associated with the corresp onding constraint. The matroid p olytop e P ( M ) ⊆ [ 0 , 1] N is deﬁned as: P ( M ) = conv { 1 S | S ∈ I } where 1 S ∈ [ 0 , 1] N is a vector suc h that for any u ∈ S , ( 1 S ) u = 1 , and for any u ∈ N \ S , ( 1 S ) u = 0 . The knapsac k p olytop e is deﬁned as P ( B ) = ( x ∈ [0 , 1] N : X u ∈N w ( u ) · x u ≤ B ) Extended Multilinear Extension. The pap er from [9] prop osed an extension of the mul- tilinear extension to address the issue that the standard multilinear extension cannot yield deterministic algorithms. F or a submo dular function f : 2 N → R , its extended multilinear extension F : [ 0 , 1] 2 N → R is deﬁned as: F ( y ) = X J ⊆ 2 N   f   [ S ∈J S   · Y S ∈J y S · Y S / ∈J (1 − y S )   Belo w we introduce some notation and prop erties related to the extended multilinear ex- tension. All the prop erties can b e found in [9]. Deﬁnition 3 (Random set) . F or a ve ctor y ∈ [0 , 1] 2 N , deﬁne the r andom set R ( y , S ) , wher e Pr[ R ( y , S ) = S ] = y S , otherwise R ( y , S ) = ∅ . F urthermor e, deﬁne R ( y ) = S S ⊆N R ( y , S ) . Observ ation 4. F or any y ∈ [0 , 1] 2 N F ( y ) = E [ f ( R ( y ))] . 6 Deﬁnition 5 (Co ordinate-wise probabilistic sum) . The c o or dinate-wise pr ob abilistic sum of m ve ctors a (1) , a (2) , . . . , a ( m ) ∈ [0 , 1] n is a ve ctor c wher e e ach entry k ∈ { 1 , . . . , n } is deﬁne d as: m M i =1 a ( i ) ! k = 1 − m Y i =1 (1 − a ( i ) k ) Deﬁnition 6 (Marginal vector of y ) . L et y ∈ [ 0 , 1] 2 N , then the mar ginal ve ctor Mar ( y ) ∈ [ 0 , 1] N is deﬁne d by: Mar u ( y ) ≜ Pr[ u ∈ R ( y )] = 1 − Y S ⊆ 2 N | u ∈ S (1 − y S ) ∀ u ∈ N Observ ation 7. L et y 1 , y 2 ∈ [0 , 1] 2 N b e ve ctors such that y 1 + y 2 ≤ 1 . Then: Mar ( y 1 ⊕ y 2 ) = Mar ( y 1 ) ⊕ Mar ( y 2 ) Observ ation 8. Given a set function f : 2 N → R and a ve ctor y ∈ [0 , 1] 2 N , we deﬁne the set function g y : 2 N → R by g y ( A ) ≜ F ( e A ∨ y ) ∀ A ⊆ N . (1) Wher e e A ∈ [0 , 1] 2 N is a ve ctor such that ( e A ) A = 1 and for any S ⊆ N , S  = A , ( e A ) S = 0 . If f is a non-ne gative submo dular function, then g y ( A ) is also a non-ne gative submo dular function. Observ ation 9. L et F b e the extende d multiline ar extension of an arbitr ary set function f : 2 N → R . Then, for every ve ctor y ∈ [ 0 , 1] 2 N , F ( e S ∨ y ) ≥ (1 − ∥ Mar( y ) ∥ ∞ ) · f ( S ) ∀ S ⊆ N ∂ 2 F ( ∂ y S ) 2 ( y ) = 0 ∀ S ⊆ N ∂ 2 F ∂ y S ∂ y T ( y ) ≤ 0 ∀ S, T ⊆ N , S ∩ T = ∅ (1 − y S ) · ∂ F ( y ) ∂ y S = F ( e S ∨ y ) − F ( y ) ∀ S ⊆ N , y S ∈ [0 , 1] F ( y ⊕ z ) = X J ⊆ 2 N   F y ∨ X S ∈ J e S ! · Y S ∈ J z S · Y S ∈ 2 N \ J (1 − z S )   ∀ z ∈ [0 , 1] 2 N Lo v ász Extension. The Lov ász extension is another imp ortant extension for set functions. Ho wev er, in this pap er, it is only used as a deﬁnition for auxiliary pro of conclusions, so its prop erties will not b e extensively co vered. F or a submo dular function f : 2 N → R , its Lo v ász extension ˆ f : [0 , 1] N → R is deﬁned as: ˆ f ( x ) = Z 1 0 f ( T λ ( x )) dλ where T λ ( x ) = { u ∈ N : x u > λ } . 7 Lemma 10. Given x ∈ [0 , 1] N and a submo dular function f : 2 N → R . L et ˆ f : [0 , 1] N → R b e its L ovász extension. L et A ( x ) ⊆ N b e a r andom set satisfying ∀ u ∈ N , Pr[ u ∈ A ( x )] = x u . Then: E [ f ( A ( x ))] ≥ ˆ f ( x ) Pr o of. Let { u 1 , u 2 , . . . , u n } = N b e ordered such that ∀ i < j , x u i ≥ x u j . Let ev ent X i indicate u i ∈ A ( x ) . Let A i = { u j | j ≤ i } . E [ f ( A ( x ))] = E " f ( ∅ ) + n X i =1 f ( A ( x ) ∩ A i | A ( x ) ∩ A i − 1 ) # = E " f ( ∅ ) + n X i =1 X i · f ( u i | A i − 1 ∩ A ( x )) # ≥ E " f ( ∅ ) + n X i =1 X i · f ( u i | A i − 1 ) # = f ( ∅ ) + n X i =1 E [ X i ] · f ( u i | A i − 1 ) = f ( ∅ ) + n X i =1 x u i · f ( u i | A i − 1 ) = f ( ∅ ) + Z 1 0 X i λ< x u i f ( u i | A i − 1 ) dλ = ˆ f ( x ) The ﬁrst equalit y decomp oses the function v alue into the incremen tal con tribution of eac h elemen t. The inequalit y follows from submo dularit y . The ﬁnal equality holds b ecause, due to the descending order of x u i , the sum (including the preceding empty set term) constitutes a diﬀerence sequence for f ( T λ ( x )) , th us equaling ˆ f ( x ) . 3 Matroid Constrain ts In this section, we in tro duce the deterministic aided measured contin uous greedy algorithm based on the extended m ultilinear extension. In [9], the authors ha v e pro vided a rounding metho d for EME v ectors when the supp ort set is suﬃciently small. Consequently , utilizing this rounding algorithm, our algorithm only needs to obtain a v ector with a suﬃcien tly small supp ort set to provide a solution for maximizing submo dular functions. Additionally , we introduce a dumm y element set D , whic h only o ccupy constraints without c hanging the function v alue, and the deﬁnition of the discrete stationary p oin t, which are essential bridges in our theoretical analysis. Deterministic-Pipage In [9], the authors provide a deterministic pipage rounding pro cedure that conv erts an EME v ector with suﬃciently small supp ort in to a feasible set in the matroid. Therefore, our algorithm only needs to compute a vector whose supp ort size is suﬃciently small in order to obtain a solution for the submo dular maximization problem. Theorem 11 (Deterministic-Pipage, Theorem 5.6 in [9]) . F or a ve ctor y ∈ [ 0 , 1] 2 N , if Mar ( y ) ∈ P ( M ) , ther e exists an algorithm that, within O ( n 2 · 2 fr ac ( y ) ) value queries and O ( n 5 log 2 n ) indep endenc e queries, r eturns a set S ∈ M satisfying f ( S ) ≥ F ( y ) . 8 Dumm y elemen ts. W e can make the subsequent analysis more conv enient b y adding at most n dummy elements to the original set N . Speciﬁcally , let ¯ N = { u 1 , . . . , u n } , ¯ I = { S ⊆ N ∪ ¯ N | S ∩ N ∈ I , | S | ≤ rank ( M ) } , and ¯ f : 2 N ∪ ¯ N → R , ¯ f ( S ) = f ( S \ ¯ N ) . The solution to the new problem ( N ∪ ¯ N , ¯ I ) , ¯ f will b e consistent with the original problem. A dditionally , w e can alw ays add dumm y elements to a set S ∈ I to make S a basis of the matroid M without c hanging the function v alue of S . The computed result will inevitably con tain some dumm y elemen ts; w e simply remov e them from the solution set to obtain a solution to the original problem with the same function v alue. In subsequen t analysis, w e will assume by default that the problem includes a suﬃcient num b er of dumm y elemen ts. Lemma 12. ¯ M = ( N ∪ ¯ N , ¯ I ) is a matr oid, and ¯ f is a submo dular function. Pr o of. F or an y A ⊆ B ⊆ N ∪ ¯ N , u ∈ ( N ∪ ¯ N ) \ B : If u ∈ ¯ N , then ¯ f ( A + u ) − ¯ f ( A ) = ¯ f ( B + u ) − ¯ f ( B ) = 0 . Otherwise, ¯ f ( A + u ) − ¯ f ( A ) = f (( A \ ¯ N ) + u ) − f ( A \ ¯ N ) ≥ f (( B \ ¯ N ) + u ) − f ( B \ ¯ N ) = ¯ f ( B + u ) − ¯ f ( B ) Therefore, ¯ f is submodular. T o pro v e ¯ M is a matroid, w e only need to prov e ¯ I satisﬁes the exc hange prop erty . F or an y A, B ∈ ¯ I with | A | < | B | . If | A ∩ ¯ N | < | B ∩ ¯ N | , then there exists a dummy element u ∈ ( B ∩ ¯ N ) \ A , so A + u ∈ ¯ I . Otherwise, consider | A \ ¯ N | = | A | − | A ∩ ¯ N | < | B | − | B ∩ ¯ N | = | B \ ¯ N | . By the deﬁnition of ¯ I , A ∈ ¯ I ⇒ A \ ¯ N ∈ I , and A \ ¯ N , B \ ¯ N ∈ I . By the matroid exc hange prop ert y , there exists u ∈ B \ ¯ N ⊆ B such that ( A \ ¯ N ) + u ∈ I , implying A + u ∈ ¯ I . The main idea of the algorithm on the extended m ultilinear extension is similar with that on the multilinear extension. First, the main algorithm ﬁnds a reference solution Z through the lo cal searc h algorithm. Then, adopting the information from Z , the main algorithm p er- forms con tinuous greedy in a direction orthogonal to Z b efore time t s and perform contin uous greedy in all direction after time t s , obtaining a fractional solution y negativ ely correlated with f ( Z ) . Finally , the b est of Rounding ( y ) and Z is an 0 . 385 − O ( ε ) approximation solution. The formal statemen t of the deterministic 0 . 385 − O ( ε ) appro ximation algorithm for non-monotone submo dular maximization sub ject to matroid constrain ts is as follows: Algorithm 1: MAIN ( M = ( N , I ) , f , t s ∈ [0 , 1] , ε ∈ (0 , 1)) 1 Z ← LocalSearch ( M , f , ε ) ; 2 y 1 ← AidedContinuousGreedy ( M , f , Z , t s , ε ) ; 3 return arg max Y ∈{ Deterministic-Pip age ( y 1 ) , Z } f ( Y ) ; 3.1 Discrete Lo cal Searc h In this subsection, we in tro duce a discrete lo cal search algorithm to obtain the reference solution Z . Although this algorithm originates from Algorithm 3 in [8], w e restate and reprov e it suc h that it can b e adapted to the non-monotone submo dular maximization problem we 9 study . The algorithm calls a deterministic combinatorial algorithm to obtain a constant-factor appro ximation of f ( O P T ) as its initial solution and con tin uously swaps elements to ensure that the curren t solution is a basis of the matroid M = ( N , I ) and the ob jective function v alue increases. Essen tially , the solution returned b y Algorithm 2 is a discrete stationary p oint. The formal statemen t is as follo ws: Algorithm 2: LocalSearch ( M = ( N , I ) , f , ε ∈ (0 , 1)) 1 Use the algorithm in Lemma 13 to ﬁnd S 0 suc h that f ( S 0 ) ≥ 0 . 305 · f ( OP T ) , set S ← S 0 ; 2 Add elements until S b ecomes a basis of M ; 3 while there exist u ∈ S , v ∈ N \ S suc h that S − u + v ∈ I and f ( v | S ) − f ( u | S − u ) ≥ ε r · f ( S 0 ) do 4 S ← S + v − u ; 5 return S ; Lemma 13 (Theorem 2.4 in [17]) . F or a non-monotone submo dular maximization pr oblem ( M , f ) , ther e exists a deterministic algorithm that r eturns an 0 . 305 − ε appr oximate solution with at most O ε ( nr ) value and indep endenc e queries. Discrete Stationary P oints. In tuitively , w e call a solution stationary if no lo cal exchange can signiﬁcantly improv e the ob jective v alue, meaning the algorithm has reac hed a stable state with respect to 1 -exc hange mo ves. W e formalize the notion of a discrete stationary p oint under 1 -exc hange. Let M = ( N , I ) b e a matroid of rank r and f : 2 N → R b e submo dular. A basis S ∈ I is called an α -appro ximate discrete stationary p oint if for ev ery pair u ∈ S and v ∈ N \ S such that S − u + v ∈ I , f ( v | S ) − f ( u | S − u ) ≤ α. When α = 0 , this coincides with the standard notion of a 1 -exchange lo cal optim um. Algorithm 2 terminates exactly when no feasible exchange impro ves the ob jective b y more than ε r · f ( S 0 ) ; hence the returned solution is an α -appro ximate discrete stationary p oint with α = ε r · f ( S 0 ) . F or notational simplicit y , throughout the sequel we refer to such α -appro ximate discrete stationary p oints simply as stationary p oints , as the additiv e error will b e explicitly accoun ted for in the analysis. Next, w e illustrate the theoretical guaran tee for Algorithm 2 as follo ws: Theorem 14. A lgorithm 2 terminates within O ( ε − 1 nr ) value queries and O ( ε − 1 nr 2 ) indep en- denc e queries, and the r eturne d set S satisﬁes S ∈ I and, for any T ∈ I , ε ∈ (0 , 1) , f ( S ) ≥ 1 2 ( f ( S ∩ T ) + f ( S ∪ T )) − ε · f ( O P T ) (2) Pr o of. Due to dummy elements, we can assume | S | = | T | = r , b oth b eing bases of M . Let { u 1 , . . . , u k } and { v 1 , . . . , v k } denote the elements in S \ T and T \ S , resp ectively . By the matroid exc hange prop ert y , we can ﬁnd a bijection h : S \ T → T \ S suc h that S − u i + h ( u i ) ∈ I for 10 eac h u i . f ( S ∪ T ) + f ( S ∩ T ) − 2 f ( S ) =( f ( S ∪ T ) − f ( S )) − ( f ( S ) − f ( S ∩ T )) ≤ k X i =1 f ( v i | S ) ! − k X i =1 f ( u i | S − u i ) ! (b y submo dularity) = k X i =1 [ f ( h ( u i ) | S ) − f ( u i | S − u i )] ≤ k · ε r · f ( S 0 ) ≤ ε · f ( O P T ) (b y loop condition) Consequen tly , the solution returned b y Algorithm 2 satisﬁes the inequality (2). In each itera- tion, w e ha ve: f ( S + v − u ) − f ( S ) = ( f ( S + v − u ) − f ( S − u )) + ( f ( S − u ) − f ( S )) ≥ ( f ( S + v ) − f ( S )) + ( f ( S − u ) − f ( S )) (b y submo dularity) = f ( v | S ) − f ( u | S − u ) ≥ ε r · f ( S 0 ) > ε 4 r · f ( O P T ) The inequality in the algorithm’s lo op condition ensures the last step. Since f ( S ) − f ( S 0 ) ≤ f ( O P T ) − f ( S 0 ) ≤ f ( O P T ) , the loop m ust terminate after at most 4 r ε iterations; otherwise, it w ould con tradict f ( S ) − f ( S 0 ) ≤ f ( O P T ) . In eac h iteration, the algorithm requires O ( n ) v alue oracle queries and O ( nr ) indep endence oracle queries. T otally , Algorithm 2 needs O ( ε − 1 nr ) v alue oracle queries and O ( ε − 1 nr 2 ) independence oracle queries. 3.2 Deterministic aided Con tin uous Greedy In this subsection, w e introduce the aided con tin uous greedy algorithm that leverages the discrete stationary p oin t returned by the lo cal search algorithm to guide the forw ard direction on contin uous greedy algorithm. Firstly , w e in tro duce a combinatorial algorithm for obtaining a forw ard direction. The con tin uous greedy pro cess then pro ceeds by taking a small step along this direction. By construction, the direction obtained at each step yields a suﬃciently large marginal gain, and thus the algorithm adv ances by up dating the current solution b y a small constan t multiple of this vector. Split Algorithm. The split algorithm serv es to extract a forw ard direction with the largest marginal gain at the current p oin t, under the constrain t that the supp ort size remains b ounded. This constrain t is crucial from a computational p ersp ectiv e: ev aluating the extended m ultilinear extension v alue incurs a cost that s cales with the size of its supp ort. The split algorithm returns a partition of an indep enden t set. As the n umber of partitions are b ounded, the size of supp ort set for the forward direction is also b ounded b y ℓ . Although the theoretical guarantees for Algorithm 3 w as prop osed in [9], for completeness, we also restate the pro of based on the Lo v ász extension, whic h has some subtle diﬀerences from [9]. 11 Algorithm 3: Split ( M = ( N , I ) , f , ℓ ) 1 Initialize T 1 ← ∅ , T 2 ← ∅ , . . . , T ℓ ← ∅ ; 2 Let T ← S ℓ i =1 T i ; 3 while T is not a b asis of M do 4 N ′ ← { u ∈ N \ T | T + u ∈ I } ; 5 ( u, j ) ← arg max ( u,j ) ∈N ′ × [ ℓ ] f ( u | T j ) ; 6 T j ← T j + u ; 7 return ( T 1 , T 2 , . . . , T ℓ ) ; Lemma 15 (Lemma 4.2 in [9]) . Given a matr oid M on the gr ound set N , a submo dular function f , and an inte ger ℓ as input, A lgorithm 3 outputs disjoint sets ( T 1 , T 2 , . . . , T ℓ ) whose union is a b asis of M , satisfying for any set O ⊆ N : ℓ X j =1 f ( T j | ∅ ) ≥  1 − 1 ℓ  · f ( O ) − 1 ℓ X j ∈ [ ℓ ] f ( T j ) (3) Pr o of. Let T = S ℓ j =1 T j . Due to dummy elements, we can assume | T | = | O | = r . By the exc hange prop erty of matroid, there exists a bijection h : O → T such that for an y u ∈ O \ T , ( T − h ( u )) + u ∈ I . F urthermore, for u ∈ O ∩ T , h ( u ) = u . Let u i ∈ T \ O b e the i -th element added to T in Algorithm 3. Deﬁne T i j = T j \ { h − 1 ( u k ) | k > i } , and let j i denote the index of the set to whic h u i b elongs, i.e., u i ∈ T j i . A ccording to the algorithm, w e state: f ( u i | T i − 1 j i ) ≥ 1 ℓ ℓ X j =1 f ( h − 1 ( u i ) | T i − 1 j ) ≥ 1 ℓ ℓ X j =1 f ( h − 1 ( u i ) | T j ) The ﬁrst inequality holds b ecause the exchange prop erty guaran tees ( T − u i ) + h − 1 ( u i ) ∈ I , so the greedy strategy of the algorithm ensures f ( u i | T i − 1 j i ) ≥ f ( h − 1 ( u i ) | T i − 1 j ) for an y j (otherwise h − 1 ( u i ) would hav e b een added to some T j at this step). The second inequality follo ws from the submo dularit y of f . Summing the ab ov e equation ov er all i gives: ℓ X j =1 f ( T j | ∅ ) = r X i =1 f ( u i | T i − 1 j i ) ≥ 1 ℓ ℓ X j =1 r X i =1 f ( h − 1 ( u i ) | T j ) ≥ 1 ℓ ℓ X j =1 f r [ i =1 { h − 1 ( u i ) } | T j ! (b y submo dularity) = 1 ℓ ℓ X j =1 ( f ( T j ∪ O ) − f ( T j )) Let A b e a random set where for each 1 ≤ j ≤ ℓ , Pr[ A = T j ∪ O ] = 1 ℓ . Then 1 ℓ P ℓ j =1 f ( T j ∪ O ) = E [ f ( A )] . Let p u = Pr[ u ∈ A ] . Since the T j are disjoint, p u ≤ 1 ℓ for all u ∈ N \ O , and p u = 1 for all u ∈ O . By Lemma 10, E [ f ( A )] ≥ ˆ f ( p ) ≥ Z 1 1 ℓ f ( T λ ( p )) dλ ≥  1 − 1 ℓ  f ( O ) 12 The ﬁrst inequalit y is from Lemma 10, the second from the deﬁnition of the Lov ász extension, and the third follows from the prop erty of p : only elements in O hav e probability > 1 ℓ of b eing in the random set. Com bining these inequalities yields (3). Corollary 16 sp ecializes Lemma 15 to the regime ℓ = 1 /ε and rewrites the inequality in terms of total marginal gain with resp ect to the empt y set. As we will show later, this form allo ws us to establish a low er b ound on the improv emen t obtained by the forward direction induced b y the split algorithm. Essen tially , Corollary 16 pro vides a quan titative guaran tee on the qualit y of the direction induced b y the split algorithm. When the function f is instantiated as the ob jective ev aluated at the current fractional solution, the left-hand side represen ts the total marginal gain contributed b y the comp onents of the forward direction. The inequalit y then implies that this direction achiev es a suﬃcien tly large increase relativ e to any indep enden t set O , and therefore constitutes a suﬃciently go o d forward direction for the contin uous greedy pro cess. Corollary 16. In A lgorithm 3, for any O ∈ I , if ℓ ≥ 1 ε , then: ℓ X j =1 f ( T j | ∅ ) ≥ (1 − 2 ε ) · f ( O ) − (1 − ε ) · f ( ∅ ) (4) Pr o of. P ℓ j =1 ( f ( T j ) − f ( ∅ )) ≥  1 − 1 ℓ  f ( O ) − 1 ℓ P ℓ j =1 f ( T j ) ⇕  1 + 1 ℓ  P ℓ j =1 ( f ( T j ) − f ( ∅ )) ≥  1 − 1 ℓ  f ( O ) − f ( ∅ ) ⇕ ℓ X j =1 ( f ( T j ) − f ( ∅ )) ≥ 1 − 1 ℓ 1 + 1 ℓ f ( O ) − 1 1 + 1 ℓ f ( ∅ ) ≥ (1 − 2 ε ) f ( O ) − (1 − ε ) f ( ∅ ) The last inequalit y uses 1 − ε 1+ ε ≥ 1 − 2 ε + ε 2 , 1 1+ ε ≥ 1 − ε − ε 2 for ε ∈ (0 , 1) and assumes f ( O ) ≥ f ( ∅ ) for a simpler co eﬃcient form. If f ( O ) < f ( ∅ ) , the righ t-hand side is negativ e. How ev er, dumm y elemen ts ensure f ( T j ) ≥ f ( ∅ ) since f ( T j ) is non-decreasing when elements are added. Con tinuous Greedy Algorithm. Inspired by [15], the authors illustrated the stationary p oin t for non-monotone submo dular maximization is probably arbitrary bad, whic h implies the solution far aw ay from the bad stationary p oint has the p oten tial to b e a go o d solution. Lev eraging this insight, we prop ose a deterministic con tinuous greedy algorithm that com bines the information of the discrete stationary p oin t through an auxiliary function. In Algorithm 4, we adopt the parameter t s to con trol the distance close to the stationary p oin t. Before t s , the feasible ascent directions is alwa ys orthogonal to the stationary p oint, i.e., alwa ys moving in the direction aw a y from the stationary p oint. After t s , w e release the limitation for ascent direction, i.e., moving in all p ossible direction in feasible domain. A dditionally , the concrete direction in each iteration is returned by the Algorithm 3 based on the auxiliary function, while limiting the num b er of co ordinates at the same time. The formal statemen t is as follows: 13 Algorithm 4: ContinuousGreedy ( M = ( N , I ) , f , Z ⊆ N , t s ∈ [0 , 1] , ε ∈ (0 , 1) ) 1 δ ← ε 3 , y 0 ← 0 ; 2 for i = 1 to 1 /δ do 3 Z i ← ( Z if i ∈ [1 , t s /δ ] ∅ if i ∈ ( t s /δ, 1 /δ ] ; 4 Deﬁne f − Z ( S ) ≜ f ( S ) − P u ∈ Z ∩ S  f ( { u } ) − f ( ∅ ) + 1  ; 5 Deﬁne g − Z ( S ) ≜ E  f − Z  R ( e S ∨ y i − 1 )  ; 6 ( T 1 , . . . , T 1 /ε ) ← Split ( M , g − Z i , 1 /ε ) ; 7 x i ← δ · P 1 /ε j =1 e T j ; 8 y i ← y i − 1 ⊕ x i ; 9 return y 1 /δ ; W e brieﬂy explain the role of the mo diﬁed set functions f − Z and g − Z used in Algorithm 4. The purp ose of these functions is purely technical: they ensure that elements in Z are nev er selected by the split pro cedure, while preserving the relative marginal v alues of all other ele- men ts. Sp eciﬁcally , for any u ∈ Z , the deﬁnition of f − Z enforces a strictly negativ e marginal con tribution for adding u to any set, and hence u is dominated by every element outside Z in the greedy selection. F or elemen ts v / ∈ Z , the marginal gains f − Z ( v | S ) coincide with f ( v | S ) for all S ⊆ N , so the b eha vior of the ob jectiv e function on N \ Z remains unc hanged. The function g − Z is the corresp onding extended m ultilinear extension ev aluated at the curren t fractional p oin t. Imp ortantly , the supp ort of f − Z and g − Z diﬀers from that of f only b y the elemen ts in Z , and ev aluating g − Z incurs the same asymptotic computational cost as ev aluating the original function, up to a constan t-factor o v erhead. Lemma 17. The set functions f − Z ( S ) and g − Z ( S ) deﬁne d in A lgorithm 4 ar e submo dular. Note we do not r e quir e these functions to b e non-ne gative. Pr o of. F or an y A ⊆ B ⊆ N , u ∈ N \ B : f − Z ( u | B ) = f ( u | B ) − [ u ∈ Z ]( f ( { u } ) − f ( ∅ ) + 1) ≤ f ( u | A ) − [ u ∈ Z ]( f ( { u } ) − f ( ∅ ) + 1) = f − Z ( u | A ) Th us f − Z is submodular. Next, g − Z ( u | B ) = E [ f − Z ( R ( e B + u ∨ y i − 1 ))] − E [ f − Z ( R ( e B ∨ y i − 1 ))] = E h f − Z ( R ( y i − 1 ) ∪ ( B + u )) − f − Z ( R ( y i − 1 ) ∪ B ) i ≤ E h f − Z ( R ( y i − 1 ) ∪ ( A + u )) − f − Z ( R ( y i − 1 ) ∪ A ) i (submo dularit y) = g − Z ( u | A ) The second equality treats b oth R ( y i − 1 ) o ccurrences as the same random v ariable. The in- equalit y uses the submo dularit y of f − Z . Th us g − Z is submodular. Lemma 18. F or any u ∈ Z , the function g − Z ( S ) deﬁne d in A lgorithm 4 satisﬁes for any set S ⊆ N − u : g − Z ( S ) > g − Z ( S + u ) . 14 Pr o of. First, we claim: F or any T ⊆ N − u , f − Z ( T ) > f − Z ( T + u ) . This follows from the deﬁnition of f − Z and submodularity: f − Z ( T + u ) − f − Z ( T ) = ( f ( T + u ) − f ( T )) + ( − 1 · ( f ( { u } ) − f ( ∅ ) + 1) ≤ ( f ( { u } ) − f ( ∅ )) − ( f ( { u } ) − f ( ∅ ) + 1) = − 1 No w, note that R ( e S ∨ y ) = R ( y ) ∪ S b ecause elements in S are alwa ys included in R ( e S ∨ y ) , and the v alues at indices corresp onding to S do not aﬀect the distribution of other elements in R ( y ) . Therefore, g − Z ( S ) = E [ f − Z ( R ( y ) ∪ S )] = X T ⊆N Pr[ R ( y ) = T ] · f − Z ( T ∪ S ) > X T ⊆N Pr[ R ( y ) = T ] · f − Z ( T ∪ ( S + u )) (by claim ab ov e) = E [ f − Z ( R ( y ) ∪ ( S + u ))] = g − Z ( S + u ) Due to the negative incremen tal gain prop erty of elements u ∈ Z , and the fact that dummy elemen ts hav e zero incremental gain and are suﬃciently n umerous, suc h u will never b e selected b y the Split (Algorithm 3) in to an y set. Corollary 19. In the i -th iter ation of A lgorithm 4, for any j ∈ [1 /ε ] , T j ∩ Z i = ∅ . Next, w e establish several structural properties of the sequence { y i } 1 /δ i =0 generated by Algo- rithm 4. First, the solution constructed by the algorithm is feasible, in particular that the ﬁnal marginal v ector lies in the matroid p olytop e. Lemma 20. Mar( y 1 /δ ) ∈ P ( M ) . Pr o of. Since the sets returned by Split form a basis of M , 1 δ Mar( x i ) ∈ P ( M ) . Therefore, their linear combination P i ∈ [1 /δ ] x i ∈ P ( M ) . Thus, Mar( y 1 /δ ) = Mar   M i ∈ [1 /δ ] x i   = M i ∈ [1 /δ ] Mar  x i  F or any u ∈ N , Mar( y 1 /δ ) u = L i ∈ [1 /δ ] Mar( x i ) u ≤  P i ∈ [1 /δ ] x i  u . By the down w ard-closed prop ert y of P ( M ) , it follo ws that Mar( y 1 /δ ) ∈ P ( M ) . A dditionally , we show that the supp ort of y i gro ws in a con trolled manner, that all marginal v alues remain b ounded, and that the con tribution of elemen ts in Z is appropriately delay ed. These b ounds allo w us to interpret the up date rule as a v alid contin uous greedy pro cess and will be rep eatedly in vok ed in the pro of of the main appro ximation result. Lemma 21. F or any inte ger 0 ≤ i ≤ 1 /δ , supp( y i ) ≤ (1 /ε ) · i ≤ 1 /ε 4 , ∥ Mar( y i ) ∥ ∞ ≤ 1 − (1 − δ ) i , and ∥ Mar( y i ) ∧ 1 Z ∥ ∞ ≤ 1 − (1 − δ ) max { 0 ,i − t s /δ } . 15 Pr o of. Since each up date from y i to y i +1 aﬀects only 1 /ε p ositions, the num b er of non-zero p ositions in y i is at most (1 /ε ) · i . A dditionally , the sets ( T 1 , T 2 , . . . , T 1 /ε ) returned by Split are disjoin t, i.e., ∥ Mar( x i ) ∥ ∞ ≤ δ . Using the property of Mar and ⊕ (Observ ation 7), for any u ∈ N : Mar( y i ) u = Mar  y i − 1  u ⊕ Mar ( x i ) u ≤ 1 −  1 − Mar( y i − 1 ) u  (1 − δ ) Recursiv ely applying this gives Mar( y i ) u ≤ 1 − (1 − δ ) i . By Corollary 19, for any u ∈ Z , i ≤ t s /δ , we hav e u / ∈ T 1 ∪ T 2 ∪ . . . T 1 /ε , so Mar( y i ) u = 0 . Then, applying the ab o ve inequality yields for u ∈ Z , i ≥ t s /δ : Mar( y i ) u ≤ 1 − (1 − δ ) i − t s /δ . Lemma 22. F or any i ≤ 1 /δ , we have 1 − (1 − δ ) i ≤ 1 − e − iδ + iδ 2 , ∥ Mar( y i ) ∥ ∞ ≤ 1 − e − iδ + δ , and ∥ Mar( y i ) ∧ 1 Z ∥ ∞ ≤ 1 − e − max { 0 ,iδ − t s } + δ . Pr o of. Pro v e by induction on i . F or i = 0 , it holds trivially . F or i = 1 , 1 − (1 − δ ) ≤ 1 − e − δ + δ 2 . Expanding e − δ : e − δ − (1 − δ + δ 2 ) = X k ≥ 0 ( − δ ) k k ! − (1 − δ + δ 2 ) = − δ 2 2 − X k ≥ 2 δ 2 k − 1 (2 k − 1)! − δ 2 k (2 k )! ! ≤ 0 So it holds for i = 1 . Assume it holds for all integers in [0 , i ] , pro v e for i + 1 : 1 − (1 − δ ) i +1 = (1 − (1 − δ ) i )(1 − δ ) + δ ≤ (1 − e − iδ + iδ 2 )(1 − δ ) + δ = 1 + iδ 2 − iδ 3 − e − iδ (1 − δ ) ≤ 1 + iδ 2 − iδ 3 − e − iδ ( e − δ − δ 2 ) (using i = 1 case) = 1 + ( iδ 2 + e − iδ δ 2 ) − e − ( i +1) δ − iδ 3 ≤ 1 + ( i + 1) δ 2 − e − ( i +1) δ (since e − iδ ≤ 1 ) Lemma 23. A lgorithm 4 r e quir es a total of O ε ( nr ) queries. Pr o of. Since R ( y ) has at most 2 frac ( y ) distinct p ossible v alues, and by Lemma 21, frac ( y i ) ≤ supp( y i ) ≤ 1 /ε 4 (a constan t dep ending only on ε ), computing g − Z ( S ) requires a constan t n umber of v alue queries. In Algorithm 4, only the Split subroutine requires query complex- it y b eyond ε dep endence. One Split execution requires O ( nr ) computations of g − Z ( S ) and indep endence queries. Therefore, Algorithm 4 requires O ε ( nr ) queries in total. Lev eraging the ab ov e lemmas for the sequence { y i } 1 /δ i =0 , w e can obtain a low er b ound on the function v alue gro wth at eac h step. Lemma 24. L et O P T i = O P T \ Z i . In A lgorithm 4, for any i ∈ [ 0 , 1 /δ ) , 1 δ h F ( y i ) − F ( y i − 1 ) i ≥ (1 − 3 ε ) ·  F ( e OP T i ∨ y i − 1 ) − F ( y i − 1 )  16 Pr o of. Note that for i ≤ t s /δ , the algorithm do es not in volv e any elements in Z . It can b e view ed as executing on the set N \ Z . Since T j ∩ Z i = ∅ , O P T i ∩ Z i = ∅ , R ( y i ) ∩ Z i = ∅ , we easily get F ( e OP T i ∨ y i ) = g − Z i ( O P T i ) , F ( e T j ∨ y i ) = g − Z i ( T j ) . Using the prop erty of Algorithm Split (Corollary 16), w e ha ve: 1 /ε X j =1  F ( e T j ∨ y i − 1 ) − F ( y i − 1 )  ≥ (1 − 2 ε ) F ( e OP T i ∨ y i − 1 ) − (1 − ε ) F ( y i − 1 ) (5) No w note: F ( y i ) = E [ f ( R ( y i ))] = E [ f ( R ( y i − 1 ⊕ x i ))] = E [ f ( R ( y i − 1 ) ∪ R ( x i ))] = X S ⊆N Pr[ R ( x i ) = S ] · E [ f ( R ( y i − 1 ) ∪ S )] = X S ⊆N Pr[ R ( x i ) = S ] · F ( e S ∨ y i − 1 ) ≥ (1 − δ ) 1 /ε F ( y i − 1 ) + 1 /ε X j =1 δ (1 − δ ) 1 /ε − 1 F ( e T j ∨ y i − 1 ) (only S = ∅ or T j ) ≥ 1 − δ ε ! F ( y i − 1 ) + δ 1 − δ ε ! 1 /ε X j =1 F ( e T j ∨ y i − 1 ) ( (1 − δ ) k ≥ 1 − k δ ) ≥ 1 − δ ε !  F ( y i − 1 ) + δ  (1 − 2 ε ) F ( e OP T i ∨ y i − 1 ) +  1 ε − 1 + ε  F ( y i − 1 )  (using (5)) ≥ (1 − δ ) F ( y i − 1 ) + δ (1 − 3 ε ) F ( e OP T i ∨ y i − 1 ) (using δ = ε 3 and simplifying) Rearranging giv es the result. Note that the low er b ound in the ab o v e lemma in v olv es F ( e OP T i ∨ y i − 1 ) . In the next lemma, w e analyze this further to relate it to the optimal solution f ( OP T ) . Lemma 25. In A lgorithm 4, for any i ∈ [1 , 1 /δ ] , A ⊆ N , we have: F ( y i ∨ e A ) ≥  e − max { 0 ,t − t s } − e − t  · max { 0 , f ( A ) − f ( A ∪ Z ) } + ( e − t − δ ) · f ( A ) wher e t = iδ . Pr o of. By Lemma 10, F ( y i ∨ e A ) ≥ ˆ f (Mar( y i ∨ e A )) . Let θ Z = e − max { 0 ,t − t s } − δ , θ = e − t − δ . By Lemma 22, for u ∈ Z , Mar( y i ) u ≤ 1 − θ Z ; for u / ∈ Z , Mar( y i ) u ≤ 1 − θ . Note that Mar( y i ∨ e A ) = Mar( y i ) ∨ 1 A b ecause R ( y i ∨ e A ) = R ( y i ) ∪ A . Using these observ ations: F ( y i ∨ e OP T ) ≥ ˆ f (Mar( y i ) ∨ 1 A ) = Z 1 0 f ( T λ (Mar( y i )) ∪ A ) dλ ≥ Z 1 − θ 1 − θ Z f ( T λ (Mar( y i )) ∪ A ) dλ + Z 1 1 − θ f ( T λ (Mar( y i )) ∪ A ) dλ ≥ Z 1 − θ 1 − θ Z f ( T λ (Mar( y i )) ∪ A ) dλ + θ · f ( A ) (since T λ = ∅ for λ > 1 − θ ) ≥ ( θ Z − θ ) · max { f ( A ) − f ( Z ∪ A ) , 0 } + θ · f ( A ) 17 The last inequality holds because when 1 − θ Z < λ ≤ 1 − θ , we hav e T λ (Mar( y i )) ⊆ N \ Z . By submo dularit y and non-negativity of f , for any B ⊆ N \ Z : f ( B ∪ ( Z \ A ) ∪ A ) − f ( B ∪ A ) ≤ f (( Z \ A ) ∪ A ) − f ( A ) ⇒ f ( B ∪ A ) ≥ f ( A ) − f ( Z ∪ A ) ⇒ Z 1 − θ 1 − θ Z f ( T λ (Mar( y i )) ∪ A ) dλ ≥ ( θ Z − θ ) · ( f ( A ) − f ( Z ∪ A )) And this v alue is also ≥ 0 due to non-negativit y . T o translate the p er-iteration impro vemen t b ounds into a b ound on the ﬁnal v alue F ( y 1 /δ ) , w e compare the discrete ev olution of F ( y i ) with an auxiliary function. Since A is arbitrary in the ab o v e lemma, we can ﬁnally express our solution F ( y 1 /δ ) in terms of O P T and Z , and then complete the entire algorithm via Theorem 11. Deﬁne g (0) = 0 , g (( i + 1) δ ) =    g ( iδ ) + δ h f ( O P T \ Z ) − (1 − e − iδ ) · f ( Z ∪ O P T ) − g ( iδ ) i if iδ < t s g ( iδ ) + δ h e − iδ · f ( O P T ) + ( e t s − iδ − e − iδ ) · max { f ( O P T ) − f ( Z ∪ OP T ) , 0 } − g ( iδ ) i if iδ ≥ t s Lemma 26. F or any i ∈ [0 , 1 /δ ] , g ( iδ ) ≤ F ( y i ) + 4 iδ ε · f ( O P T ) Pr o of. W e pro ve b y induction on i . F or i = 0 , g (0) = 0 ≤ F ( y 0 ) . Assume the lemma holds for i , pro ve for i + 1 . Deﬁne the increment of g as: g ′ ( iδ ) =    f ( O P T \ Z ) − (1 − e − iδ ) · f ( Z ∪ O P T ) if iδ < t s e − iδ · f ( O P T ) + ( e t s − iδ − e − iδ ) · max { f ( O P T ) − f ( Z ∪ OP T ) , 0 } if iδ ≥ t s By deﬁnition, g ′ ( iδ ) ≤ f ( O P T ) and g (( i + 1) δ ) = δ [ g ′ ( iδ ) − g ( iδ )] + g ( iδ ) By Lemmas 24 and 25, we ha v e: F ( y i +1 ) ≥ F ( y i ) + δ h (1 − 3 ε )( g ′ ( iδ ) − δ · f ( O P T )) − F ( y i ) i (6) ≥ F ( y i ) + δ h (1 − 3 ε ) g ′ ( iδ ) − δ · f ( O P T ) − F ( y i ) i (7) Therefore: g (( i + 1) δ ) = g ( iδ ) + δ ( g ′ ( iδ ) − g ( iδ )) = (1 − δ ) g ( iδ ) + δ g ′ ( iδ ) ≤ (1 − δ )[ F ( y i ) + 4 iδ ε · f ( O P T )] + δ g ′ ( iδ ) (induction h yp.) ≤ F ( y i +1 ) − δ (1 − 3 ε ) g ′ ( iδ ) + δ 2 f ( O P T ) + (1 − δ ) · 4 iδ ε · f ( O P T ) + δ g ′ ( iδ ) (using 6) = F ( y i +1 ) + (1 − δ )4 iδ ε · f ( O P T ) + 3 εδ g ′ ( iδ ) + δ 2 f ( O P T ) ≤ F ( y i +1 ) + 4(( i + 1) δ ) ε · f ( O P T ) (since g ′ ( iδ ) ≤ f ( O P T ) , δ < ε ) 18 Theorem 27. The solution r eturne d by A lgorithm 4 satisﬁes the fol lowing ine quality: F ( y 1 /δ ) ≥ e t s − 1  (2 − t s − e − t s − 4 ε ) f ( O P T ) − (1 − e − t s ) f ( Z ∩ O P T ) − (2 − t s − 2 e − t s ) f ( Z ∪ O P T )  Pr o of. The deﬁnition of g ( iδ ) is consistent with that in [7]. Its Corollary A.5 pro v es: g (1) ≥ e t s − 1  (2 − t s − e − t s ) f ( O P T ) − (1 − e − t s ) f ( Z ∩ O P T ) − (2 − t s − 2 e − t s ) f ( Z ∪ O P T )  By Lemma 26, F ( y 1 /δ ) ≥ g (1) − 4 εf ( O P T ) ≥ e t s − 1  (2 − t s − e − t s − 4 ε ) f ( O P T ) − (1 − e − t s ) f ( Z ∩ O P T ) − (2 − t s − 2 e − t s ) f ( Z ∪ O P T )  3.3 Pro of of the Main Result In the previous subsections, we established the theoretical guarantees for the t w o main com- p onen ts of our algorithm. First, the Split algorithm returns a set Z satisfying the structural inequalities in Theorem 14. Second, the contin uous greedy phase pro duces a fractional vector whose v alue satisﬁes the b ound in Theorem 27. Finally , b y Theorem 11, the Deterministic- Pip age pro cedure conv erts any fractional vector y with Mar ( y ) ∈ P ( M ) into a feasible set S ∈ M such that f ( S ) ≥ F ( y ) . Therefore, to prov e the appro ximation guaran tee of our algorithm, it suﬃces to establish a lo wer b ound on the v alue returned by Algorithm 1. W e no w pro ve the main theorem. Pro of of Theorem 1. By Theorem 11, the Deterministic-Pip a ge algorithm conv erts a fractional v ector y with Mar ( y ) ∈ P ( M ) into a feasible set S ∈ M satisfying f ( S ) ≥ F ( y ) . Therefore, it suﬃces to prov e a lo wer b ound on the fractional v alue returned b y Algorithm 1. The feasibilit y of the solutions pro duced by the algorithm and the query complexit y b ound follo w from Theorem 14, Lemma 20, and Lemma 23. Hence, w e only need to analyze the appro ximation ratio. The lo wer b ound for the solution of our algorithm follo ws essen tially the same analysis as in [6]. The algorithm returns the maximum of tw o candidate solutions. W e ﬁrst list the inequalities satisﬁed by these t wo quantities. 19 By Theorem 14, for every A ∈ I , f ( Z ) ≥ 1 2 f ( Z ∪ A ) + 1 2 f ( Z ∩ A ) − ε · f ( O P T ) . Substituting A = O P T and A = O P T ∩ Z giv es f ( Z ) ≥ 1 2 f ( Z ∪ O P T ) + 1 2 f ( Z ∩ O P T ) − ε · f ( O P T ) , (8) f ( Z ) ≥ f ( Z ∩ O P T ) − ε · f ( O P T ) . (9) By Theorem 27, F ( y 1 /δ ) ≥ e t s − 1  (2 − t s − e − t s − 4 ε ) f ( O P T ) − (1 − e − t s ) f ( Z ∩ O P T ) − (2 − t s − 2 e − t s ) f ( Z ∪ O P T )  . (10) Inequalities (8), (9), and (10) are all v alid low er bounds on the ob jectiv e v alue returned by the algorithm. Therefore, an y con vex com bination of them is also a v alid low er b ound. Let ALG denote the fractional solution returned b y the algorithm. F or an y p 1 , p 2 , p 3 ≥ 0 with p 1 + p 2 + p 3 = 1 , w e obtain F ( ALG ) ≥ 1 2 p 1 ( f ( Z ∪ O P T ) + f ( Z ∩ O P T ) + εf ( O P T )) + p 2 ( f ( Z ∩ O P T ) + εf ( O P T )) + p 3 e t s − 1  (2 − t s − e − t s − 4 ε ) f ( O P T ) − (1 − e − t s ) f ( Z ∩ O P T ) − (2 − t s − 2 e − t s ) f ( Z ∪ O P T )  . W e choose p 1 , p 2 , p 3 and t s so that the co eﬃcien ts of f ( Z ∪ O P T ) and f ( Z ∩ O P T ) are non-negativ e while maximizing the co eﬃcient of f ( OP T ) . As sho wn in Section 3.1 of [6], the optimal parameters are p 1 = 0 . 205 , p 2 = 0 . 025 , p 3 = 0 . 070 , t s = 0 . 372 , whic h yields F ( ALG ) ≥ (0 . 385 − O ( ε )) f ( O P T ) . Applying Theorem 11 completes the pro of. 4 Knapsac k Constrain ts In this section, w e present our deterministic algorithm for the knapsac k constraint and ana- lyze its approximation ratio and query complexity . Our algorithm consists of three comp onents: en umerate elemen ts, optimization, and rounding. The algorithm ﬁrst enumerates all subsets E i of the ground set whose size is at most ε − 2 . Then, for eac h E i , w e assume E i has b een 20 selected, that is, we consider f ( · | E i ) as the new function and (1 − ε ) B as the capacity . W e apply a measured con tinuous greedy algorithm on the extended multilinear extension to obtain a vector y i ∈ [0 , 1] 2 N . Next, we apply a rounding algorithm to y i to obtain a set S i . Finally , w e select the set with the largest v alue among all S i as our output. The ab o v e pro cedure is summarized in Algorithm 5. Algorithm 5: Deterministic Knapsac k ( f , w , B ) 1 for e ach set E i ∈ N with | E i | ≤ ε − 2 do 2 y i ← D M C G ( f ( · ∪ E i ) , w , (1 − ε )( B − w ( E i )) , ε ) 3 S i ← R ounding ( f , w , y i ) 4 return ar g max i f ( S i ∪ E i ) 4.1 En umerate Elemen ts W e b egin our analysis with the elemen t enumeration step. F ollo wing the approach in [14], b y en umerating all subsets of size at most O ( ε − 2 ) , w e eﬀectively ﬁx the heavy elements and reduce the remaining problem to one consisting only of ligh tw eight elemen ts, each con tributing at most an ε fraction of the total w eigh t. W e then sho w that, in this reduced instance, the knapsac k capacity can b e safely scaled down by a factor of α , and under this reduced capacity , the v alue of the optimal solution decreases by at most α − O ( ε ) . This reduction has t w o purp oses: ﬁrst, it makes the knapsack constrain t b eha v e more lik e a cardinalit y constraint during the optimization phase; second, it provides slac k for the subsequen t rounding pro cedure to ensure feasibility . Lemma 28 (Corollary 4.16 in [14]) . F or maximizing a submo dular function f : 2 N → R + under a knapsack c onstr aint with c ap acity B and a weight function w : N → R + , ther e exists a subset E ⊆ N satisfying the fol lowing pr op erties: 1. Its c ar dinality is b ounde d by | E | ≤ ε − 2 , and henc e E c an b e identiﬁe d by enumer ating at most ε − 2 elements. 2. F or every element o ∈ OPT \ E , the mar ginal c ontribution satisﬁes f ( o | E ) ≤ ε 2 f (OPT) . 3. L et B ig = { u ∈ OPT | w ( u ) ≥ ε ( B − w ( E )) }\ E , then f (OPT \ B ig ) ≥ (1 − ε ) f ( O P T ) . Pr o of. W e consider a subset E constructed as follo ws. Initially , E is empty . W e then iterate o ver the elemen ts of the optimal solution OPT . F or eac h elemen t o ∈ OPT , if f ( o | E ) ≥ ε 2 f (OPT) , then we pick the item o into E . W e ﬁrst observe that if | E | > ε − 2 , then b y the construction rule of E we would ha v e f ( E ) > | E | · ε 2 f (OPT) ≥ ε − 2 · ε 2 f (OPT) = f (OPT) . This is imp ossible, since f ( E ) ≤ f ( O P T ) denotes the optimal v alue. Therefore, w e m ust ha v e | E | ≤ ε − 2 . 21 Next, consider the second prop ert y . Since f is submo dular, the marginal contribution of an y element can only decrease as the set E grows. Consequen tly , once ev ery element of OPT has b een examined exactly once, submo dularity guaran tees that its marginal con tribution in all subsequen t steps cannot increase, and hence no remaining element can satisfy the inequality ab o v e. Therefore, for all o ∈ OPT \ E , w e ha ve f ( o | E ) ≤ ε 2 f (OPT) . Finally , consider the third prop ert y . Let B ig = { b 1 , b 2 , . . . , b t } denote the subset of elements in OPT with w eights at least ε ( B − w ( E )) . W e analyze the pro cess of removing these elemen ts from OPT one b y one. F or eac h b i ∈ B ig , the loss incurred is b ounded b y f ( b i | OPT \ B ig ∪ { b i +1 , . . . , b t } ) ≤ f ( b i | OPT \ B ig ) ≤ f ( b i | E ) ≤ ε 2 f (OPT) , where the last inequality follows from the prop erty that no elemen t outside E has marginal con tribution larger than ε 2 f (OPT) . Since there are at most ε − 1 suc h elements, the total loss is bounded b y εf (OPT) . Moreo ver, we pro ve that if we assume such an elemen t E has already b een selected and further reduce the remaining capacit y to a α fraction of the original, then the v alue of the optimal solution decreases b y at most a factor of ( α − ε 2 ) . Lemma 29. Given the pr oblem of maximzing submo dular function f : 2 N → R + under a knapsack c onstr aint with c ap acity B and a weight function w : N → R + . If for any u ∈ N , we have f ( u | ∅ ) ≤ ε 2 f (OPT) and w ( u ) ≤ εB , then we have max w ( S ) ≤ αB f ( S ) ≥ ( α − ε 2 ) f (OPT) . Pr o of. W e consider the following iterative pro cedure. Let O 0 = OPT . At the i -th round, w e select the element o i ∈ O i − 1 with the low est curren t densit y , i.e., o i = arg min u ∈ O i − 1 f ( u | O i − 1 − u ) w ( u ) , and remov e it to obtain O i = O i − 1 \ { o i } , con tinuing until w ( O t ) ≤ αB for some t . Since o i has the lo west densit y , it follo ws that f ( o i | O i − 1 − o i ) ≤ f ( O i − 1 ) · w ( o i ) w ( O i − 1 ) . Otherwise, summing ov er all u ∈ O i − 1 , w e w ould obtain f ( O i − 1 ) ≥ X u ∈ O i − 1 f ( u | O i − 1 − u ) > f ( O i − 1 ) X u ∈ O i − 1 w ( u ) w ( O i − 1 ) = f ( O i − 1 ) , whic h is a contradiction. Rearranging terms, w e then ha ve f ( O i ) w ( O i ) ≥ f ( O i − 1 ) w ( O i − 1 ) . 22 By applying this inequalit y iterativ ely , we obtain f ( O t − 1 ) w ( O t − 1 ) ≥ f ( O t − 2 ) w ( O t − 2 ) ≥ · · · ≥ f (OPT) w (OPT) . It follo ws that f ( O t ) ≥ f ( O t − 1 ) − f ( o t | O t ) ≥ f (OPT) w ( O t − 1 ) w (OPT) − f ( o t | O t ) ≥ αf (OPT) − f ( o t | O t ) ≥ αf (OPT) − f ( o t | ∅ ) ≥ ( α − ε 2 ) · f (OPT) . Consequen tly , our algorithm ﬁrst en umerates all subsets E i of size at most ε − 2 . This guar- an tees that the desired set E will b e included among the enumerated candidates. F or each suc h set E i , we solv e the corresp onding derived optimization problem deﬁned ab ov e using an ap- propriate optimization algorithm. If this algorithm returns an appro ximate fractional solution for the derived problem, then, when applied to the correct set E , it yields a fractional solution that ac hieves the desired approximation guarantee. 4.2 Deterministic Measured Con tin uous Greedy W e now describ e our optimization comp onent, whic h relies on a nov el split algorithm namely , KnapsackSplit subroutine. Diﬀerent from the matroid split algorithm, we replace the marginal gain f ( u | T j ) with the marginal density f ( u | T j ) /w ( u ) . Throughout the remain- der of this section, w e set ℓ = 1 /ε and, without loss of generalit y , assume that 1 /ε is an integer. F or the Knapsa ckSplit algorithm, w e establish the following result: Lemma 30. Given a submo dular function f : 2 N → R + , a knapsack c onstr aint with c ap acity B and a weight function w : N → R + , then for any fe asible set Q , if f ( u | ∅ ) ≤ ε 2 f ( Q ) for al l u ∈ N , the KnapsackSplit A lgorithm pr o duc es disjoint sets T 1 , T 2 , . . . , T ℓ ⊆ N such that their union T = ∪ ℓ j =1 T j that satisfying ∪ ℓ j =1 T j ≤ B , and then we have ℓ X j =1 f ( T j | ∅ ) ≥ max    1 ℓ ℓ X j =1 ( f ( Q ∪ T j ) − f ( T j )) − ε 2 f ( O P T ) , 0    . This result is analogous to the prov ed for the Split algorithm in [9], except for an additional ε 2 f ( Q ) term. The pro of strategy also diﬀers from that of Split . In the original analysis, the argumen t relies on the basis exchange prop erty of matroids, whic h establishes a bijection b et w een the selected set T and an y feasible set Q . Under a knapsack constrain t, such a bijection no longer exists. Instead, we order the elements in to a sequence and partition them into several segmen ts so that they can b e paired accordingly . As a result, at most one element q ∈ Q with f ( q | T ) ≥ 0 ma y remain unpaired. As sho wn ab ov e, the marginal contribution of such an element is at most ε 2 f ( Q ) . W e further sho w that this additional term is negligible in the o verall analysis of the optimization algorithm. 23 Algorithm 6: KnapsackSplit ( f , w , B , ℓ ) 1 Initialize: T 1 ← ∅ , T 2 ← ∅ , . . . , T ℓ ← ∅ . 2 Use T to denote the union ∪ ℓ j =1 T j . 3 while w ( T ) ≤ B do 4 Let N ′ ← { u ∈ N \ T | w ( T ∪ { u } ) ≤ B } . 5 Let ( u, j ) ∈ ar g max ( u,j ) ∈N ′ × [ ℓ ] f ( u | T j ) w ( u ) . 6 T j ← T j ∪ { u } . 7 return ( T 1 , · · · , T ℓ ) . Algorithm 7: KnapsackDMCG( g , w , B , ε ) 1 Let δ ← ε 3 . 2 Let y (0) ← 0 . 3 for i = 1 to 1 /δ do 4 Deﬁne g : 2 N → R ≥ 0 to be g ( S ) = F ( e S ∨ y ( t )) . 5 ( T 1 , . . . , T 1 /ε ) ← Split ( g , w , B , 1 /ε ) 6 y ( t + δ ) ← y ( t ) ⊕ ( δ · P 1 /ϵ j =1 e T j ) . 7 return y (1) . Pr o of. W e ﬁrst aim to establish a corresp ondence b etw een the elemen ts in Q and those in T = S i T i . W e arbitrarily order the elemen ts in Q as q 1 , q 2 , . . . , q x , and denote the elemen ts of T according to their order of added as t 1 , t 2 , . . . , t y , where h i indicates that the elemen t t i w as added to the set T h i . W e additionally denote by T ( i ) j the set of elements in T j at the momen t when the element t i is added. W e represen t each element as an interv al on the real line [0 , B ] , with length equal to its w eight. W e ﬁrst place the elemen ts of Q consecutively according to the ﬁxed order: the elemen t q 1 corresp onds to the in terv al [0 , w ( q 1 )] , q 2 corresp onds to the in terv al [ w ( q 1 ) , w ( q 1 ) + w ( q 2 )] , and in general, q i corresp onds to the in terv al I Q i =  i − 1 X k =1 w ( q k ) , i X k =1 w ( q k )  . Similarly , we place the elements of T = S i T i on the same in terv al [0 , B ] according to their order of insertion. Sp eciﬁcally , the element t 1 corresp onds to the interv al [0 , w ( t 1 )] , the element t 2 corresp onds to the in terv al [ w ( t 1 ) , w ( t 1 ) + w ( t 2 )] , and in general, the elemen t t i corresp onds to the interv al I T k =  i − 1 X k =1 w ( t ℓ ) , i X k =1 w ( t ℓ )  . Based on the interv al represen tations constructed ab ov e, w e consider the pairwise intersec- tions b et ween interv als asso ciated with elements in Q and those asso ciated with elements in T . Sp eciﬁcally , each p oin t in [0 , B ] lies in exactly one interv al corresp onding to an element of Q and in at most one interv al corresp onding to an element of T , which naturally induces a pairing b et w een elements of Q and elements of T through their ov erlapping interv als. F or α indexing 24 elemen ts of Q and β indexing elements of T , w e denote by w α,β the length of the intersection b et w een the corresp onding in terv als, w α,β =    I Q α ∩ I T β    . q 1 q 2 q 3 . . . q x t 1 t 2 t 3 . . . t y B F or each α ∈ { 1 , 2 , . . . , x } and β ∈ { 1 , 2 , . . . , y } , we hav e f ( t β | T ( β − 1) h β ) w ( t β ) ≥ 1 ℓ ℓ X j =1 f ( q α | T ( β − 1) j ) w ( q α ) ≥ 1 ℓ ℓ X j =1 f ( q α | T j ∪ Q α − 1 ) w ( q α ) . Multiplying b oth sides by w α,β and summing o ver all α ∈ { 1 , . . . , x } and β ∈ { 1 , . . . , y } , w e obtain y X β =1 x X α =1 w α,β w ( t β ) f ( t β | T ( β − 1) h β ) ≥ 1 ℓ ℓ X j =1 x X α =1 y X β =1 w α,β w ( q α ) f ( q α | T j ∪ Q α − 1 ) . On the left-hand side, by construction w e ha v e P x α =1 w α,β ≤ w ( t β ) for eac h β . Th us, the left-hand side is lo wer b ounded b y y X β =1 f ( t β | T ( β − 1) h β ) = ℓ X j =1 f ( T j ) . On the right-hand side, the term P y β =1 w α,β admits a simple characterization dep ending on the p osition of α relativ e to γ . F or all α < γ , the in terv al I Q α is fully cov ered by the union of in terv als corresp onding to T , and hence P y β =1 w α,β = w ( q α ) . F or α = γ , the interv al I Q α is only partially cov ered, which implies P y β =1 w α,β ≤ w ( q α ) . F or α > γ , the in terv al I Q α has empty in tersection with the in terv als corresponding to T and lies en tirely to their righ t, implying w ( q α ) + w ( T ) ≤ B . Th us, f ( o α | T i ) ≤ 0 , ∀ i = 1 , 2 , . . . , ℓ, otherwise this element could be added to T i . Summing and scaling, w e obtain: x X α = γ +1 ℓ X ℓ =1 f ( q α | T j ∪ Q α − 1 ) ≤ x X α = γ +1 ℓ X ℓ =1 f ( q α | T j ) ≤ 0 Consequen tly , w e hav e: 25 1 ℓ ℓ X j =1 x X α =1 y X β =1 w α,β w ( q α ) f ( q α | T j ∪ Q α − 1 ) = 1 ℓ γ − 1 X α =1 ℓ X j =1 y X β =1 w α,β w ( q α ) f ( q α | T j ∪ Q α − 1 ) + 1 ℓ ℓ X j =1 y X β =1 w γ ,β w ( q γ ) f ( q γ | T j ∪ Q γ − 1 ) = 1 ℓ γ X α =1 ℓ X j =1 f ( q α | T j ∪ Q α − 1 ) −   1 − y X β =1 w γ ,β w ( q γ )   1 ℓ ℓ X j =1 f ( q γ | T j ∪ Q γ − 1 ) ≥ 1 ℓ x X α =1 ℓ X j =1 f ( q α | T j ∪ Q α − 1 ) −   1 − y X β =1 w γ ,β w ( q γ )   1 ℓ ℓ X j =1 f ( q γ | T j ∪ Q γ − 1 ) ≥ 1 ℓ x X α =1 ℓ X j =1 f ( q α | T j ∪ Q α − 1 ) −   1 − y X β =1 w γ ,β w ( q γ )   1 ℓ ℓ X j =1 f ( q γ | ∅ ) = 1 ℓ ℓ X j =1 ( f ( Q ∪ T j ) − f ( T j )) − ε 2 f ( Q ) With the p er-iteration progress guaran tee in place, w e can pro ve that the algorithm returns a con tinuous solution with a guaran teed approximation ratio. Theorem 31. Given a submo dular function f : 2 N → R + , a knapsack c onstr aint with c ap acity B and a weight function w : N → R + , the KnapsackDMCG algorithm r eturns a fr actional solution y ∈ [0 , 1] 2 N with P u ∈N Mar u ( y ) w ( u ) ≤ B , and F ( y (1)) ≥ ( 1 e − O ( ε )) f (OPT) (11) wher e F is the extende d multiline ar extension of f . Besides, for al l i ∈ [0 , 1 /δ ] , we have frac( y ( iδ )) ≤ 1 /ε · i ≤ ε − 4 . Pr o of. Firstly , we brieﬂy explain why (11) holds, as this part of the pro of is iden tical to the argumen t in Section 3 (the matroid case) and can b e directly deriv ed from [9]. By submo dularity and the choice ℓ = 1 /ε , w e obtain 1 /ε X j =1 g ( T j | ∅ ) ≥ (1 − O ( ε )) g (OPT) − ε 1 /ε X j =1 g ( T j ) . Plugging this inequality in to the iterative pro cess yields 1 /ε X j =1  F ( e T j ∨ y (( i − 1) δ )) − F ( y (( i − 1) δ ))  ≥ (1 − O ( ε )) · F ( e OPT ∨ y (( i − 1) δ )) − (1 − ε ) · F ( y (( i − 1) δ )) . Expanding the up date rule of the algorithm giv es 1 δ ( F ( y ( iδ )) − F ( y (( i − 1) δ ))) ≥ (1 − O ( ε )) · F ( e OPT ∨ y (( i − 1) δ )) − F ( y (( i − 1) δ )) . 26 Finally , using ∥ Mar( y ( iδ )) ∥ ≤ 1 − (1 − δ ) i together with Observ ation 9 we can acquire F ( e OPT ∨ y (( i − 1) δ )) ≥ (1 − δ ) i − 1 · f (OPT) , and solving this recurrence ov er the in terv al [0 , 1] yields F ( y (1)) ≥  1 e − O ( ε )  f (OPT) , whic h establishes (11). Besides,it is easy to see that in each iteration, when the algorithm up dates y ( t ) to y ( t + δ ) , it increases at most 1 /ε co ordinates. Since the algorithm p erforms at most 1 /ε 3 iterations in total, w e ha ve frac( y ( iδ )) ≤ (1 /ε ) · i . No w it suﬃces to sho w that X u ∈N Mar u ( y ) w ( u ) ≤ B . Let T ( i ) j denote the set T i pro duced in the j -th iteration of the algorithm. The ﬁnal vector y can be written as y = 1 /δ M j =1  δ · 1 /ε X i =1 e T ( i ) j  . F or eac h ﬁxed j , the KnapsackSplit algorithm guaran tees that the total weigh t of the selected sets does not exceed the capacit y , implying Mar  δ · 1 /ε X i =1 e T ( i ) j  = δ 1 /ε X i =1 w ( T ( i ) j ) ≤ δ B . By the additivity of the op erator Mar( · ) ov er the direct sum, we obtain Mar( y ) = 1 /δ M j =1 Mar  δ · 1 /ε X i =1 e T ( i ) j  ≤ 1 /δ X j =1 δ B = B . Algorithm 8: Relax ( y , u ) 1 Up date y u ← Mar u ( y ) 2 for every non-empty set S ⊆ N − u do 3 y S ← 1 − (1 − y S + u )(1 − y S ) . 4 y S + u ← 0 . 5 return y . 4.3 Rounding Finally , we apply a rounding algorithm to the fractional solution y obtained by running the optimization algorithm on the function f ( · ∪ E ) . 27 Algorithm 9: R ounding ( f , w , y ) 1 while R  = N do 2 S ← R ∩ { u ∈ N ′ | y { u } ∈ (0 , 1) } . 3 while | S | < 2 do 4 Cho ose u ∈ N ′ \ R . 5 y ← Relax ( y , u ) . 6 R ← R ∪ { u } . 7 Up date S ← R ∩ { u ∈ N ′ | y { u } ∈ (0 , 1) } . 8 Pic k distinct u, v ∈ S . 9 Deﬁne g ( t ) : = F  y + t  e { u } w ( u ) − e { v } w ( v )  . 10 Let t max = min { (1 − y { u } ) w ( u ) , y { v } w ( v ) } and t min = max {− y { u } w ( u ) , ( y { v } − 1) w ( v ) } . 11 Let t ⋆ ∈ arg max t ∈{ t min ,t max } g ( t ) . 12 y ← y + t ⋆  e { u } w ( u ) − e { v } w ( v )  . 13 return arg max  f ( R ) , f  R \ { u ∈ N | y { u } ∈ (0 , 1) }   . Our rounding algorithm is inspired b y the Pipage Rounding paradigm and emplo ys the Relax algorithm of [9] as a key subroutine. The Relax algorithm transforms a vector x ∈ [0 , 1] 2 n b y aggregating all comp onents corresp onding to sets that contain a given elemen t u in to the co ordinate e u . This transformation preserv es the marginal v ector Mar( x ) , increases the supp ort size by at most one, and do es not decrease the objective v alue. Lemma 32 (Lemma 5.1 in [9]) . L et x ∈ [0 , 1] 2 N , u ∈ N and z = Relax ( x , u ) ∈ [0 , 1] 2 N . Computing z r e quir es O (frac( x )) time, and the new ve ctor satisﬁes: • frac( z ) ≤ 1 + frac( x ) . • Mar( x ) = Mar( z ) , and x S = 0 for al l sets S that c ontain u , but ar e not the singleton set { u } . • F ( z ) ≥ F ( x ) . Our rounding pro cedure iterativ ely reduces the n umber of fractional elements. At each step, w e consider the elements whose corresp onding co ordinates in y are fractional after applying Relax . If few er than t wo suc h elements remain, w e Relax additional elements so that exactly t wo fractional elemen ts are presen t, denoted b y u and v . W e then examine the extended m ultilinear extension along the direction g ( t ) = F  y + t  e { u } w ( u ) − e { v } w ( v )  . W e can see that g is con vex. 28 Lemma 33 (Conv exity along exc hange directions) . L et f : 2 N → R b e a submo dular function and let F denote its multiline ar extension. F or any y ∈ [0 , 1] N and any two distinct elements u, v ∈ N , the function g ( t ) = F  y + t  e { u } w ( u ) − e { v } w ( v )  is c onvex in t over its fe asible interval. Pr o of. It’s easy to know that g ′′ ( t ) = 1 w ( u ) 2 ∂ 2 F ( z ) ∂ z 2 { u }       z = y + t  e { u } w ( u ) − e { v } w ( v )  + 1 w ( v ) 2 ∂ 2 F ( z ) ∂ z 2 { v }       z = y + t  e { u } w ( u ) − e { v } w ( v )  − 2 w ( u ) w ( v ) ∂ 2 F ( z ) ∂ z { u } ∂ z { v }      z = y + t  e { u } w ( u ) − e { v } w ( v )  . By Observ ation 8 we hav e g ′′ ( t ) = − 2 w ( u ) w ( v ) ∂ 2 F ( z ) ∂ z { u } ∂ z { v }      z = y + t  e { u } w ( u ) − e { v } w ( v )  ≥ 0 . Our algorithm then up dates y to the v ector corresp onding to the endp oint of g that ac hiev es the larger function v alue. This pro cedure is rep eated iterativ ely until all elemen ts hav e b een pro cessed by Relax , leaving at most one fractional co ordinate. At this p oin t, the algorithm compares the tw o candidate sets: one consisting of all elements rounded to 1, and the other consisting of these elements together with the remaining fractional element, and selects the set with the larger function v alue. The rounding algorithm comes with the follo wing p erformance guaran tee: Theorem 34. Given a submo dular function f : 2 N → R + , a knapsack c onstr aint with w ( u ) ≤ εB for al l u ∈ N and a fr actional solution y satisfying P u ∈N Mar u ( y ) w ( u ) ≤ B , our Rounding algorithm r eturns a discr ete solution S ⊆ N such that f ( S ) ≥ F ( y ) and w ( S ) ≤ (1 + ε ) B , and the algorithm uses O ( n · 2 frac( y ) ) queries. Pr o of. W e ﬁrst analyze the b ehavior of the algorithm when y is up dated, sp eciﬁcally during lines 9–12. By con vexit y , one of the t wo endp oin ts of this in terv al achiev es a v alue at least as large as the curren t one, allowing us to round either u or v to an in tegral v alue without decreasing the ob jective. Moreov er, mo ving along this direction preserv es the weigh ted sum X u ∈N Mar u ( y ) · w ( u ) , Therefore, the knapsack capacit y is maintained. 29 During the execution, applying Relax ma y increase frac( y ) b y 1, which also increases | S | b y 1. Eac h step along a con v ex exchange direction decreases b oth frac( y ) and | S | by 1. Since | S | ≤ 2 is alwa ys main tained, it follo ws that frac( y ) can increase by at most 2 relativ e to its initial v alue. A t the ﬁnal iteration, it is p ossible that only a single element remains fractional. In this case, the current ob jective is a con vex com bination of the v alues obtained by either selecting or discarding this elemen t. Hence, w e can choose the better integral option, ensuring f ( S ) ≥ F ( y ) . Finally , since each rounding step preserv es the w eighted sum and only one elemen t may remain fractional at the end, we ha v e w ( S ) ≤ (1 + ε ) B . The total num b er of queries is O ( n · 2 frac( y ) ) , as each ev aluation of F requires 2 frac( y ) queries and there are at most n elemen ts to process. This ﬁnal decision ma y cause the knapsac k capacity to b e exceeded; how ev er, the violation is b ounded by at most εB . Since the rounding pro cedure is executed with an initial capacit y of (1 − ε ) B , suc h a b ounded capacit y violation can b e safely tolerated. 4.4 Pro of of the Main Result In light of the abov e descriptions, w e are now ready to combine all the preceding results to pro ve Theorem 2. Pr o of of The or em 2. Algorithm 5 ﬁrst enumerates all sets E i with | E i | ≤ ε − 2 , which ensures that the set constructed in Lemma 2, denoted b y E , is included among the en umerated candi- dates. At this p oin t, our algorithm then applies an optimization algorithm to the resulting prob- lem instance with ob jective function g ( S ) = f ( S ∪ E i ) and knapsack capacit y (1 − ε )( B − w ( E )) . Then we obtain a vector y for G ( · ) , where G is the extended m ultilinear extension of g . Due to the prop erty of E , this guaran tees that G ( y ) ≥  1 e − O ( ε )  g (OPT) ≥  1 e − ε  f (OPT) , and X u ∈N Mar u ( y ) · w ( u ) ≤ (1 − ε )( B − w ( E )) . W e then apply a rounding pro cedure, which returns tw o discrete solutions R and R + u suc h that max { g ( R ) , g ( R + u ) } ≥ g ( y ) and w ( R + u ) ≤ B − w ( E ) . W e select the one with the larger v alue and denote it by S . 30 Finally , our algorithm outputs the set S i ∪ E i with the largest function v alue among all candidates, which in particular includes the set S ∪ E discussed ab o v e. Since w ( S ∪ E ) ≤ B and f ( S ∪ E ) = g ( S ) ≥ g ( y ) ≥  1 e − ε  f (OPT) , b oth the feasibility and the appro ximation ratio are satisﬁed. F or the query complexity , the enumeration step ﬁrst incurs a multiplicativ e ov erhead of 2 O ( ε − 2 ) . In the optimization phase, w e alwa ys maintain frac( y ) ≤ ε − 4 . The pro cess runs for ε − 3 rounds, and in eac h round the algorithm ev aluates the function for n 2 candidate directions. Since querying the v alue of g ( y ) requires 2 frac( y ) v alue queries to g , and frac( y ) ≤ ε − 4 throughout the algorithm, each suc h ev aluation costs at most 2 ε − 4 queries. Therefore, the total n um b er of queries in this phase is O ( n 2 · 2 ε − 4 ) . Then, the rounding algorithm tak es a fractional solution with frac( y ) ≤ ε − 4 as input.Each ev aluation requires 2 frac( y ) ≤ 2 ε − 4 queries, resulting in O ( n · 2 ε − 4 ) queries in total. Combining all parts, the ov erall n umber of v alue queries is n O ( ε − 2 ) ·  O ( n 2 · 2 ε − 4 ) + O ( n · 2 ε − 4 )  = O ε ( n ε − 2 ) . 5 Conclusion In this work, based on the optimization–then–rounding paradigm o v er the extended m ulti- linear extension, we design deterministic algorithms for maximizing non-monotone submo dular functions under matroid and knapsac k constrain ts. Our algorithms ac hieve improv ed appro xi- mation ratios compared with the b est previously kno wn deterministic results. F or the future, the extended multilinear extension framew ork still has broader p otential for further study , although many related questions remain challenging. One p ossible direction is to derandomize the aided contin uous greedy algorithm for knapsac k constraints, which w ould lead to a deterministic algorithm with an approximation ratio of 0 . 385 . Another direction is to extend the constraint to a multi-dimensional knapsack. W e hav e v eriﬁed that in the optimization phase, the m ultiplicativ e up dates technique can b e applied within the extended m ultilinear extension framework. Nevertheless, a deterministic rounding algorithm tailored for the m ulti-dimensional knapsack constraint is still required. Suc h a rounding pro cedure must sim ultaneously maintain feasibilit y for all knapsack constrain ts while carefully con trolling the n umber of fractional co ordinates in the v ector. 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Deterministic Algorithm for Non-monotone Submodular Maximization under Matroid and Knapsack Constraints

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