A Parameterized Perspective on $P_2$-Packings

A P arameterized P ersp ectiv e on P 2 -P ac kings Jianer Chen 2 , 3 Henning F ernau 1 Dan Ning 2 Daniel Raible 1 Jianxin W ang 2 1 Universit¨ at T rier , FB IV—Abteilung Informatik, 54286 T rier, German y , { fernau,r aible } @informati k.uni-trier.de 2 School of Information Science and Engineering, Central South Universit y , Changsha, Hu an 4100 83, P .R. China, alina1028@ hotmail.com,jxwa ng@mail.csu.edu.cn 3 Department of Computer Science, T exas A&M Universit y , College Station, T exas 77843, U SA, chen@cs.tamu.e du Abstract. W e study (vertex-disjoin t) P 2 -packings in graphs und er a parameterized p ersp ective. Starting from a m ax imal P 2 -packing P of size j we use extremal arguments for determining how many vertices of P app ear in some P 2 -packing of size ( j + 1). W e basically can ’reuse’ 2 . 5 j vertices . W e also presen t a kernelizatio n algorithm th at giv es a k ernel of size b oun d ed by 7 k . With these tw o results w e build an algori thm whic h constructs a P 2 -packing of size k in time O ∗ (2 . 482 3 k ). 1 In tro duction and Definitions Motivation. W e consider a generaliza tion of the matching pr o blem in graphs. A matching is a maximum ca rdinality set of vertex disjoint edg es. O ur problem is a generalizatio n in the sense that the ter m edge may be r eplaced b y 2-edg e-path called P 2 . More formally , we study the following problem, called P 2 -p acking: Giv en: A gr aph G = ( V , E ), a nd the parameter k . W e ask: Is there a s et of k vertex-disjoint P 2 ’s in G ? P . Hell a nd D. K irkpatrick [8,6] prov ed N P -co mpleteness for this problem. P 2 - p acking attracts attention as it is N P -ha rd whereas the matching problem, which is P 1 -p acking , is p o ly-time solv able. Also there is a primal-dual rela tion to tot al edge cover sho wn by H. F ernau and D. F. Ma nlov e [4]. Recall that an e dge c over is a set of edges E C ⊆ E that cov er all vertices o f a given graph G = ( V , E ). An edge co ver is called total if every compo nent in G [ E C ] has at least tw o edges. By matching techniques, the problem of finding an edge cov er of size at most k is p oly- time solv able. How ever, the following Gallai-type identit y s hows that finding total edge c overs of s ize at most k is N P -har d: The sum of the num b er of P 2 ’s in a ma ximum P 2 -packing and the size of a minimum total edge cov er equals n = | V | . There is a lso a rela tio n to test co ver . The input to t his problem is a hyper - graph H = ( G, E ) and one wis hes to iden tify a subset E ′ ⊆ E (the t est c over ) such that, for an y distinct i, j ∈ V , there is an e ′ ∈ E ′ with | e ′ ∩ { i.j } | = 1. Test cover mo dels identification problems: Giv en a set of ind ividua ls and a s et of binary attributes we se arch for a minim um subset of attributes that identifies 2 each individual distinctly . Applications, as mentioned in K. M. J. Bontridder et al. [1], ra ng e fr om fault testing and diagnosis, patter n recognitio n to biological ident ifica tio n. K. M. J. Bontridder et al. could show for the case TCP 2, where for all e ∈ E w e hav e | e | ≤ 2, the subseq uent tw o statements. Fir st, if H has a test cover of size τ , then there is a P 2 -packing o f size n − τ − 1 that leav es at least one vertex is olated. Second, if H ha s a maximal P 2 -packing of size π that leaves at least one vertex isolated, then ther e is a test cov er of size n − π − 1. This a lso establishes a clo se r elation b etw een test cover and tot al edge cover . So, we can employ our a lg orithms to solve the TCP2 case o f test cover , by using an initial catalytic bra nch that determines one vertex that s hould be isolated. Discussion of R elate d Work. R. Hassin a nd S. Rubinstein [5] found a ra ndo mized 35 67 -approximation. K. M. J. Bontridder et al. [1 ] s tudied P 2 -packing a ls o in the context of approximation, wher e they co nsidered a serie s of heur istics H ℓ . H ℓ starts from a ma ximal P 2 -packing P and attempts to improv e it b y replacing any ℓ P 2 ’s by ℓ + 1 P 2 ’s not con tained y et in P . The corresp onding approximation ratios ρ ℓ are as follows: ρ 0 = 1 3 , ρ 1 = 1 2 , ρ 2 = 5 9 , ρ 3 = 7 11 and ρ ℓ = 2 3 for ℓ ≥ 4. As an y P 2 -p acking instance can be tra nsformed to fit in 3-set p acking o ne can use Y. Liu et al. [9] algor ithm which needs O ∗ (4 . 61 3 k ) steps. The fir s t paper to individually study P 2 -p acking under a parameter ized view was E. Pr ieto and C. Slop er [10]. The authors were able to prov e a 15 k -kernel. Via a clever ’midpo int ’ s earch on the kernel they could achieve a r un time of O ∗ (3 . 403 3 k ). Our Contributions. The t wo main algor ithmic achiev ements of this pap er are : (1) a new 7 k -kernel for P 2 -P a cking , (2) an algo r ithm which solves this problem in O ∗ (2 . 482 3 k ). This alg orithm makes use of a new theorem that says that w e basi- cally can r euse 2 . 5 j vertices of a maximal P 2 -packing with size j . This improv es a similar result for gener al 3-set p acking [9] where only 2 j elemen ts a re reusable. This theor em is prov en by making extensive use of extremal com binator ial ar- guments. Another no velt y is that in this algorithm, the dynamic progra mming phase (use d to inductively augment maximal P 2 -packings) is interleav ed with kernelization. This pays off not only heuristically but also asymptotically by a sp ecific for m of co mbinatorial analys is. Thereby we can completely skip the time consuming color-co ding which was needed in Liu et al. [9] for 3-set p acking . W e b elieve that t he idea of saving colors by extre ma l co mbin a torial arg ument s could be applied in o ther situations, as well. Some Notations and Definitions. W e only consider undirected graphs G = ( V , E ). F or a subgraph H o f G , denote by N ( H ) the set of vertices that are not in H but adjacent to at lea st one vertex on H , i.e., N ( H ) = ( S v ∈ H N ( { v } )) \ H . The subgr aph H is adjac ent to a vertex v if v ∈ N ( H ) . A P 2 in G is a path which consists of three v ertices and tw o edges. F or an y p a th p of this kind we consider the vertices as num be red s uch that p = p 1 p 2 p 3 (where the roles o f p 1 and p 3 might be in terchanged). F or a path p , V ( p ) ( E ( p ), resp.) deno tes the set of vertices (edg es, resp.) on p . Lik ewis e , for a set of paths P , V ( P ) := S p ∈P V ( p ) ( E ( P ) := S p ∈P E ( p ), resp.). 3 2 Kernelization W e ar e going to improv e o n the e a rlier 1 5 k -kernel of E. Prieto and C. Slop er by allowing lo ca l improv ements on a maximal P 2 -packing, but otherwise using the ideas of E. P rieto and C. Slop er. Ther efore, we firs t revise the necess ary notions and lemmas from their pa pe r [10]. 2.1 Essential Prerequisi tes Definition 1. A double cr own de c omp osition of a gr aph G is a de c omp osition ( H, C, R ) of the vertic es in G such that 1. H (t he he ad) sep ar ates C and R ; 2. C = C 0 ∪ C ′ ∪ C ′′ (the cr own) is an indep endent set such that | C ′ | = | H | , | C ′′ | = | H | , and ther e exist a p erfe ct matching b etwe en C ′ and H , and a p erfe ct matching b etwe en C ′′ and H . Definition 2. A fat cr own de c omp osition of a gr aph G is a de c omp osition ( H , C, R ) of the vertic es in G such t hat 1. H (t he he ad) sep ar ates C and R ; 2. the induc e d sub gr aph G ( C ) is a c ol le ction of p airwise disjoint K 2 ’s; 3. ther e is a p erfe ct matching M b etwe en H and a subset of vertic es in C such that e ach c onne cte d c omp onent in C has at m ost one vertex i n M . E. Prieto and C. Slop er co uld show the following lemmas. Lemma 1. A gr aph G with a double cr own ( H , C, R ) has a P 2 -p acking of size k if and only if the gr aph G − H − C has a P 2 -p acking of size k − | H | . Lemma 2. A gr aph G with a fat cr own ( H , C, R ) h as a P 2 -p acking of size k if and only if the gr aph G − H − C has a P 2 -p acking of size k − | H | . Lemma 3. A gr aph G with an indep endent set I , wher e | I | ≥ 2 | N ( I ) | , has a double cr own de c omp osition ( H, C, R ) , wher e H ⊆ N ( I ) , which c an b e c on- structe d in li ne ar time. Lemma 4. A gr aph G with a c ol le ction J o f indep endent K 2 ’s, wher e | J | ≥ | N ( J ) | , has a fat cr own de c omp osition ( H, C, R ) , wher e H ⊆ N ( J ) , which c an b e c onstr u cte d in line ar time. 2.2 A Smaller Ke rnel Let G be a gr aph, a nd let P = { L 1 , . . . , L t } b e a maximal P 2 -packing in G , where e a ch L i is a subgr aph in G that is isomor phic to P 2 , and t < k . Then each connected comp onent of the g raph G − P is either a single vertex or a single edge. Let Q 0 be the s et o f all vertices such that each vertex in Q 0 makes a connected comp onent of G − P (each vertex in Q 0 will b e ca lled a Q 0 -vertex ). Let Q 1 be 4 s s s ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ❝ ❝ ✲ ❝ s s ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ❝ s s s s ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ❝ ❝ ✲ s s ❝ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ s ❝ Fig. 1. Reducing the n umber of Q 0 -vertices s s s ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ❝ ❝ ❝ ❝ ✲ s s ❝ ◗ ◗ ◗✑ ✑ ✑ s s s s s s s ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ❝ ❝ ❝ ❝ ✲ s ❝ s ◗ ◗ ◗ ✑ ✑ ✑ s s s s Fig. 2. Reducing the n umber of Q 1 -edges the set o f all edg e s such tha t each edge in Q 1 makes a c onnected comp onent of G − P (each edge in Q 1 will be called a Q 1 -e dge ). Our kernelization algor ithm starts with the following pro cess, which tries to reduce the num be r of Q 0 -vertices a nd the num ber of Q 1 -edges, by applying the following rules : 1 Rule 1. If a P 2 -copy L i in P has tw o vertices that each is adjacent to a different Q 0 -vertex, then apply the pro ces ses des c rib ed in Figure 1 to decrease the num b e r of Q 0 -vertices by 2 (and increa se the num b er of Q 1 -edges by 1). Rule 2. If a P 2 -copy L i in P has tw o vertices that each is adjace nt to a different Q 1 -edge, then apply the pro cesses describ ed in Fig ure 2 to decrease the n umber of Q 1 -edges by 2 (and increa se the size of t he maximal P 2 -packing by 1). Note that these r ule s cannot b e applied forever. The num b er of consecutive applications of Rule 1 is b ounded by n / 2 since each applicatio n of Rule 1 re duce s the num b er of Q 0 -vertices b y 2 ; and the total num b er o f applicatio ns of Rule 2 is bo unded b y k s ince each application of Rule 2 increase s the num ber of P 2 -copies in the P 2 -packing b y 1 . W e also remark that during the applica tions of these rules, the resulting P 2 -packing P may beco me non-maxima l. In this case, we simply first make P max ima l a gain, using any prop er gr e e dy a lgorithm, b efor e we further apply the r ules. Therefore, w e m ust r each a p o int , in p olynomia l time, where none of the rules ab ov e is applicable. At this p oint, the maximal P 2 -packing P has the following prop erties: F or each L i of the P 2 -copies in P , Prop ert y 1. If mor e than one Q 0 -vertices ar e adjacen t to L i , then all these Q 0 -vertices must b e adjacen t to the s ame (and unique) vertex in L i . 1 W e ha ve u sed solid circles and thic k lines for vertices and edges, resp ectively , in th e P 2 -packing P , and hollow circles and th in lines for vertices and edges n ot in P . In particular, tw o h ollow circles linked by a thin line represents a Q 1 -edge. 5 Prop ert y 2. If mor e than one vertex in L i are adjacent to Q 0 -vertices, then all these vertices in L i m ust be adjacent to the same (and unique) Q 0 -vertex. Prop ert y 3. If more than one Q 1 -edges are a dja c e nt to L i , then a ll these Q 1 -edges must b e adjace nt to the same (and unique) vertex in L i . Prop ert y 4. If more than o ne vertex in L i are adjacent to Q 1 -edges, then all these vertices in L i m ust be adjacent to the same (and unique) Q 1 -edge. Theorem 1 . L et P = { L 1 , L 2 , . . . , L t } b e a maximal P 2 -p acking on which Rules 1-2 ar e not applic able, wher e t ≤ k − 1 . If t he numb er of Q 0 -vertic es is lar ger than 2 k − 3 , then ther e is a double cr own, c onstructible in line ar time. Pr o of. W e par tition the Q 0 -vertices into t wo groups: the g roup Q ′ 0 that consis ts of all the Q 0 -vertices such that eac h Q 0 -vertex in Q ′ 0 has at least tw o different neighbors in a single P 2 -copy L i of P ; and Q ′′ 0 = Q 0 − Q ′ 0 . Without loss of generality , let L 1 = { L 1 , . . . , L d } be the collection of P 2 -copies in P such tha t each L i in L 1 has at least tw o vertices tha t are adjacent to the same vertex in Q ′ 0 . By Prop erty 2, at most one vertex in Q ′ 0 can b e adjacen t to each P 2 -copy L i in L 1 . There fo re, | Q ′ 0 | ≤ |L 1 | = d , whic h also implies that Q ′′ 0 is not empt y . Moreov er , | Q ′′ 0 | = | Q 0 | − | Q ′ 0 | ≥ 2 k − 2 − d ≥ 2( k − 1 − d ) ≥ 2 t − 2 d. By prop er ty 2 again, no vertex in Q ′′ 0 can be adjacent to any P 2 -copy L i in L 1 . Therefore, the neighbor s of the vertices in Q ′′ 0 are all contained in the colle ction L 2 = { L d +1 , . . . , L t } . By definition, ther e is a t most one vertex in ea ch L i in L 2 that is a djacent to Q 0 -vertices. Therefor e, the total n umber | N ( Q ′′ 0 ) | of neighbo rs of Q ′′ 0 is b ounded by t − d . No te tha t Q ′′ 0 is an indep endent set, a nd | Q ′′ 0 | ≥ 2 t − 2 d = 2( t − d ) ≥ 2 · | N ( Q ′′ 0 ) | By Lemma 3, the graph has a double crown that can be constructed in linear time. ⊓ ⊔ The pro of of the following theor e m is quite similar to that o f Theorem 1. Theorem 2 . L et P = { L 1 , L 2 , . . . , L t } b e a maximal P 2 -p acking on which Rules 1-2 ar e n ot applic able, wher e t ≤ k − 1 . If the num b er of Q 1 -e dges is lar ger than k − 1 , then t her e is a fat cr own, which c an b e c onstru cte d in line ar t ime. Pr o of. W e partition the Q 1 -edges in to tw o g roups: the group Q ′ 1 that consists of all the Q 1 -edges such that each Q 1 -edge in Q ′ 1 has a t leas t tw o different neighbor s in a single P 2 -copy L i of P ; and Q ′′ 1 = Q 1 − Q ′ 1 . Without loss of gener ality , let L 1 = { L 1 , . . . , L d } b e the collection of P 2 -copies in P such that each L i in L 1 has a t lea st tw o vertices that ar e a dja c ent to the same edge in Q ′ 1 . By Pr o p erty 4, at most one edge in Q ′ 1 can be a dja c ent to each P 2 -copy L i in L 1 . T he r efore, | Q ′ 1 | ≤ |L 1 | = d , which also implies that Q ′′ 1 is not empty . More over, | Q ′′ 1 | = | Q 1 | − | Q ′ 1 | ≥ k − 1 − d ≥ t − d. 6 By proper t y 4 aga in, no edge in Q ′′ 1 can be adjacen t to any P 2 -copy L i in L 1 . Therefore, the ne ig hbors of the edges in Q ′′ 1 are all con tained in the collection L 2 = { L d +1 , . . . , L t } . By definition, ther e is a t most one vertex in ea ch L i in L 2 that is adjace nt to Q 1 -edges. Therefor e, the total num ber | N ( Q ′′ 1 ) | of neighbor s of Q ′′ 1 is bounded by t − d . Note that Q ′′ 1 is a c ollection of independent K 2 ’s, and | Q ′′ 1 | ≥ t − d ≥ | N ( Q ′′ 1 ) | By Lemma 4, the g raph ha s a fat crown that can be constr ucted in linea r time. ⊓ ⊔ Based on all these facts , our kernelization a lgorithm g o es like this: we start with a maximal P 2 -packing P , and rep ea tedly a pply Rules 1-2 (and keeping P maximal) un til neither Rule 1 no r Rule 2 is applicable. At this p oint, if the nu mber of Q 0 -vertices is lar ger than 2 k − 3, then by Theorem 1 , we genera te a double crown that, by Lemma 1, leads to a larger P 2 -packing. On the other hand, if the n umber of Q 1 -edges is lar ger than k − 1, then by Theorem 2, we genera te a fat crown that, by Lemma 2, leads to a larger P 2 -packing. By repeating this pro cess po lynomial many times, either we will end up with a P 2 -packing of size at least k , or we e nd up with a ma ximal P 2 -packing P of size les s than k on whic h neither Rule 1 nor Rule 2 is applicable, the num b er of Q 0 -vertices is bounded by 2 k − 3, and the num ber of Q 1 -edges is b ounded b y k − 1 (whic h implies tha t there a re a t most 2 k − 2 v ertices in Q 1 ). The vertices in the sets Q 0 and Q 1 , plus the at most 3 k − 3 vertices in the P 2 -packing, give a g raph of a t m o s t 7 k − 8 vertices. Theorem 3 . P 2 -p acking admits a kernel wi th at mo st 7 k vertic es. W e men tion here that the (more general results) of H. F ernau and D. Manlo ve [4 ] can b e improv ed for the parametric dual (in the se ns e of the mentioned Ga llai- t yp e ident ity) tot al edge cover : Theorem 4 . tot al edge co ver admits a kernel with at most 1 . 5 k d vertic es. Pr o of. Since we aim at a tota l edge cov er, the larg est num b er of v er tice s that can be covered by k edges is 1 . 5 k (namely , if the edge c ov er is a P 2 -packing). Hence, if the gr a ph contains more than 1 . 5 k v ertices, we can reject. This leav es us with a kernel w ith at most 1 . 5 k vertices. ⊓ ⊔ Corollary 1. T rivial ly, P 2 -p acking do es not admit a kernel with less than 3 k vertic es. tot al edge cover do es not admit a kernel with less than α a k d vertic es for any α d < (7 / 6) , u n less P = N P . Pr o of. A P 2 -packing of size k is only poss ible in a graph with at least 3 k vertices. Due to Theorem 3 and [2, Theorem 3.1], there do es no t exist a kernel o f s ize α d k d for tot al edge cover under the assumption that P = N P if (7 − 1)( α d − 1) < 1. ⊓ ⊔ 7 3 Com binatorial Prop erties of P 2 -pac kings W e consider the following setting. Le t P b e a ma ximal P 2 -packing of size j of a given g raph G = ( V , E ). W e will ar gue in this sectio n that, whenever a P 2 -packing o f size ( j + 1) exists, then there is a lso one, called Q , that uses at least 2 . 5 j out of the 3 j vertices of P . This c o mbinatorial prop erty of Q (among others) will b e used in the next section by the inductiv e step of o ur algo rithm for P 2 -p acking . W e emplo y extremal combinatorial ar guments to achiev e our results, deriving more a nd more prope r ties that Q could p o ssess, without risking to miss any P 2 -packing of size ( j + 1). So, among all P 2 -packings of size ( j + 1), we will consider those packings Q that maximize X p ∈P X q ∈Q 1 [ E ( p )= E ( q )] , (1) where 1 [ ] is the indicato r function. W e ca ll these Q (1) . In Q (1) we find those packings from Q who ’r euse’ the m a x imu m n umber of P 2 ’s in P . F rom Liu et al. [9] we know: Lemma 5. | V ( p ) ∩ V ( Q ) | ≥ 2 for any p ∈ P and Q ∈ Q (1) . Pr o of. If there is p ∈ P with | V ( p ) ∩ V ( Q ) | = 1 , then replace the intersecting path of Q by p . In the case where | V ( p ) ∩ V ( Q ) | = 0 , simply replac e an arbitr ary q ∈ Q\ P , that m ust exist by pigeon-hole, by p . In b oth cases, we obtain a packing Q ′ of the same size a s Q , but |P ∩ Q ′ | = |P ∩ Q| + 1, contradicting Q ∈ Q (1) . ⊓ ⊔ A slightly shar pe r version is the next as s ertion: Corollary 2. If Q ∈ Q (1) , then for any p ∈ P with p 6∈ Q , t her e ar e q 1 , q 2 ∈ Q with | V ( p ) ∩ V ( q i ) | ≥ 1 ( i = 1 , 2 ). Pr o of. Supp ose it exists p ∈ P and only one q ∈ Q with | V ( p ) ∩ V ( q ) | ≥ 2. Then Q \ { q } ∪ { p } improv es on priority (1), contradicting Q ∈ Q (1) . ⊓ ⊔ F urthermore, fro m the set Q (1) we only collect those P 2 -packings Q ′ , which maximize the following se cond prop erty: X p ∈P X q ∈Q ′ | E ( p ) ∩ E ( q ) | . (2) The set o f the re ma ining P 2 -packings will b e called Q (2) . So, in Q (2) are those packings who cover the maximum num be r o f edges in E ( P ). W e define P i ( Q ) := { p ∈ P | i = | p ∩ V ( Q ) |} . A vertex v ∈ V is a Q -endp oint if there is a unique q = q 1 . . . q 3 ∈ Q such that v = q 1 or v = q 3 . A vertex v is called Q -midp oint if there is a q = q 1 q 2 q 3 ∈ Q with q 2 = v . Definition 3. 1. We c al l q = q 1 q 2 q 3 ∈ Q foldable on p = p 1 p 2 p 3 ∈ P if, for q 2 ∈ V ( p ) ∩ V ( q ) , we have p s = q 2 , s ∈ { 1 , 2 , 3 } , and either p s +1 6∈ V ( Q ) or p s − 1 6∈ V ( Q ) , se e Figur e 3(a) . 8 P S f r a g r e p l a c e m e n t s q 1 p 1 q 3 (a) q is foldable on p . P S f r a g r e p l a c e m e n t s q 1 p 1 q 3 (b) ( q 1 , p 1 )- folding. P S f r a g r e p l a c e m e n t s q 1 p 1 q 3 (c) q is shiftable on p . P S f r a g r e p l a c e m e n t s q 1 p 1 q 3 (d) ( q 3 , p 1 )- shifting. Fig. 3. The bla ck vertices and so lid edges indicate the P 2 -packing P . The p o ly - gons contain the P 2 ’s of the packing Q . 2. If q is foldabl e on p , then substitut ing q by q \ { q i } ∪ { p s ± 1 } with i ∈ { 1 , 3 } , wil l b e c al le d ( q i , p s ± 1 )-folding , se e Figur e 3(b). 3. We c al l q = q 1 q 2 q 3 ∈ Q shiftable with r esp e ct to q 1 ( q 3 , re sp.) on p = p 1 p 2 p 3 ∈ P if the fol lowing h olds: q 1 ∈ V ( p ) ∩ V ( q ) ( q 3 ∈ V ( p ) ∩ V ( q ) , r esp.) and either p s +1 6∈ V ( Q ) or p s − 1 6∈ V ( Q ) wher e p s = q 1 ( p s = q 3 , r esp.) and s ∈ { 1 , 2 , 3 } , se e Figur e 3(c). 4. If q is shiftable on p with r esp e ct to t ∈ { q 1 , q 3 } , then s ubstituting q by q \ { g } ∪ { p s +1 } (or by q \ { g } ∪ { p s − 1 } , r esp.), g ∈ { q 1 , q 3 } \ { t } , wil l b e c al le d ( g , p s +1 )-shifting ( ( g , p s − 1 )-shifting , r esp.), se e Figur e 3(d) . Lemma 6. If q = q 1 q 2 q 3 ∈ Q with Q ∈ Q (2) is shiftable on p ∈ P with r esp e ct to q 1 (or q 3 , r esp.), t hen t her e is some p ′ ∈ P with p ′ 6 = p such that { q 3 , q 2 } ∈ E ( p ) (or { q 2 , q 1 } ∈ E ( p ) , r esp.). Pr o of. W e examine the ca s e where V ( p ) ∩ V ( q ) = { q 1 } and, w.l.o.g., p s +1 6∈ V ( Q ). Now assume the contrary . The n by ( q 3 , p s +1 )-shifting, we obtain a P 2 - packing Q ′ . Comparing Q and Q ′ with resp ect to priority 1, Q ′ is no worse than Q . But Q ′ improv es on priority 2, as w e g ain { p s , p s +1 } . But this contradicts Q ∈ Q (2) . The case for V ( p ) ∩ V ( q ) = { q 3 } follows analo gously . ⊓ ⊔ Lemma 7. If Q ∈ Q (2) , then no q ∈ Q is foldable. Pr o of. Supp ose so me q ∈ Q is foldable on p and, w.l.o .g., p s +1 6∈ V ( Q ) a nd q 1 6∈ V ( P ). Then by ( q 1 , p s +1 )-folding q we could improve o n prio rity 2, cont r a dicting Q ∈ Q (2) . ⊓ ⊔ Suppo se there is a path p with | V ( p ) ∩ V ( Q ) | = 2. Then p sha res exactly one vertex p q ′ , p q ′′ with paths q ′ , q ′′ ∈ Q due to Cor o llary 2. In the following p q ′ and p q ′′ will always refer to the tw o cut vertices of the paths q ′ , q ′′ ∈ Q which cut a path p with | V ( p ) ∩ V ( Q ) | = 2. Lemma 8. L et Q ∈ Q (2) . Consider p ∈ P with | V ( p ) ∩ V ( Q ) | = 2 and neithe r p q ′ nor p q ′′ ar e Q - endp oints. The n o ne o f q ′ , q ′′ is foldabl e. Pr o of. Let i, j ∈ { 1 , 2 , 3 } suc h that p q ′ = p i and p q ′′ = p j . Then for f ∈ { 1 , 2 , 3 } \ { i, j } , w e have p f 6∈ V ( Q ). W.l.o.g ., { p i , p f } ∈ E ( p ). Then q ′ is ( q ′ 1 , p f )-foldable. ⊓ ⊔ 9 Corollary 3. L et Q ∈ Q (2) and p ∈ P with | V ( p ) ∩ V ( Q ) | = 2 . Then one of p q ′ , p q ′′ must b e a Q -en dp oint. Pr o of. Assume the c o ntrary . Lemmas 7 and 8 lead to a contradiction. ⊓ ⊔ Theorem 5 . L et P b e a maximal P 2 -p acking of size j . If ther e is a P 2 -p acking of size ( j + 1) , then ther e is also a p acking Q ∈ Q 2 such that | V ( P ) ∩ V ( Q ) | ≥ 2 . 5 j . Pr o of. Supp ose there is a path p ∈ P with | V ( p ) ∩ V ( Q ) | = 2. By Cor ollary 3, w.l.o.g.. p q ′ is a Q -endp oint. F or p q ′′ there are tw o p ossibilities: a) p q ′′ is also a Q -endp oint. Let { p f } = V ( p ) \ { p q ′ , p q ′′ } . Then w.l.o.g. p q ′ is a path neighbor of p f . Therefore p q ′ is shiftable. b) p q ′′ is a Q -midp o int. Claim. p q ′′ 6 = p 2 . Suppo se the con trary . The n w.l.o.g ., p q ′ = p 1 and thus q ′′ is folda ble o n p by a ( q ′′ 1 , p 3 )-folding. This contradicts Lemma 7. The claim follows. W.l.o.g., we assume p q ′′ = p 1 . Then it follows that p q ′ = p 2 , as o therwise a ( q ′′ 1 , p 2 )-folding would contradict Le mma 7 a gain. F rom p q ′ = p 2 and p 3 6∈ V ( Q ) w e ca n derive that a lso in this case p q ′ is shiftable. W e now examine for b oth ca ses the implications of the shiftabilit y of p q ′ . W.l.o.g., we supp os e that p q ′ = q ′ 1 . Due to Lemma 6 there is a p ′ ∈ P with { q ′ 3 , q ′ 2 } ∈ E ( p ′ ). F rom Corollary 2, it fo llows that ther e m ust b e a ¯ q ∈ Q \ { q ′ } with | V ( p ′ ) ∩ V ( ¯ q ) | = 1. Hence, | V ( p ′ ) ∩ V ( Q ) | = 3. Note that q ′ is the only path in Q with | V ( q ′ ) ∩ V ( p ′ ) | = 2. Summarizing, w e ca n say that for any p ∈ P with | V ( p ) ∩ V ( Q ) | = 2 we find a distinct p ′ ∈ P (via q ′ ) such that | V ( p ′ ) ∩ V ( Q ) | = 3. So, there is a total injection γ fro m P 2 ( Q ) to P 3 ( Q ). F rom |P 2 ( Q ) ∪ P 3 ( Q ) | = j and the existence of γ we derive |P 2 ( Q ) | ≤ 0 . 5 j . This implies | V ( P ) ∩ V ( Q ) | = 2 |P 2 ( Q ) | + 3 |P 3 ( Q ) | ≥ 2 . 5 j . ⊓ ⊔ 4 The Algorit hm W e like to p oint out the following t wo facts ab out P 2 -packings. First, if a gr aph has a P 2 -packing P = { p 1 , . . . , p k } , then it suffices to know the set of midp oints M P = { p 1 2 , . . . , p k 2 } to co nstruct a P 2 -packing of size k (which is p oss ibly P ) in po ly-time. This fact was discovered by E. P rieto and C. Slop er [10] and basically can b e achiev ed b y bipartite ma tching tec hniques. Second, it a lso suffices to know the set of endp oint pairs E P = { ( p 1 1 , p 1 3 ) , . . . , ( p k 1 , p k 3 ) } to construct a P 2 -packing of size k in poly- time. T his is due to Lemma 3.3 of Jia et al. [7] as any P 2 -packing instance also can b e viewed a s a 3-set p a cking instance. 4.1 Correctness Algo rithm 1. Steps one to six of Algor ithm 1 a re used for finding fat and double crowns. First we build a maximal P 2 -packing a nd lo cally improv e it via R ule 1 and Rule 2 . If afterwards we hav e | Q 0 | > 2 j − 3 we construct a double cro wn where j := |P | . If | Q 1 | > j − 1 w e find a fat cr own. These tw o actions are directly justified by Lemmas 1 and 2. If w e do not succee d an ymor e in finding either one 10 Algorithm 1 An Algor ithm fo r P 2 -p acking . 1: P = ∅ . 2: Greedily augment P to a maximal P 2 -packing. 3: Apply Rule 1 and Rule 2 exhaustively and call th e resulting p ac kin g P . 4: if There is a fat crown or double cro wn ( C , H, R ) with C ⊆ V \ V ( P ) then 5: k ← k − | H | . G ← G − H − C . 6: Goto 1. 7: else if k ≤ 0 then 8: return YES 9: else 10: T ry to construct a P 2 -packing P ′ F rom P with |P | + 1 = |P ′ | using A lgorithm 2. 11: if Step 10 failed then 12: return NO . 13: else 14: P ← P ′ . 15: Goto 2. Algorithm 2 An Algor ithm fo r a ugmenting a maximal P 2 -packing P . 1: j ← |P | . 2: for ℓ =0 to 0 . 3251 j do 3: for all S i ⊆ V ( P ), S o ⊆ V \ V ( P ) with | S i | = ( j + 1) − ℓ and | S o | = ℓ do 4: T ry to constru ct a P 2 -packing P ′ with S i ∪ S o as midp oints. 5: if Step 4 succeeded then 6: return P ′ . 7: for ¯ ℓ = 0 to 0 . 1749 j + 3 do 8: for all B i ⊆ V ( P ), B o ⊆ V \ V ( P ) with | B i | = 2( j + 1) − ¯ ℓ and | B o | = ¯ ℓ do 9: for all p ossible endp oint p airs ( e 1 1 , e 1 2 ) , . . . , ( e j +1 1 , e j +1 2 ) from B i ∪ B o do 10: T ry to constru ct a P 2 -packing P ′ with ( e 1 1 , e 1 2 ) , . . . , ( e j +1 1 , e j +1 2 ) as endp oint pairs. 11: if Step 10 succeeded then 12: return P ′ . 13: return failure. of the tw o cr own types we immediately can rely on | V ( G ) | ≤ 7 j . The next step tries to constr uct a new P 2 -packing P ′ from P such that P ′ comprises one mor e P 2 than P . F or this we inv oke Algorithm 2. Algo rithm 2. If a P 2 -packing P ′ with |P ′ | = j + 1 exists we can partion the midpo int s M P ′ in a part whic h lies within V ( P ) and one which lies outside. W e call them M i P ′ := M P ′ ∩ V ( P ) and M o P ′ := M P ′ ∩ O , resp ectively with O := V ( P ′ ) \ V ( P ). Theor em 5 yields | O | ≤ 0 . 5 j + 3 and thus |M o P ′ | ≤ 0 . 5 j + 3. Basically , we can find an integer ℓ with 0 ≤ ℓ ≤ 0 . 5 j + 3 such that |M i P ′ | = ( j + 1 ) − ℓ and |M o P ′ | = ℓ . In step 3 we run through ev ery such ℓ un til we reach 0 . 32 51 j . F or an y choice of ℓ in step 4 we cycle through all p os sibilities o f choosing sets S i ⊆ V ( P ) and S o ⊆ V \ V ( P ) such tha t | S i | = ( j + 1) − ℓ and | S o | = ℓ . Here S i and S o are candidates for M i P ′ and M o P ′ , re sp ectively . F or any choice of S i and S o we try 11 to co nstruct a P 2 -packing. If w e succee d once we ca n r e tur n the desir ed la r ger P 2 -packing. Otherwis e we rea ch the p oint where ℓ = 0 . 325 1 j . At this point we change our strategy . Instead of lo oking for the midpo ints of P ′ we fo cus on the endpo ints. W e do so b ecause this will improv e the run time as we will see la ter . O is the disjoint union of M o P ′ and the endp oints o f P ′ which do not lie in V ( P ) which we call E o P ′ . At this p oint we must hav e |M o P ′ | > 0 . 3 251 j and therefore | E o P ′ | < 0 . 17 49 j + 3. Now there must b e an integer ¯ ℓ with 0 ≤ ¯ ℓ ≤ 0 . 1 785 j + 3 such that | E o P ′ | = ¯ ℓ a nd the num b er of endp oints within V ( P ) (called E i P ′ ) must be 2( j + 1 ) − ¯ ℓ . In step 7 w e itera te through ¯ ℓ . In the next step we cycle thr ough all candidate sets for E o P and E i P which ar e ca lled B i and B o in the algo rithm. In step 9 w e consider all p o ssibilities ( e 1 1 , e 1 2 ) , . . . , ( e j +1 1 , e j +1 2 ) of how to pair the vertices in B i ∪ B o . A pair of endp oints ( e s r , e s r +1 ) means that b oth ver- tices sho uld app ear in the same P 2 of P ′ . Finally , w e try to co ns truct P ′ from ( e 1 1 , e 1 2 ) , . . . , ( e j +1 1 , e j +1 2 ) by co mputing a matching according to [7]. 4.2 Running Tim e Viewed separately , Algorithm 1 runs in p oly-time, a s fat and do uble crowns c a n be constructed in linear time (Lemmas 3 a nd 4). The o nly exp o ne ntial run time contribution comes fro m Algor ithm 2 . Here the r un times of a ll steps e x cept steps 3 and 4 are poly nomial in k . F or an y ℓ we exe cute step 3 a t most  3 j ( j +1) − ℓ  4 j ℓ  ∈ O   3 j j − ℓ  4 j ℓ   times. Likewise s tep 8 can be upp erb ounded by O   3 j 2 j − ℓ  4 j ℓ   . Lemma 9. F or any inte ger z with 0 ≤ z ≤ 0 . 5 j − 1 t he fol lowing h olds: 1 .  3 j j − z  4 j z  <  3 j j − ( z +1)  4 j z +1  . 2 .  3 j 2 j − z  4 j z  <  3 j 2 j − ( z +1)  4 j z +1  . Pr o of. 1. W e have  3 j j − ( z +1)  4 j z +1  −  3 j j − z  4 j z  = (3 j )!(4 j )!(( j − z )(4 j − z ) − (2 j + z +1)( z +1)) ( j − z )!(2 j + z +1)!( z +1)!(4 j − z )! . Now it is enough to sho w ( j − z )(4 j − z ) − (2 j + z + 1 )( z + 1 )) > 0 which ev a luates to 4 j 2 − 7 j z − 2 j − 2 z − 1 > 0 . F or the given z this alwa ys is tr ue. 2. W e have  3 j 2 j − ( z +1)  4 j z +1  −  3 j 2 j − z  4 j z  = (3 j )!(4 j )!((2 j − z )(4 j − z ) − ( j + z +1)( z +1)) (2 j − z )!( j + z +1)!( z +1)!(4 j − z )! . Then ((2 j − z )(4 j − z ) − ( j + z + 1)( z + 1)) = 8 j 2 − 7 j z − j − 2 z − 1 which for the given z is gr eater than zero. ⊓ ⊔ With Lemma 9 step 3 is upp erb o unded by O   3 j (0 . 6749) j  4 j 0 . 3251 j   and step 8 by O   3 j 1 . 8251 j  4 j 0 . 1749 j   . Both are dominated by O (15 . 285 j ). Notice the a s ymptotic sp eed-up w e achiev e b y c hanging the str ategy . If we would skip the sear ch fo r the endp oints, we would have to c ount ℓ up to 0 . 5 j in step 3. Then,  3 j 0 . 5 j  4 j 0 . 5 j  ∈ O (1 7 . 44 j ) which is also not a big impro vement c o mpared to a brute force search for the midp oints o n the 7 k -kernel, taking O ∗ (17 . 66 k ) steps. W e conclude: Theorem 6 . P 2 -p acking c an b e solve d in time O ∗ (2 . 482 3 k ) . 12 5 F uture work It would be nice to derive smaller kernels than 7 k or 1 . 5 k for P 2 -p acking or tot al edge cover , res p., in view of the mentioned low er b ound results [2]. W e try to apply extremal com binatoria l methods to sav e co lors for r elated problems, like P d -packings for d ≥ 3. First results seem to be pro mis ing. So, a deta iled com binatoria l (extr emal s tructure) s tudy of (say graph) structure under the persp ective of a specific com binatoria l pro blem seems to pay off not only for kernelization (a s expla ined with muc h detail in [3]), but also for iterative augmentation (and p ossibly co mpr ession). It would b e a lso interesting to w or k o n exa ct alg orithms for maximum P 2 - p acking . By using dynamic progr amming, this pro blem can be so lved in time O ∗ (2 n ). By Theorem 4, tot al edge cover ca n b e solved in time O ∗ (2 1 . 5 k ) ⊆ O ∗ (2 . 829 k ). Impro ving on exact algo rithmics would also impro ve on the pa- rameterized algor ithm for tot al edge co ver . Likewise, finding for example a search-tree algor ithm fo r tot al edge cover would b e in ter esting. References 1. K. M. J. De Bontridder, B. V. H alld´ orsson, M. M. H alld´ orsson, J. K. Lenstra, R. Ra vi, and L. Stougie. Appro x imation algorithms fo r the test cove r problem. Math. Pr o gr., Ser. B , 98:477–491, 2003. 2. J. Chen, H. F ernau, Y. A. Kanj, and G. Xia. Parametric dualit y and kernelization: lo wer b ounds and u pp er b ou n ds on kernel size. SI AM Journal o n Computing , 37:1077 –1108, 2007. 3. V. Es tivill-Castro, M. R. F ello ws, M. A. Langsto n, and F. A. Rosamond. FPT is P-time extremal structu re I. I n Algorithms and Complexity in Durham A CiD , vol u me 4 of T exts i n A l gorithmics , pages 1–41. K ing’s College Pub lications, 2005. 4. H. F ernau and D . F. Manlo ve. V ertex and edge cov ers with clustering p rop erties: Complexit y and algorithms. In A lgorithms and Complexity in Durham A Ci D , page s 69–84. King’s College, London, 2006. 5. R. Hassin and S. Rubinstein. An approximation algorithm for maximum triangle packing. Discr ete Applie d Mathematics , 154:97 1–979; 2620 [Erratum], 2006. 6. P . Hell an d D. G. Kirkpatrick. Star factors and star packings. T echnical Rep ort 82- 6, Computing Science, Simon F raser U niversi ty , Burn aby , B.C. V5A1S6, Canada, 1982. 7. W. Jia, C. Zhang, and J. Chen. A n efficient p arameterized algorithm for m - set packing. Jo urnal of Algorithms , 50:1 06–117, 2004. 8. D. G. Kirkpatric k and P . Hell. On the completeness of a general ized matc hing problem. In ACM Symp osi um on The ory of Comput ing STOC , pages 240–245, 1978. 9. Y. Liu, S . Lu, J. Chen, and S.-H. S ze. Greedy lo calization and color-cod ing: im- prov ed matching and packing algorithms. In International W orkshop on Pa r am- eterize d and Exact Computation IW PEC , volume 4169 of LNC S , pages 84–95. Springer, 2006. 10. E. Prieto and C. Slop er. Lo oking at the stars. In International Workshop on Par am eterize d and Exact Computation IW PEC , v olume 3162 of LNCS , pages 138– 148. Springer, 2004. ar q1 p1 ar p1 q3