A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

A W eakly Robust PT AS for Minim um Clique P artition in Unit Disk Graphs Imran A. Pirw ani ∗ Mohammad R. Sala v atipour † Abstract W e consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) int o a minim um num be r of cliques. The problem is NP-ha rd and v a r ious constant f acto r ap- proximations are known, with the curr ent best r atio o f 3. Our main r e s ult is a we akly r obust po lynomial time approximation sc heme (PT AS) f or UDGs expressed with edge- lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is no t a UDG; for the ca se (i) that it computes a clique partition, we show that it is g uaranteed to b e within (1 + ε ) ratio of the o ptim um if the input is UDG; how ever if the input is no t a UDG it either co mputes a clique partition a s in ca s e (i) with no guar antee on the quality of the cliq ue partition or detec ts that it is not a UDG. Noting that recognition of UDG’s is NP-hard even if we ar e given edg e lengths, our PT AS is a w eakly- robust algo rithm. Our a lgorithm ca n b e transfor med into a n O  log ∗ n ε O (1)  time distributed PT AS. W e consider a weigh ted version of the clique par tition problem on vertex weigh ted UDGs that generalize s the pr oblem. W e note some key distinctions with the un weigh ted version, where ideas useful in obtaining a P T AS breakdown. Y et, surprisingly , it admits a (2 + ε )-approximation algorithm for the weigh ted case where the gr aph is expres sed, say , as an adjacency matrix. This improv es on the b est known 8-appr oximation for the un weighte d ca se for UDGs ex pressed in standard form. Keyw ords: Computational Geometry , Approximation Algo rithms. 1 In tro duc tion A standard net w ork mo del for homogeneous netw orks is th e unit disk gr ap h (UDG). A graph G = ( V , E ) is a UDG if there is a mapp ing f : V 7→ R 2 suc h that k f ( u ) − f ( v ) k 2 ≤ 1 ⇔ { u, v } ∈ E ; f ( u ) 1 mo dels the p ositio n of the no de u while the unit disk cen tered at f ( u ) mo dels the range of radio comm unication. Two no des u and v are said to b e able to dir ectly communicate if they lie in the unit disks placed at eac h others’ cente rs. There is a v ast collection of literature on algorithmic problems studied on UDGs. See th e surv ey [2]. Clustering of a set of p oin ts is an imp ortant subroutine in m an y algo rithmic and practical applications an d there are many kinds of clus terings d ep endin g up on the application. A typical ob jectiv e in clustering is to minimize the num b er of “groups ” suc h that eac h “group” (cluster) ∗ Department of Computing Science, Universit y of Alb erta, Edmonton, Alb erta T6G 2E8, Canada. Email: pirwani@cs .ualberta.ca . S upp orted by Alb erta Ingenuit y . † Department of Computing Science, Universit y of Alb erta, Edmonton, Alb erta T6G 2E8, Canada. Email: mreza@cs.u alberta.ca . Supp orted by NS ERC and Alb erta In genuity . 1 f ( . ) is called a realization of G . N ote th at G may n ot come with a realization. 1 satisfies a set of criteria. Mutual p ro ximit y of p oin ts in a cluster is one suc h criterion, while p oint s in a cluster formin g a clique in the underlyin g netw ork is an extreme form of m utual p ro ximit y . W e study an optimization problem related to clustering, called the minimum clique p artition p roblem on this UDGs. Minim um clique part ition on unit disk graphs (MCP): Giv en a u nit disk grap h , G = ( V , E ), partition V in to a smallest n umber of cliques. Despite b eing theoretically interesting, MCP h as b een useful for other p roblems. F or example, [17] sho ws h o w to us e a small-sized clique partition of a UDG to construct a large collection of disjoin t (almost) dominating sets. Th ey [18] also sh o w ho w to obtain a go o d qualit y realization of UDGs, and an imp ortan t ingredien t in their tec hnique was to construct a small-sized clique partition of the graph. It is shown [12] ho w to use a small-sized clique partition to ob tain sparse spanners with b ounded dilation, wh ic h also p ermit guaranteed geographic r outing on a related class of graphs. [14] emp lo y MC P to obtain an O (log ∗ n ) time distr ib uted algorithm w h ic h is an O (log n )-appro ximation for the facil it y lo cation p roblem on UDGs w ithout geometry; they also giv e an O (1) time d istr ibuted O (1)-appro ximation to the facilit y lo cation problem on UDGs with geometry also using MCP . Recent ly , [15] sh o ws h ow to obtain a firs t O (1) approximat ion to the domatic partition pr oblem on UDGs using MCP . On general graph s, the clique-partition p r oblem is equ iv al ent to the minimum graph coloring on the complement graph wh ic h is not appr o ximable within n 1 − ε , for any ε > 0, unless P=NP [22]. MCP has b een studied for sp ecia l graph classes. It is shown to b e MaxSNP-hard for cubic graphs and NP-complete for planar cubic graphs [5]; they also giv e a 5 / 4-appro ximation algorithm for graphs with maxim um degree at most 3. MCP is NP-hard for a sub class of UDGs, called unit c oin gr aphs , wh er e th e inte riors of the asso ciated d isks are p airwise disj oint [6]. Go o d appr o xi- mations, ho w eve r, are p ossible on UDGs. The b est kn o wn app ro ximation is due to [6] wh o giv e a 3-appro ximation via a partitioning the v ertices in to co- comparabilit y graphs, and solving the problem exactly on them. They giv e a 2-appro ximation algorithm for coin graphs . MCP has also b een studied on UDGs exp ressed in stand ard form. F or UDGs exp ressed in general form [18] give an 8-appro ximation algorithm. Our Results a nd T echniques: In this pap er w e p resen t a w eakly-robust 2 PT AS for MCP on a giv en UDG. F or ease of exp osition, first we p ro v e this (in S ection 2.1) wh en the UDG is give n with a realization, f ( . ). Th e holy-grail is a PT AS when the UDG is expressed in standard form, sa y , as an adjacency matrix. How ev er, falling short of p ro ving this, we s ho w (in Section 2) h o w to get a PT AS when the input UDG is expressed in standard f orm along with asso cia ted edge-lengths corresp onding to some (un kno wn) realization. The algorithm is wea kly-robu s t in the sense that it either (i) computes a clique p artition of the input graph or (ii) gives a certificate that the inpu t graph is not a UDG. If the inp ut is indeed a UDG then the algorithm return s a clique partition (case (i)) wh ich is a (1 + ε )-a pp ro ximation (for a giv en ǫ > 0). Ho w ev er, if the inp u t is n ot a UDG, the algorithm either computes a clique partition but w ith no guaran tee on the qualit y of the solution or returns that it is not a UDG. Therefore, this 2 An algorithm is called r obust if it either compu tes an answ er or declares that th e input is not from the restricted domain; if the algorithm computes an answ er th en it is correct [20]. W e call our algorithm we akly-r obust in that it alw ays computes a clique partition or declares th at the inp ut is n ot from the restricted d omain (i.e. not a UDG); if the inpu t hap p ens to b e a UD G t hen th e answer is a (1 + ǫ )-app roximate clique p artition. Otherwise, it still retu rns a clique partition but there is no guarantee on t h e quality of th e clique partition. 2 algorithm sh ou ld b e seen as a w eakly-robust PT AS. The generation of a p olynomial -sized certificate whic h pr o v es why the input graph is not a UDG sh ould b e seen in the con text of the negativ e r esult of [1 ] w hic h sa ys that ev en if edge lengths are giv en, UDG r ecognition is NP-hard . W e show (in Section 4) ho w th is algorithm can b e mo d ified to run in O ( log ∗ n ε O (1) ) distribu ted round s. In S ection 3 we explore a we ight ed v ersion of MCP where we are giv en a vertex we ighte d UDG. In this formulatio n, the weig ht of a clique is the w eigh t of a hea viest v ertex in it, and th e weigh t of a clique partition is the sum of the w eigh ts of the cliques in it. W e note some k ey d istinctions b et wee n the w eight ed and the unw eig hted versions of the problem and sho w that the ideas that help in obtaining a PT AS d o n ot help in the we ight ed case. Y et, sur prisingly , we sh o w that the problem admits a (2 + ε )-appro ximation algorithm for the we ighte d case using only adjac ency . This result should b e contraste d with the un wei ghte d case where it is not clear as to ho w to remov e the dep end ence on the u se of edge-lengths, wh ic h w as cru cially exploited in deriving a PT AS. W e u se O PT to denote an optimum clique partition and opt to denote the size (or, in Section 3, w eigh t) of an optim um clique partition. W e also u se n and m to denote the num b er of p oin ts (i.e. no des of G = ( V , E )) and the num b er of edges, resp ectiv ely . 2 A W eakly-Robust PT AS for UDG Expressed with Edge-lengths F or simp licit y , we first describ e an algorithm w hen the input is giv en with a geometric realizat ion. 2.1 A PT AS for UDGs With a Geometric R ealization W e assume the inpu t UDG is expressed with geometry of its p oin ts. Using a randomly sh ifted grid whose cell size is k × k (for k = k ( ε )) w e partition the plane. Since the diameter of th e conv ex hull of eac h clique is at most 1, for large v alues of k , a fixed clique is cut b y this grid (and therefore b elongs to at most four cells) with p robabilit y at most 2 k . Ther efore, if w e could efficien tly compute an optimal clique partition in eac h k × k cell, then taking th e u nion of these cliques yields a s olution whose exp ect ed size at most (1 + ε )opt. W e can easily rep eat this pro cess O (log n ) times to obtain a solution with size at most (1 + ε )opt w.h .p. W e call the algorithm MinCP1 , form alized b elo w. Theorem 1. A lgorithm MinCP1 (given b elow) r eturns, in p oly-time, a clique p artition of size at most (1 + ε ) opt w.h.p. Algorithm 1 MinCP1( G, ε ) 1: Let k = ⌈ 16 ε ⌉ . Place a grid whose squ ares h a v e size k × k , on the plane. Call it G 0 , 0 . 2: Pic k ( a, b ) ∈ [0 , k ) × [0 , k ) uniformly at random. 3: Shift G 0 , 0 b y ( a, b ) to get G a,b whic h is a grid shifted a un its to th e right and b u n its ab o v e. G a,b induces a random partition of V into p oin ts in k × k r egions. 4: for all k × k regions of G a,b do 5: Ob tain an optimal partition C i for p oint- set P in the k × k square. 6: Let C a,b = S t i =1 C i b e the union of clique p artitions obtained for the p oin ts in eac h k × k square. 7: Rep eat “Step 2–6” ⌈ log n ⌉ times and retur n the smallest C a,b o v er th e ⌈ log n ⌉ indep enden t trials. W e b egin with a simple observ ation. 3 Observ at ion 2. Th e diameter of the c onvex hul l of every clique is at most 1 . In the next subsection we argue ho w to p er f orm “Step 5” of the algorithm MinCP 1 efficien tly . Assuming th is, we pro ve Theorem 1. F or a random shift G a,b and a clique C , w e sa y that G a,b “cuts” C if some line of G a,b crosses an edge of C . It is easy to see that: Pr h C is cut b y G a,b i ≤ Pr h a vertical or horizon tal l ine of G a,b crosses an edge of C i ≤ 2 k Th us , the exp ecte d n umb er of cliques in an optimal partition that are “cut” by G a,b is at most 2 k · opt. So, b y Mark o v’s in equalit y , with probability at least 1 / 2 there are n o more than 4 k · opt cliques cut b y G a,b . Ther efore, if w e compute an optimal s olution for eac h of the k × k grid cells and tak e the u nion of them, w ith probab ility at least 1 / 2 w e get an excess of at most 4 × 4 k · opt cliques with r esp ect to optim um since eac h clique that is “cut” b y the grid can b e counted up to four times. If we rep eat this pro cess for ⌈ log n ⌉ indep enden t rand om trials, we get that w ith probab ility at least 1 − 1 n the size of the solution we obtain is at most opt + 16 k · opt ≤ (1 + ε ) · opt. 2.1.1 Optimal Clique P artition of a UDG in a k × k Square Unlik e optimization problems su c h as maxim um (weig hted) ind ep endent set and minimum d omi- nating set, wh ere one can “guess” only a small-sized s ubset of p oin ts to obtain an optimal solution, the combinato rial complexit y of any s in gle clique in an optimal solution can b e h igh. Therefore, it is u nclear as to how to “guess” ev en few cliques, eac h of whic h ma y b e large. A result of C ap o yleas et al. [4 ] comes to our aid; a ve rsion of their result sa ys th at there exists an optimal clique partition where the con v ex hulls of the cliques are p air-wise non-o v erlappin g. This phenomenon of sep ar a- bility of an optimal partition, coupled with the fact that the size of an optimal partition in a sm all region is small, allo ws us to circum ve nt the ab o ve difficult y . Th e follo wing simple lemma b ound s the size of an optimal solution of an instance of b ounded diameter. Lemma 3. Any set of p o ints P i n a k × k squar e has a clique p artition of size O ( k 2 ) . Pr o of. Place a grid whose cells hav e size 1 / 2 × 1 / 2. This grid induces a v ertex partition where eac h blo c k in the p artition consists of the p oints that s hare a common grid cell (and therefore form a clique). W e state a v arian t of a result by Cap oyl eas et al. [4] according to wh ic h there exists an optimal clique partition wh ere the con v ex hulls of the cliques are non-ov erlapping, th at is, for an y p air of cliques in an optimal p artition, ther e is a straigh t line wh ic h separates them. 3 Theorem 4 ([4]) . F or a clique p artition in which the c onvex hul ls of the cliques ar e p airwise non- overlapping, ther e is a str aight line l ij that sep ar ates a p air of cliques C i , C j such that al l vertic es of C i ar e on one side of l ij , and al l the vertic es of C j ar e on the other side of l ij . (se e Figur e 1). F urthermor e, this p artition c an b e c omp ute d in p oly-time. The general structure of the algorithm f or computing optimal solution of a k × k cell is as follo ws. In order to reduce the searc h space for separator lines, one can fi nd a charact erization of the separator lines with some extra prop erties. Let C i , C j b e a pair of cliques eac h h a ving at least 3 W e gav e a pro of of this theorem [19] b efore it was brought t o our attention that Cap oyleas, Rote, and W o eginger [4] prov ed this muc h earlier in a different context. 4 k k C j C i x y C i C j l ij l ′ ij x y (a) (b) Figure 1: (a) An optimal clique partition of UDG p oin ts in a b ound ed region; eac h light conv ex shap e corresp onds to a clique in the clique partition. The hea vy line-segmen ts represent segments of the corresp onding separators. (b) A close-up view of C i and C j . A separator lin e, l ij is sho wn whic h separates C i and C j , corresp ondin g to the segmen t in (a). Note that l ′ ij is also a separator for C i and C j and l ′ ij is passing through p oints x and y in C i . t w o p oints. Let L ij b e the (infinite) set of d istinct separator lines. Since C i and C j are con v ex, there exists at least one line in L ij that goes through tw o p oints of C i (or C j ) (see Figure 1(b)). Therefore, giv en t wo cliques C i and C j in a clique p artition (with pairwise non-o v erlapping parts) there is a separator line l ij that goes through t wo v ertices of one of them, sa y u, v ∈ C i suc h that all the vertice s of C j are on one side of this line and all the vertic es of C i are on the other sid e or on the line. Since there are O ( k 2 ) cliques in an optimal partition of k × k cell, th ere are O ( k 4 ) pairs of cliques in the partition and their con v ex h ulls are p airwise n on-o v erlapping. I n fact, a more carefu l analysis sho ws that th e dual graph of the regions is planar (see Figure 1(a)); thus there are O ( k 2 ) distinct straigh t lines, eac h of whic h separate a pair of cliques in our optimal solution. F or every clique C i , the separator lines l ij (for all v alues of j ) defin e a conv ex region that contai ns clique C i . S o once we guess this set of O ( k 2 ) lines, these con v ex regions d efine the cliques. W e will try all p ossible (non-equiv ale nt) sets of O ( k 2 ) separator lines and c hec k if eac h of the con v ex regions indeed defi nes a clique and if we obtain a clique partition. This can b e p erformed in O ( n k 2 ) time (see [4] for more details). 2.2 A PT AS for UDGs With Edge-Lengths Only W e w eak en our assumption on ha ving access to geometry; w e assume only edge-lengths are kno wn with resp ect to a feasible (unkno wn) realization of the UDG. W e p ro v e that, Theorem 5. Given a gr aph G with asso c iate d (r at ional) e dge-lengths and ε > 0 , ther e is a p ol y- nomial time algorithm which either c omputes a clique p artition of G or gives a c ertific ate that G is not a UDG. If G is a UDG, the size of the clique p artitio n c ompute d is a (1 + ε ) -app r oximation of the optimum clique p artition (but ther e is no guar ante e on the size of the clique p art ition if the input gr aph is not UDG). The high leve l id ea of the algorithm is as follo ws. As in the geomet ric case, w e fi rst decomp ose the graph into b ounded d iameter regions and sho w that if w e can compute the optimum clique partition of eac h r egion then the union of these clique p artitions is within (1 + ε ) fraction of the optim um. There are t wo main difficulties here for whic h we need n ew id eas. The first ma jor 5 difference is that, we cannot us e the r andom sh ift argument as in the geometric case. T o o v ercome this, we use a ball gro wing tec hnique that yields b ounded diameter regions. T his is ins pired by [13] who giv e lo cal PT AS for weigh ted ind ep end ent set, and minim um d ominating set for UDGs without geometry . T h e second ma jor d ifference is that, ev en if w e h a v e the set of p oints b elonging to a b ounded region (a ball) it is un clear as to ho w to use the separation theorem to obtain an optimal s olution for this instance. Note that w e are not gu aranteed to ha v e a UDG as inpu t. W e sho w th at w e can either compu te a clique p artition for eac h sub graph induced by a ball, or give a certificate that the subgraph is not UDG. If it is a UDG, then our clique partition is optimal b ut if it is not a UDG there is no guaran tee on its size. Let B r ( v ) = { u : d ( u, v ) ≤ r } , where by d ( u, v ) we mean the num b er of edges on a sh ortest path from u to v . So, B r ( v ) can b e computed usin g a breadth-firs t searc h (BFS) tree ro oted at v . W e describ e our d ecomp ositio n algorithm wh ic h p artitions the graph into b oun d ed diameter subgraphs in Algorithm 2. W e w ill describ e a pr o cedure, called OPT-CP whic h, giv en a graph ind uced by the v ertices of B r ( v ) an d a parameter ℓ = p oly( r ), ru ns in time | B r ( v ) | O ( ℓ 2 ) ≤ n O ( ℓ 2 ) and either pro du ces a certificate th at B r ( v ) is not a UDG or computes a clique p artition of B r ( v ); this clique partition is optim um if B r ( v ) is a UDG. W e only call this pro cedure for “small” v alues of r . Algorithm 2 MinCP2( G, ε ) 1: C ← ∅ ; β ← ⌈ c 0 1 ε log 1 ε ⌉ ; ℓ ← c 1 β 2 . { where c 0 is the constan t in Lemma 9, and c 1 is the constan t in inequalit y (1). } 2: while V 6 = ∅ do 3: Pick an arbitrary v ertex v ∈ V 4: r ← 0 { Let C r ( v ) denote a clique partition of B r ( v ) compu ted b y calling OPT -CP } 5: w hile | C r +2 ( v ) | > (1 + ε ) · | C r ( v ) | do 6: r ← r + 1 7: if ( r > β ) or (OPT-CP( B r ( v )) return s “not a UDG”) the n 8: return “ G is not a UDG” and pro duce B r ( v ) as the certificate 9: C ← C ∪ C r +2 ( v ) 10: V ← V \ B r +2 ( v ) 11: return C as our clique partition Clearly , if the algorithm returns C on “Step 11”, it is a clique partition. Let us assume that eac h b all B r ( v ) w e consider in d uces a UDG and that the p ro cedure OPT-CP r eturns an optimal clique partition C r ( v ) for ball B r ( v ). W e show that in this case |C | ≤ (1 + ε )opt . W e also sh o w that for any iteration of the outer “while–lo op”, “Step 5” of MinCP2 is executed in time p olynomial in n , b y using edge-lengths instead of Euclidean co ordinates. F or an iteration i of the outer lo op, let v i b e th e vertex c hosen in “Step 3” and let r ∗ i b e th e v alue of r for which the “wh ile-loop” on “Step 5” terminates, that is, | C r ∗ i +2 ( v i ) | ≤ (1 + ε ) · | C r ∗ i ( v i ) | . Let k b e the maxim um num b er of iterations of the outer lo op. T he follo wing lemmas sho w that tw o distinct balls grown around ve rtices are f ar from eac h other, that the union of th e optimal solutions to th e b alls form a lo wer-boun d on th e cost of the en tire instance, and that the cost of C and opt is within a factor (1 + ε ) of opt. Lemma 6. F or every i 6 = j , every p air v ∈ B r ∗ i ( v i ) and u ∈ B r ∗ j ( v j ) ar e non-adjac ent. 6 Pr o of. Without loss of generalit y , let i < j . Therefore, ev ery ve rtex in B r ∗ j ( v j ) is at a leve l larger than r ∗ i + 2 of the BFS tree r o oted at v i , otherwise it would h a v e b een part of the ball B r ∗ i +2 ( v i ) th us remo v ed from V . Note that in a BFS tree ro oted at v i , there cannot b e an edge b et w een a lev el r an d r ′ with r ′ ≥ r + 2. Thus th ere cannot b e an edge b etw een a no d e in v ∈ B r ∗ i ( v i ), whic h has lev el at most r ∗ i and a no d e u ∈ B r ∗ j ( v j ), wh ic h w ould b een at a lev el at least r ∗ i + 3 in the BFS tree ro oted at v i . Next, we derive a low er-b ound on opt. Lemma 7. opt ≥ k X i =1 | C r ∗ i ( v i ) | Pr o of. Note that B r ∗ i ( v i ) is obtained by constructing a BFS tree ro oted at ve rtex v i up to some depth r ∗ i . According to Lemma 6, there is no edge b et we en an y t w o no d es v ∈ B r ∗ i ( v i ) and u ∈ B r ∗ j ( v j ). S o, no single clique in an optim um solution can con tain v ertices from distinct B r ∗ i ( v i ) and B r ∗ j ( v j ). Consider the subset of cliques in an optimal clique partition of G that inte rsect B r ∗ i ( v i ) and call this su bset OPT i . The argument ab o v e sh o ws that OPT i is disjoint from OPT j . Also, eac h OPT i con tains all the v ertices in B r ∗ i ( v i ). Sin ce C r ∗ i ( v i ) is an optimal clique partition for B r ∗ i ( v i ), | OPT i | ≥ | C r ∗ i ( v i ) | . The lemma immediately follo ws by observing that OPT i and OPT j are disjoint . The next lemma relates the cost of our solution to opt. Lemma 8. If | C r ∗ i +2 ( v i ) | ≤ (1 + ε ) · | C r ∗ i ( v i ) | , then      k [ i =1 C r ∗ i +2 ( v i )      ≤ (1 + ε ) · opt Pr o of.      k [ i =1 C r ∗ i +2 ( v i )      = k X i =1   C r ∗ i +2 ( v i )   ≤ (1 + ε ) · k X i =1   C r ∗ i ( v i )   ≤ (1 + ε ) · opt Finally , we s h o w that th e inner “while-lo op” terminates in ˜ O ( 1 ε ), so r ∗ i ∈ ˜ O ( 1 ε ). Obviously , the “while-loop” on “S tep 5” terminates even tually , s o r ∗ i exists. By definition of r ∗ i , for all smaller v alues of r < r ∗ i : | C r ( v i ) | > (1 + ε ) · | C r − 2 ( v i ) | . Since diameter of B r ( v i ) is O ( r ), if B r ( v ) is a UDG, there is a realizat ion of it in whic h all the p oints fit int o a r × r grid. T h us, | C r ( v i ) | ∈ O ( r 2 ). So for some α ∈ O (1) : α · r 2 > | C r ( v i ) | > (1 + ε ) · | C r − 2 ( v i ) | > . . . > (1 + ε ) r 2 · | C 0 ( v i ) | = O (  √ 1 + ε  r ) , when r is ev en (for o dd v alues of r w e obtain | C r ( v i ) | > (1 + ε ) r − 1 2 · | C 1 ( v i ) | ≥ O (  √ 1 + ε  r − 1 ). Therefore w e h av e: Lemma 9. Ther e is a c onstant c 0 > 0 such that for e ach i : r ∗ i ≤ c 0 /ε · log 1 /ε . In the next su bsection, we sho w that the algo rithm O P T-CP, giv en B r ( v ) and an upp er b ound ℓ on | C r ( v ) | , either computes a clique partition or d eclares th at the graph is not UDG; the size of the partition is optimal if B r ( v ) is a UDG. T he algorithm run s in time n O ( ℓ 2 ) . By the ab ov e argumen ts, if B r ( v ) is a UDG then, there is a constan t c 1 > 0 suc h that: | C r ( v ) | = O ( r ∗ i 2 ) ≤ c 1 · c 2 0 ε 2 log 2 1 ε . (1) 7 W e can set ℓ = ⌈ c 1 c 2 0 ε 2 log 2 1 ε ⌉ for an y inv ocation of OPT-CP as an upp er b ound, where c 1 is the constan t in O ( r ∗ i 2 ). So, the run ning time of the algorithm is n ˜ O (1 /ε 4 ) . 2.3 An Optimal Clique P artition for B r ( v ) Here w e p resen t the algorithm O PT-CP that giv en B r ( v ) (henceforth referred to as G ′ ) and an upp er b ound ℓ on the size of an optimal solution for G ′ , either computes a clique p artition of it or detects that it is not a UDG; if G ′ is a UDG then the partition is optimal. The algorithm r u ns in time n O ( ℓ 2 ) . Since, by Lemm a 9, ℓ is a constant in eac h call to this algorithm, the run ning time of OPT-CP is p olynomial in n . Our algorithm is based on th e separation theorem [4]. Even though w e do not hav e a realization of the n o des on the plane, assumin g that G ′ is a UDG, w e sho w h ow to apply the sep ar ation the or em [4] as in the geometric s etting. W e u se n o de/p oint to refer to a v ertex of G ′ and/or its corresp onding p oin t on the p lane f or some realization of G ′ . W e will use the follo wing tec hnical lemma. Lemma 10. Supp os e we have four mutual ly adjac e nt no des p, a, b, r and their p airwise distanc es with r esp e ct to some r e alization on the Euclide an plane. Then ther e is a p oly-time pr o c e dur e that c an de cide if p and r ar e on the same side of the line that go es thr ough a and b or ar e on differ ent sides. Pr o of. First, we describ e ho w to detect if the quadrilateral on these four p oint s is con ve x or conca v e. If the quadr ilateral is conca v e, then one of the p oin ts will b e inside the triangle formed by the other three. There are three p ossible cases: r is inside, p is in side, or one of a or b is in side (see Figure 2(c)-(e)). There are four triangles eac h of whic h is o ve r three of these four p oin ts. The quadrilateral is conca v e if th e su m of the areas of three of these triangles is equal to th e area of the fourth triangle. Equiv alen tly , it is conv ex if sum of areas of t wo of the triangles is equal to the sum of areas of the other tw o. Giv en a triangle with edge lengths x, y , z , usin g Heron’s formula, the area of the triangle is equal to p 2( x 2 y 2 + y 2 z 2 + z 2 x 2 ) − ( x 4 + y 4 + z 4 )) / 4. So th e area of a triangle is of the form √ A where A is a p olynomial in terms of lengths of the ed ges of the triangle. Supp ose th at the areas of the four triangles ov er these four p oint s are √ A 1 , √ A 2 , √ A 3 , and √ A 4 . W e need to c hec k if the sum of t w o is equal to the sum of th e other tw o and we would like to do this without computing the squ are ro ots of n umb ers. F or instance, sup p ose w e wan t to verify √ A 1 + √ A 2 = √ A 3 + √ A 4 . F or this to hold, we m ust hav e A 1 + A 2 + 2 √ A 1 A 2 = A 3 + A 4 + 2 √ A 3 A 4 . V erifying this is equiv alent to verifying D + √ A 1 A 2 = √ A 3 A 4 where D = 1 2 ( A 1 + A 2 − A 3 − A 4 ). T aking the sq u are of b oth sides, we n eed to hav e D 2 + A 1 A 2 + 2 D √ A 1 A 2 = A 3 A 4 , which is the same as 1 4 ( A 3 A 4 − D 2 − A 1 A 2 ) 2 = A 1 A 2 D 2 . Thus b y comparing t w o p olynomials of edge-lengths (and without computing square ro ots) we can c hec k if the quadrilateral is con v ex or conca v e. Supp ose the quadrilateral is conv ex. I f r and p are on t wo opp osite corners (see Figure 2(a)), then r and p are on d ifferen t sides. In this case | r p | + | ab | > | r a | + | bp | and | r p | + | ab | > | r b | + | ap | . If r p is one of its sides (see Figure 2 (b)), then | r p | + | ab | is not the largest of the ab ov e three pairs of sums. No w sup p ose that th e qu ad r ilateral is conca v e. The only case in whic h r and p are on tw o s id es of line ab is when one of a or b is inside the triangle obtained by the other three (see Figure 2(e)). In this case, the area of the largest triangle is the one that do es not con tain a or b . Thus, if we compute the square of the areas of the four triangle, w e can d etect this case to o. 8 r a p b r a p b r a p b r a p b r a p b (a) (b) (c) (d) (e) Figure 2: The five n on-isomorphic configur ations needed to consider for a quad r ilateral on four p oints in Lemma 10 Assume that G ′ is a UDG and h as an optim um clique partition of size α ≤ ℓ . T h e cliques fall in t w o categories: small (h aving at most 2 α − 2 p oin ts), and large (having at least 2 α − 1 p oin ts). W e fo cus only on findin g the large cliques since it is easy to guess all th e sm all cliques. Supp ose for eac h pair C i , C j ∈ OPT of large cliques, w e guess their resp ectiv e representa tiv es, c i and c j . F urther, s u pp ose that we also guess a separating lin e l ij correctly which go es through p oin ts u ij and v ij . F or a p oint p that is adjacen t to c i or c j w e w ant to efficien tly test if p is on the the same side of line l ij as c i (the p ositive side), or on c j ’s sid e (the ne gative sid e), u sing only edge-lengths. Without loss of generalit y , let b oth u ij and v ij b elong to clique C i . F or ev ery no de p differen t fr om the representat ive s: • S upp ose p is adjacent to all of c i , u ij , v ij , c j . O bserve that we also hav e the edges c i u ij and c i v ij . Giv en the edge-lengths of all the six edges among the four v ertices c i , u ij , v ij , p us in g Lemma 10 we can decide if in a r ealizat ion of these four p oin ts, th e line going through u ij , v ij separates the t wo p oin ts p and c i or not. If p and c i are on the s ame side, w e sa y p is on the p ositiv e side of l ij for C i . Else, it is on the p ositiv e side of l ij for C j . • S upp ose p is adjacen t to c i (and also to u ij and v ij ) but n ot to c j . Giv en the edge-lengths of all the six edges among the fou r vertic es c i , u ij , v ij , p us in g Lemma 10 we can d ecide if in a realizatio n of these four p oints, the line going through u ij , v ij separates the t wo p oin ts p and c i or n ot. If p and c i are on the s ame side, we say p is on the p ositiv e side of l ij for C i . Else, it is on the p ositive side of l ij for C j . F or eac h C i and all the lines l ij , consider the set of no des that are on the p ositive side of all these lines with resp ect to C i ; we place these n o des in C i . After obtaining the large and the small cliques, w e obtain s ets C 1 , . . . , C α . At the end w e c hec k if eac h C i forms a clique and if their un ion co v ers all the p oin ts. Th e num b er of guesses for repr esentati ves is n O ( α ) and the n umb er of guesses for the separator lines is n O ( α 2 ) . So there are a total of n O ( α 2 ) configurations that w e consider. Clearly , if G ′ is a UDG then some set of guesses is a correct one, allo wing us to obtain an optim um clique partition. If G ′ is not a UDG, w e may still find a clique partition of G ′ . Ho w eve r, if w e fail to obtain a clique partition in our searc h then the subgraph is a certificate that G ′ is not a UDG. 9 3 (2 + ε ) -A ppro ximation for W eigh ted Clique P artition using Ad- jacency In this section we consider a generalizatio n of the minimum clique partition on UDGs, wh ic h we call minimum weig hte d clique p artition (MW CP). Giv en a no de-we ighte d grap h G ( V , E ) with vertex w eigh t wt ( v ), the weig ht of a clique C is d efined as the w eigh t of the hea viest vertex in it. F or a clique partition C = { C 1 , C 2 , . . . , C t } , the weigh t of C is defined as sum of the wei ghts of the cliques in C , i.e. w t ( C ) = w t  S t i =1 C i  = P t i =1 wt ( C i ). The pr ob lem is, give n G in standard form, sa y , as an adjacency matrix, constru ct a clique partition C = { C 1 , C 2 , . . . , C t } while minimizing wt ( C ). The weig hted ve rsion of the pr oblem as it is defi n ed ab o ve has also b een studied in differen t con texts. See [7, 3, 8] for study of weig hted clique-partition on inte rv al graph s and circular arc graphs. Observe that MW CP distinguishes itself fr om MCP in t wo imp ortan t w a ys: (i) T he sep ar ability prop erty w hic h wa s crucially used earlier to devise a PT AS do es not hold in the wei ghte d case, and (ii) the num b er of cliques in an optimal solution for a UDG in a region of b ounded radius is not b ound ed by the d iameter of the r egion an ymore, i.e. it is easy to construct examples of wei ghte d UDGs in a b ounded region where an optimal weigh ted clique p artition cont ains an unb ounded (in terms of region diameter) num b er of cliques. In addition, examples where t w o cliques in an optimal solution are n ot separable, that is, their con ve x hulls ov erlap, is easy to construct. (S ee the examples giv en in Figure 3.) T o the b est of our kn o wledge, MW CP has n ot b een inv estigat ed b efore on UDGs. W e, ho w ev er, n ote that a simple mo d ification to the algorithm b y [18] also yields a factor-8 app ro ximation to the we ight ed case, a generaliza tion which they do not consid er. Here, a 1 b 1 p 1 p 2 α α 2 α 4 α 2 i α 2 t α α 2 α 4 α 2 i α 2 t (a) (b) Figure 3: (a) Tw o o v erlapping w eigh ted cliques, A = { a 1 , . . . , a k } and B = { b 1 , . . . , b k } are sh o wn, a i , b i are indep end en t for all i . The hea vy p olygon h as vertice s w eigh ted k wh ile the dashed ones are weigh ted 1. opt = k + 1 while an y separable partition m ust pay a cost of at least 2 k . (b) A UDG whic h is a matc hing b et wee n t w o cliques f or w hic h OPT con tains t cliques. T h e wei ght is less than 2 · α . w e giv e an algorithm which runs in time O ( n poly (1 /ε ) ) for a giv en ε > 0 and computes a (2 + ε )- appro ximation to MW CP for UDGs expressed in standard form, for example, as an adjacency matrix. Ou r algo rithm is wea kly robust in that it either pro d uces a clique partition or pr o duces a p olynomial-sized certificate pro ving that the input is n ot a UDG. When the input is a UDG, th e algorithm return s a clique p artition and it is guarant eed to b e a (2 + ε )-appro ximation; but if the input is not UDG there is n o guaran tee on the qualit y of the clique partition (if it compu tes one). Theorem 11. Given a gr ap h G expr esse d in stand ar d form, and ε > 0 , ther e is a p olynomial 10 time algorithm which e i ther c omp utes a clique p artition of G or gives a c ertific at e that G is not a UDG. If G is a UDG, the weight of the clique p artition c ompute d is a (2 + ε ) -appr oxim ation of the minimum weighte d cliqu e p artition (but ther e i s no guar ante e on the weight of the cliqu e p artition if the input gr aph is not UDG). Our algorithm will b orr o w some id eas dev elop ed in Section 2 and in [18 ]. The high leve l idea of the algorithm is as follo ws. Similar to the algorithm in Section 2, we fi rst decomp ose the graph int o b ound ed diameter regions and sho w that if we can compute a (2 + ε )-a pp r o ximate clique partition of eac h region then the u nion of these clique partitions is within (2 + ε ) fr action of opt. W e will emplo y a similar ball gro wing tec hniqu e (as in Section 2) that will giv e us b oun d ed diameter regions. W e then sh o w that w e can either compu te a clique partition or giv e a certificate that the su bgraph is n ot a UDG. If the subgraph is a UDG, then our clique partition is with in a factor (2 + ε ) of the optimal. F or the case of b ounded diameter region, although the optimum solution may hav e a large num b er of cliques, we can sho w that there is a clique partition with small num b er of cliques whose cost is within (1 + ε )-fac tor of the optimum solution. First we describ e the m ain algorithm. Then in S u bsection 3.1 we sh o w that for eac h subgraph B r ( v ) (of b oun ded diameter) there is a near optimal clique partition with ˜ O ( r 2 ) cliques. T h en in Su bsection 3.2 w e sh o w how to find such a near optimal clique partition. Let us denote the we ight of the optimum clique partition of G b y opt. As b efore, let B r ( v ) = { u : d ( u, v ) ≤ r } , called the ball of (unw eigh ted) distance r around v , b e the set of vertic es th at are at most r hop s from v in G . O ur d ecomp osition algorithm d escrib ed b elo w (see Algorithm 3) is similar to Algorithm 2 and partitions th e graph into b ounded diameter su bgraphs b elow. Th e pro cedur e CP, giv en a graph indu ced by the v ertices of B r ( v ) and a p arameter ℓ = p oly ( r ), run s in time n O ( ℓ 2 ) ) and either give s a certificate that B r ( v ) is not a UDG or computes a clique partition of B r ( v ); this clique partition is within a factor (2 + ε ) of the optim um if B r ( v ) is a UDG. W e on ly call this p ro cedure for constan t v alues of r . In the follo wing, let 0 < γ ≤ √ 9+4 ε − 3 2 b e a rational n umb er. See Algorithm 3. Algorithm 3 MinCP( G, γ ) 1: C ← ∅ ; β ← ⌈ c 0 1 γ log 1 γ ⌉ ; ℓ ← c 1 β 2 . { where c 0 is the constan t in Lemma 15, and c 1 is the constan t in inequalit y (2). } 2: while V 6 = ∅ do 3: v ← arg max u { wt ( u ) } 4: r ← 0 { Let C r ( v ) denote a factor-(2 + γ ) p artition of B r ( v ) compu ted b y calling CP } 5: w hile wt ( C r +2 ( v )) > (1 + γ ) · wt ( C r ( v )) do 6: r ← r + 1 7: if ( r > β ) or (CP( B r ( v ) , ℓ ) returns “not a UDG”) then 8: return “ G is not a UDG” and pro duce B r ( v ) as the certificate 9: C ← C ∪ C r +2 ( v ) 10: V ← V \ B r +2 ( v ) 11: return C as our clique partition Let k is the m axim um n umb er of iterations of the outer “while-lo op”. The pro of of the follo wing Lemma is ident ical to th e pro of of Lemma 6. Lemma 12. Every two vertic es v ∈ B r ∗ i ( v i ) and u ∈ B r ∗ j ( v j ) ar e non-adjac ent. 11 The follo wing lemma sho ws a lo we r-b ound for opt. Lemma 13. (2 + γ ) · opt ≥ wt k [ i =1 C r ∗ i ( v i ) ! Pr o of. Note that B r ∗ i ( v i ) is obtained by constructing a BFS tree ro oted at ve rtex v i up to some depth r ∗ i . Since the algorithm remov es a s u p er-set, B r ∗ i +2 ( v i ), wh ic h has tw o more lev els of th e BFS tree, usin g the previous lemma there is no edge b et w een an y t w o n o des v ∈ B r ∗ i ( v i ) and u ∈ B r ∗ j ( v j ) for an y pair i 6 = j . So, no single clique in an optimum solution can con tain v ertices from distinct B r ∗ i ( v i ) and B r ∗ j ( v j ). Consider the subset of cliques in an optimal clique partition of G that inte rsect B r ∗ i ( v i ) and call this subset O PT i . The argum en t ab o ve shows that OPT i is d isjoin t from OPT j . Also, eac h OPT i con tains all the vertices in B r ∗ i ( v i ). S ince C r ∗ i ( v i ) is a factor-(2 + γ ) approximati on for B r ∗ i ( v i ), (2 + γ ) · wt ( OPT i ) ≥ w t  C r ∗ i ( v i )  . The lemma immediately follo ws by observin g that OPT i and OPT j are disjoint . W e can relate the cost of our clique partition to opt as follo ws. Lemma 14. If wt  C r ∗ i +2 ( v i )  ≤ (1 + γ ) wt  C r ∗ i ( v i )  , then wt k [ i =1 C r ∗ i +2 ( v i ) ! ≤ (2 + ε ) opt. Pr o of. wt k [ i =1 C r ∗ i +2 ( v i ) ! = k X i =1 wt  C r ∗ i +2 ( v i )  ≤ (1 + γ ) · k X i =1 wt  C r ∗ i ( v i )  ≤ (2 + γ )(1 + γ ) · opt , where the last inequalit y u ses Lemma 13. Next, w e s ho w that the inner “while-lo op” terminates in ˜ O ( 1 γ ), th at is eac h r ∗ i is b ounded b y ˜ O ( 1 γ ). This is similar to the p ro of of Lemma 9. Since th e while lo op terminates, r ∗ i exists and b y defi nition of r ∗ i , it must b e the case that for all smaller v alues of r < r ∗ i , wt ( C r ( v i )) > (1 + γ ) · wt ( C r − 2 ( v i )). Because the diameter of B r ( v i ) is O ( r ), if B r ( v ) is a UDG, there is a realizatio n of it in wh ic h all the p oin ts fit in to a r × r grid. Also, since v i is a h eavie st vertex in the (residual) graph, there is a clique partition whose w eigh t is at most α · wt ( v i ) · r 2 . Therefore, wt ( C r ( v i )) < α · wt ( v i ) · r 2 , for some constan t α . So: α · wt ( v i ) · r 2 > wt ( C r ( v i )) > (1+ γ ) · wt ( C r − 2 ( v i )) > . . . > (1+ γ ) r 2 · wt ( C 0 ( v i )) = wt ( v i ) ·  p 1 + γ  r , whic h implies α · r 2 >  √ 1 + γ  r , for the case that r is eve n. If r is o dd w e obtain α · r 2 >  √ 1 + γ  r − 1 . Thus, the follo wing lemma easily follo ws: Lemma 15. Ther e is a c onstant c 0 > 0 such that for e ach i : r ∗ i ≤ c 0 /γ · log 1 /γ . In S ubsection 3.2, we sho w the algorithm CP that given B r ( v ) and an upp er b ound ℓ on | C r ( v ) | , either computes a clique partition (w h ic h is within a factor 2 + γ of opt if B r ( v ) is a UDG) or detects that the graph is not UDG; the algorithm r uns in time n O ( ℓ 2 ) . By the ab ov e arguments, if B r ( v ) is a unit disk graph then there is a constant c 1 > 0 suc h that: | C r ( v ) | = O ( r ∗ i 2 ) ≤ c 1 · c 2 0 γ 2 log 2 1 γ (2) 12 W e can set ℓ = ⌈ c 1 c 2 0 γ 2 log 2 1 γ ⌉ for an y inv o cation of OPT-CP as an upp er b ou n d, where c 1 is the constan t in O ( r ∗ i 2 ). So, the run ning time of the algorithm is n ˜ O (1 /ε 4 ) . 3.1 Existence of a Small Clique Partition of B r ( v ) havin g Near-optimal W eight Unlik e the unw eigh ted case, an optimal weig hted clique partition in a small r egion ma y con tain a large num b er of cliques. Y et, there exists a partition whose weigh t is within a factor (1 + γ 2 ) of the minim um wei ght whic h con tains few cliques (where by “few” we mean ℓ as in Algorithm 3). The existence of a ligh t and small partition allo ws u s to en umerate them in the same manner in the algorithm of subs ection 2.3, yielding a (2 + γ )-appro ximation for the problem instance in a ball of small radius. In the follo wing, let r ∈ ˜ O ( 1 γ ); we fo cu s on the subprob lem that lies in s ome B r ( v ). Recall that an y ball of radius r can b e partitioned in to O ( r 2 ) cliques (Lemma 3). W e b egin w ith a simple lemma which states that for any clique partition C , if the set of v ertices can b e b e co v ered b y another clique partition C ′ con taining x cliques then the sum of the weigh ts of the x cliques in C ′ is not significan tly more than the w eigh t of the hea viest clique in C . Lemma 16. F or any c ol le ction of disjoint cliques C = { C 1 , C 2 , . . . , C t } having weights such that wt ( C 1 ) ≥ wt ( C 2 ) ≥ . . . ≥ w t ( C t ) supp ose the vertic es of C c an b e p artitione d into x cliques C ′ = { C ′ 1 , C ′ 2 , . . . , C ′ x } . Then wt ( C ′ ) = wt ( S x l =1 C ′ l ) = P x l =1 wt ( C ′ l ) ≤ x · wt ( C 1 ) Pr o of. Without loss of generalit y , let wt ( C ′ 1 ) ≥ wt ( C ′ 2 ) ≥ . . . ≥ wt ( C ′ x ). Since C ′ partitions v ertices in C , wt ( C ′ 1 ) = wt ( C 1 ). Since |C ′ | = x , w t ( C ′ ) = P x l =1 wt ( C ′ l ) ≤ x · wt ( C ′ 1 ) = x · wt ( C 1 ). In an optimal partition of a ball of r adius r , the sum of the w eigh ts of the lighter cliques is not significan tly more th an its w eigh t. Lemma 17. L et C = { C 1 , C 2 , . . . , C t } b e an optimal cliqu e p artition and let wt ( C 1 ) ≥ wt ( C 2 ) ≥ . . . ≥ wt ( C t ) . Supp ose ther e is another clique p artition C ′ = { C ′ 1 , . . . , C ′ x } of the vertic es of C . Then, for every 1 ≤ i < t : ( x − 1) · wt ( C i ) ≥ P t l = i +1 wt ( C l ) . Pr o of. By wa y of cont radiction, su pp ose there exists an ind ex 1 ≤ j < t such th at ( x − 1) · wt ( C j ) < P t l = j +1 wt ( C l ). Because S t l =1 C l can b e co v ered b y C ′ , so can S t l = j C l . Let the 2 ≤ x ′ ≤ x b e the sm allest index su c h that C ′ j = { C ′ 1 , C ′ 2 , . . . , C ′ x ′ } co ve rs S t l = j C l . On the other hand, ( x ′ − 1) · wt ( C j ) ≤ ( x − 1) · wt ( C j ) < P t l = j +1 wt ( C l ), whic h imp lies opt = t X l =1 wt ( C l ) > j X l =1 wt ( C l ) + ( x ′ − 1) · wt ( C j ) = j − 1 X l =1 wt ( C l ) + x ′ wt ( C j ) . (3) By Lemma 16, wt  C ′ j  ≤ x ′ · wt ( C j ). Th is, combined with in equalit y (3) implies opt > P j − 1 l =1 wt ( C l )+ P x ′ l =1 wt ( C ′ l ). T herefore the cliques in C ′′ = { C 1 , C 2 , . . . , C j − 1 , C ′ 1 , C ′ 2 , . . . , C ′ x ′ } co ve r all the no des of cliques in C and has cost smaller than op t. If a vertex b elongs to t wo or more cliques in C ′′ w e remo v e it from all but one of them to obtain a clique p artition with cost no more than cost of C ′′ whic h is s m aller than opt. This completes the pro of. 13 W e now are ready to pr o v e the main result of this section which states that for any optimal w eigh ted clique p artition of a ball of radius r , there exists another clique partition whose weigh t is arb itrarily close to th e wei ght of the optimal partition, but has O ( r 2 ) cliques in it. Sin ce the radius of the ball within w h ic h the subp roblem lies is small, r ∈ ˜ O ( 1 γ ), this means that if we we re to enumerate all the clique p artitions of the subp roblem up to O ( r 2 ), we will see one wh ose w eigh t is arbitrarily close to th e w eight of an optimal clique. Cho osing a ligh test one from amongst all suc h cliques guarante es that w e will c ho ose a one whose w eigh t is arbitrarily close to th e optimal w eigh t. Lemma 18. L et γ > 0 and r ∈ ˜ O (1 /γ ) b e two c onsta nts. L et C = { C 1 , C 2 , . . . , C t } b e an optimal weighte d clique p artition of B r ( v ) and let C ′ = { C ′ 1 , . . . , C ′ x } b e another clique p ar tition of vertic es of C with x ∈ O ( r 2 ) . L et wt ( C 1 ) ≥ wt ( C 2 ) ≥ . . . ≥ wt ( C t ) . Then, ther e i s a p artition of ve rtic es of C into at most j + x cliqu es for some c onstant j = j ( γ ) , with c ost at most (1 + γ 2 ) opt. Pr o of. Without loss of generalit y , we assume that b oth x and t are at least tw o (as if B r ( v ) is a clique w e are done). Consider an arb itrary v alue of j ≤ t . Since S t l =1 C l can b e co vered by x cliques in C ′ , there is an in dex x ′ (2 ≤ x ′ ≤ x ) such that S t l = j C l can b e co v ered by C ′ j = { C ′ 1 , C ′ 2 , . . . , C ′ x ′ } . By applying Lemma 17 rep eatedly: opt ≥ j X l =1 wt ( C l ) ≥ 1 x ′ − 1 j X l =2 wt ( C l ) ! + j X l =2 wt ( C l ) ≥ . . . ≥  x ′ x ′ − 1  j − 1 · wt ( C j ) ⇒ opt ( x ′ − 1) j − 1 x ′ j − 2 ≥ x ′ · wt ( C j ) (4) Using inequalit y (4): opt + opt · ( x ′ − 1) j − 1 x ′ j − 2 ≥ j − 1 X l =1 wt ( C l ) + x ′ · wt ( C j ) ≥ j − 1 X l =1 wt ( C l ) + x ′ X l =1 wt  C ′ l  , (5) where the second inequalit y follo ws by applying Lemma 16. Let C ′′ = { C 1 , . . . , C j − 1 , C ′ 1 , . . . , C ′ x ′ } . Th us , the cliques in C ′′ co v er all th e vertic es of C and has total cost at most  1 + ( x ′ − 1) j − 1 x ′ j − 2  opt by inequalit y (5). If a v ertex b elongs to t w o or m ore cliques in C ′′ w e remov e it from all bu t one of them arbitrarily to obtain a clique partition of size j − 1 + x ′ and whose total cost is upp er b oun ded by  1 + ( x ′ − 1) j − 1 x ′ j − 2  opt. Note th at, ( x ′ − 1) j − 1 x ′ j − 2 = ( x ′ − 1)  x ′ − 1 x ′  j − 2 and 0 < x ′ − 1 x ′ < 1 (b ecause x ′ ≥ 2). Since r ∈ ˜ O ( 1 γ ) and x ′ ≤ x ∈ O ( r 2 ), for an appropr iate choic e of j = j ( γ ), ( x ′ − 1)  x ′ − 1 x ′  j − 2 < γ / 2. Th us we obtain a clique partition with j + x − 1 cliques and cost at most (1 + γ / 2) · opt . Th is pro ve s the lemma. 3.2 (2 + γ ) -Approxim ation for MWCP in B r ( v ) Finally , we s h o w h ow to compute a (2 + γ )-appro ximate MWCP of the graph B r ( v ) f or any given γ . F or an edge ordering L = ( e 1 , e 2 , . . . , e m ) of a graph G w ith m edges, let G L [ i ] denote the edge indu ced sub graph with edge-set { e i , e i +1 , . . . , e m } . F or eac h e i , let N L [ i ] denote the common neigh b orh o o d of the end -p oints of e i in G L [ i ]. An edge orderin g L = ( e 1 , e 2 , . . . , e m ) is a CNEEO if 14 for ev ery e i in L , N L ( i ) induces a co-bipartite graph in G . It is kno wn [20] that every UDG graph admits a co-bipartite edge elimination ordering (CNEEO). In the follo wing, let G v denote B r ( v ). W e state a lemma of [18]. Lemma 19. [18] L et C b e a clique in G v , and let L b e a CNEEO of G v . Then, ther e is an i , 1 ≤ i ≤ m , such that N L [ i ] c onta ins C . Assume that G v can b e partitioned in to α ≤ ℓ = ˜ O (1 /γ 2 ) cliques, O = { O 1 , O 2 , . . . , O α } , such that wt ( O ) ≤ (1 + γ 2 ) · wt ( OPT v ), wh ere OPT v is an optimal weigh ted clique partition of G v . Note that by Lemma 18 this is tru e for subgraph B r ( v ). Su pp ose that we are given the upp er b oun d ℓ ; w e will try all p ossib le v alues of α . Without loss of generalit y , let w t ( O 1 ) ≥ wt ( O 2 ) ≥ . . . ≥ w t ( O α ). Observe that, w ithout loss of generalit y , we can assume O i is a maximal clique in S α j = i O j . The implication of the ab ov e lemma is that ev en though w e d o not kno w O 1 , h en ce we do n ot know O , we do know that for every CNEEO L of G v , there is an e i suc h that N L [ i ] can b e partitioned in to at most tw o cliques that fully cov er O 1 . Since O 1 is a h ea viest clique, the t w o cliques that co v er the s u bgraph N L [ i ] pay a cost of at most 2 · wt ( O 1 ). T his suggests an algorithm that guesses an edge sequence ( f 1 , f 2 , . . . , f α ) of G v . T hen, th e algorithm computes L , a CNEEO of G v . T he algorithm’s fi rst guess is “goo d ” if f 1 is an edge in O 1 that o ccurs fir st in L . Su pp ose that this is the case and sup p ose that f 1 has r ank i in L . Th en, O 1 is con tained in N L [ i ], and we co v er N L [ i ] with at most tw o cliques. Call these C ′ 1 and C ′′ 1 and wt ( C ′ 1 ) + wt ( C ′′ 1 ) ≤ 2 · wt ( O 1 ). So, when we remo v e N L [ i ] from G v , we get a UDG whic h can b e partitioned in to at most α − 1 cliques, namely , O ′ = { O 2 , . . . , O α } . W e then again construct a CNEEO, L ′ , of G ′ v = G v \ N L [ i ]. Just like b efore, our guess f 2 is “go o d ” if f 2 is an edge in O 2 and o ccur s firs t in L ′ . Let i ′ b e th e rank of f 2 in L ′ , we see that N L ′ [ i ′ ] fully con tains O 2 , and we again co v er it w ith at most 2 cliques. Next, delete N L ′ [ i ′ ] from G ′ v to get a graph whic h can b e partitioned into α − 2 cliques, and so on. See Algorithm 4 for details. Algorithm 4 CP( G v , ℓ ) 1: C ← V ; min ← wt ( C ); 2: for all α ≤ ℓ do 3: for all α -edge sequence ( f 1 , f 2 , . . . , f α ) of G v do 4: G 0 ← G v 5: for j = 1 to α do 6: Compute a CNEEO L of G j − 1 7: i ← rank of f j in L 8: P artition N L [ i ] in to tw o cliques C ′ j and C ′′ j 9: G j ← G j − 1 \ N L [ i ] 10: if G α = ∅ and wt  S α j =1 { C ′ j , C ′′ j }  < min then 11: C ← S α j =1 { C ′ j , C ′′ j } ; min ← wt ( C ); 12: return C Note that while Lemm a 19 allo ws us to co v er any clique with at most 2 cliques, it do es not find the clique. I n the algorithm, note that if at an y p oin t, the algorithm is unable to construct a CNEEO, we can declare that the grap h G v is not a UDG. Also, if for all in vocations of the algorithm by an external algorithm that guesses the v alue of opt we are un ab le to find a clique partition, then again we can d eclare that G v is not a UDG. 15 4 O (lo g ∗ n ) -round Distributed PT AS for UDGs with Edge-Lengths In this s ection, we giv e details of a d istributed PT AS for MCP wh ic h run s in O ( log ∗ n ε O (1) ) round s of distributed computation und er th e LO C AL mo del of computation [16]. Th e mo del of computation that w e emplo y assu mes a synchr onous system wh ere c ommunic ation b et w een neighb orin g n o des tak es place in sync hr onous roun ds u s ing messages of unbou n ded size [16]. So, in a single round of comm un ication, any no de acquires the subgraph (in f ormation p ertaining to the set of no des, edges, the states of lo cal v ariables, etc.) within its immediate neigh b orh o od . So, after k r ounds of comm unication, any no de acquires complete kno wledge ab out its k -neigh b orhoo d. Observe that in Algorithm Min C P2, the radius r of any b all B v ( r ) is b ounded ab o v e b y ˜ O (1 /ε ), while the cen ter, v , is an arb itrary vertex. Since the radius of any ball is “small”, the maxim um n umb er of rounds of distributed computation th at the sequential algorithm n eeds b efore terminating the “while-lo op” is also “small”. Therefore, for any p air of balls B u ( r i ) an d B v ( r j ), suc h that d ( u, v ) ∈ ω (1 /ε ) , one should b e able to r un p art of the sequen tial algorithm in parallel, as they surely are indep enden t of eac h other. W e b orr o w some ideas from [10] an d find r egions that are far apart su c h that w e can ru n the sequen tial algorithm in those regions in parallel. See Algorithm 5 for details. Algorithm 5 Distr-MCP-UDG( G , ε ) 1: β ← ⌈ c 0 1 ε log 1 ε ⌉ ; ℓ ← c 1 β 2 ; all v ertices are unmarke d. { c 0 is the constan t in Lemma 9 and c 1 is the constan t inequalit y (1). } 2: Construct a maximal subset, V c ⊂ V , suc h that for any pair u, v ∈ V c , d ( u, v ) > β . Construct a graph G c = ( V c , E c ), where E c = {{ u, v } : u, v ∈ V c , d G ( u, v ) ≤ 4 β } . W e call V c , the set of le ad ers . 3: Prop er color G c using ∆( G c ) + 1 colors, where ∆( G c ) is the maximum degree of G c . 4: Ev ery v ∈ V \ V c , “assigns” itself to a nearest leader u ∈ V c , with ties broken arbitrarily , and colors itself the same color as the leader. 5: for i = 1 to ∆( G c ) + 1 do 6: F or eac h leader j with color i let G j i b e the s ubgraph induced by the vertice s assigned to leader j . 7: for all G j i in pa rallel do 8: Consider a fixed ordering on th e unmarke d vertic es of G j i ; 9: Run th e sequential b all gro wing algorithm on th e next (in th is ord ering) u n mark ed v ertex v ∈ G j i , we compute B r ( v ); Note that B r ( v ) might con tain v ertices of differen t colors (fr om outside G j i ). 10: Compute (u sing the sequen tial algorithm) the optimal clique-partition of B r ( v ) and “mark” all those vertic es It should b e p oin ted out that adapting the algo rithm of [10] for maximum i ndep endent se t and minimum dominating set to our s etting is not trivial. Th e r eason is th at MCP is a p artition of the en tire v ertex set and p artitioning jus t a subset w ell enough will not d o. Sp ecifically [10] c ho oses a subset of vertice s up on wh ic h th eir ball-gro wing algorithm is r un; it suffices for their purp oses to disp en s e with the remaining sub set of v ertices that w ere not pic ke d by their b all-gro wing algorithm. If w e had follo wed a similar scheme th en w e wo uld surely get a go o d clique partition on a s u bset of v ertices; ho we ver, it is unclear as to h o w to obtain a go o d partition of the remaining sub set in 16 terms of the optimal s ize for the original pr oblem ins tance ov er th e entire ve rtex set. As a means to circum ve nt this issue, w e fi rst construct a “cru d e” partition of the v ertex set, in s tead of just a subset of v ertices as done in [10]. 4.1 Analysis W e no w s ho w that th e algorithm constru cts a (1 + ε )-appro ximation to MCP on UDGs giv en on ly rational edge-lengths in O (log ∗ n ) round s of distribu ted computation under the LO C A L mo del; we first s ho w correctness of the algorithm, follo w ed b y b ounding th e num b er of comm unication roun d s. Correctness: W e pro ve that our algorithm is correct by showing that an y execution of Distr- MCP-UDG can b e tu r ned into a sequential execution of MinCP2. As stated, every v ertex has the same color as its leader; let a leader v ertex b e its o wn leader. W e fir st sho w that the distance of ev ery verte x to its leader is small. Lemma 20. F or any vertex v / ∈ V c , ther e is a vertex u ∈ V c such that d G ( u, v ) ≤ β . Pr o of. Sup p ose n ot. So there is a v whose distance to ev ery u ∈ V c is more than β . But then V ′ c = V c ∪ { v } has the prop ert y that for all x, y ∈ V ′ c , d G ( x, y ) > β , con tradicting the maximalit y of V c . Next we sh o w that for an y pair of v ertices u, v of the same color but with d ifferen t leaders, the minim um distance b etw een them is large enough so th at a p air of balls of radius at m ost β o ve r them will b e disj oint, where β is defined in MinCP2 and Distr-MCP-UDG. Lemma 21. Co nsider two le aders x, y of the same c olo r, say i , and any two vertic es u ∈ G x i and v ∈ G y i (note that we might have u = x or v = y ). Then for al l values of r c onsider e d in the b al l gr ow ing algorithm, B r ( u ) and B r ( v ) ar e disjoint. Pr o of. Since x and y ha v e the same color d ( x, y ) > 4 β . By Lemma 20, any vertex in either of G x i or G y i is at a distance of at most β from the resp ectiv e leader; so, d G ( u, v ) > 2 β . The lemma follo ws easily by noting the fact that r ≤ β in the ball gro wing algorithm. W e are no w ready to pro ve the correctness of Distr-MCP-UDG by sh o wing an equiv alence b et wee n an y execution of it to some execution of MinCP2. Lemma 22. Any exe cution of Di str-M CP-UD G fr om “Step 5” to “Step 10” c a n b e c onverte d to a valid e xe cution of MinCP 2. Pr o of. Consider an arb itrary execution of Distr-MCP-UDG. Supp ose that V 1 , V 2 , V 3 , . . . is a s e- quence of disj oin t sets of the v ertices of V such that w e r un the ball gro wing algorithm in p arallel (during Distr-MCP-UDG) on ve rtices of V 1 (and th us we compu te an optimal clique partition on eac h v ertex of V 1 in parallel) then we do this for ve rtices in V 2 , and so on. Note that the v ertices in V i all h a v e the same color and eac h has a different leader. Cons ider an arbitrary ordering π i of the vertice s in eac h V i and supp ose that w e run MinCP2 algorithm on v ertices of V 1 based on ordering π 1 , then on v ertices of V 2 based on ordering π 2 , and so on. S ince the v ertices in eac h V i ha v e distinct leaders, by Lemma 21, the balls gro wn aroun d them are disjoint. It should b e easy to see that the balls grown by algorithm Min C P2 is exactly the same as the on es computed by Distr-MCP-UDG. 17 The follo wing r esult follo ws immed iately as a corollary to Lemma 22. Corollary 23. Given an ε > 0 , Distr-M CP-UDG c onstructs a clique p ar tition of the input gr aph G with asso ciate d e dge-lengths, or pr o duc es a c ertific ate that G is not a UD G. If G is a U D G then the size of the p artitio n is within (1 + ε ) of the optimum cliqu e p artition. Running Time: W e no w sh o w that the algorithm runs in O ( log ∗ n ε O (1) ) distribu ted r ounds u n der the LO C AL mo del of computation. Lemma 24. “Step 2” r e quir es O ( β · log ∗ n ) r ounds of c omm unic ation. Pr o of. Obser ve that the result of “Step 2” is identica l to constructing a maximal indep endent set (MIS) in G β . Note th at G β is also a UDG where the new unit is β . As a result, G β is a su b class of gr owth-b ounde d gr ap hs [9] where all the distances are scaled by β ; computation of MIS on G β tak es O ( β · log ∗ n ) r ounds [21] while the construction of G β tak es β roun ds. Hence, th e num b er of rounds needed by “Step 2” can b e b ounded by O ( β · log ∗ n ). It is easy to see that constru cting G c requires at most 4 β comm unication r ounds. Ne xt, w e sho w that the m axim um degree of G c , ∆( G c ) is b oun ded by a constan t. Lemma 25. ∆( G c ) ∈ O (1) Pr o of. Let v b e a v ertex of G c ha ving maximum degree. Note that all its n eigh b orin g vertice s in G c lie in a d isk of radius at most 4 β . Also note th at du e to “Step 2” the min im um distance b et w een an y pair of ve rtices in G c is more than β . As a result, any d isk of d iameter β con tains at most 1 v ertex of G c . Using standard pac king arguments of th e under lyin g sp ace, a crude up p er b ound on the num b er of v ertices of G c in a disk of radius at most 4 β is 256 v ertices; this also upp er b ounds the degree of v . Next, we b oun d the n umber of roun ds needed for “Step 3” Lemma 26. “Step 3” r e quir es O ( β · log ∗ n ) r ounds of c omm unic ation. Pr o of. F or graph s whose maxim um d egree is ∆, a ∆ + 1 prop er coloring r equires O (∆ + log ∗ n ) rounds [11 ]. Since ∆( G c ) ∈ O (1) (Lemma 25), and the fact that distances in G c are scaled by a factor of 4 β as compared to the distances in G , a ∆( G c ) + 1 prop er coloring of G c can b e obtained in O ( β · log ∗ n ) rounds. “Step 4” requir es at most β rounds of comm unication; according to Lemma 20, for ev ery v / ∈ V c , there is some u ∈ V c that is at a d istance at most β from it. Th e identit y and color of such a v ertex can b e obtained in β round s. W e can n o w b oun d the num b er of roun ds that Distr-MCP-UDG requires. First, note that for any iteration, i , of “Step 7”, only knowle dge of a subgraph u p to radiu s β is required, and an y n o de can obtain kn o wledge of the subgraph u p to radius β from it in β round s of communicatio n. So, for an y ve rtex in G j i obtains kno wledge ab out the “mark ed/unmarked” status of all the v ertices in G j i in β round s of comm unication. Sin ce the diameter of eac h G j i is at most 2 β , the num b er of balls to grow in “Step 9.” is at most O ( β 2 ). Therefore: Theorem 27. Distr-MCP - UDG r e quir es O ( β · log ∗ n ) r ounds of c ommunic ation under the LO C A L mo del of c omputation. 18 5 Concluding Remarks Recall that the weak est assump tion that we n eeded to obtain a PT AS for un weig hted clique partition problem was that all the edge lengths are giv en. This information wa s crucially used in obtaining a r obust PT AS. In the case of w eigh ted clique partition, w e ga ve a (2 + ε )-appro ximation algorithm without the u se of edge-lengths (using only the adjacency information). It will b e int eresting to see if a PT AS exists for the unw eigh ted case bu t with r eliance only on adjacency . It is also unclear if a PT AS is p ossible eve n with the use of geometry in the weigh ted case. Recall that the PT AS giv en in Sections 2 crucially us es the idea of separabilit y of an optimal clique partition. Ho we ver, in the wei ghte d case, even though a near optimal clique partition in a sm all region has few cliques, th er e are examples where any separable partition p a ys a cost at least factor-2 to th at of a near optimal partition. W e giv e an example in Figure 3(a). In the example sho wn in Figure 3(a) t w o cliques of optimal w eigh t are sh o wn: one of th em, A , whose v ertices are the v ertices of the k -go n sho wn in dashed-hea vy lines, and the other, B , wh ose vertice s are the v ertices of the k -gon shown in solid-hea vy lines. The example is that for k = 7. The vertice s of A are lab eled a 1 , a 2 , . . . , a k in a counte r-clo c kwise fashion. The v ertices of B are lab eled such th at b i is diametrically opp osite to a i . T he distance b et w een a i and b i is more than 1 while the distance b et wee n a i and b j , i 6 = j is at most 1. So, there is an edge b et w een a i to ev ery a l and to eve ry b j , j 6 = i . This is also the case for b i . In the figure, th e edges inciden t to a 1 are sho wn b y solid-ligh t lines. Also, the dashed arc sho ws part of th e unit disk b oun dary that is cen tered at a 1 – note that it do es n ot include b 1 . Let th e w eigh ts of ve rtices in A b e k and the w eigh ts of v ertices in B b e 1. Clearly , opt ≤ k + 1. Ho w eve r, an y sep arable clique partition pays a cost of at least 2 k : if vertices in A m ust all b elong to a common clique, then every vertex in B m ust b elong to a distin ct clique in a separable clique partition. Also, note that as-p er separabilit y , a line going through { p 1 , p 2 } separates t w o cliques ha ving weigh t 2 k also. Note that our results only apply in the Euclidean plane; they do not generalize. In particular, Cap o yleas et al. [4] giv e an “un separable” ins tance in R 3 . Our result in the w eigh ted case also is restricted to the plane; the concept of c o-bip artite neighb orh o o d e dge elimination or dering (CNEEO) do es not generalize to R 3 . Ac kno wledgmen ts W e thank Sriram P emmara ju, Lorna Stew art, and Zoy a Svitkina for helpful d iscussions. Our thanks to an anon ymous s ource for p oint ing out the result of Cap oyl eas et al. [4]. References [1] J. Aspnes, D. K. Golden b erg, and Y. R. Y ang. On the compu tational complexit y of sensor net w ork lo calizati on. In ALGOSENSORS ’04: First International Workshop on Algorithmic Asp e cts of Wir eless Se nsor Networks , pages 32–44 , T urku , Finland , 2004. Spr inger-V erlag . [2] B. Balasundaram an d S. Butenko. Optimization prob lems in unit-disk graph s . I n Christo dou- los A. Floudas and P anos M. Pa rd alos, editors, E ncyclop e dia of Optimization , pages 2832– 2844. Springer, 2009. 19 [3] L. Becc hetti, P . Kortewe g, A. Marc hetti-Spaccamela , M. Skutella, L. Stougie, and A. Vitaletti. Latency constrained aggrega tion in sensor net works. In Y ossi Azar and Thomas Erlebac h, editors, ESA , volume 4168 of L e ctur e Notes in Computer Scienc e , pages 88–99. Sp r inger, 2006. [4] V. Cap oyle as, G. Rote, and G. J. W o eginger. Geometric clus terings. J. Algor ithms , 12(2):341 – 356, 1991. [5] M. R. Cerioli, L. F aria, T. O. F erreira, C. A. J. Martinhon, F. Protti, and B. Reed. P artition in to cliques for cu bic graphs: Planar case, complexity and appro ximation. Discr ete Applie d Mathematics , 156(12) :2270–2278, 2008. [6] M.R. Cerioli, L. F aria, T.O. F erreira, and F. Protti. On min imum clique p artition and maxi- m um ind ep endent set on unit disk graphs and p enn y graphs: complexit y and appro ximation. Ele ctr onic Notes in Discr ete M athematics , 18:7 3–79, 2004. [7] G. Finke, V. Jost, M. Queyranne, and A. Seb¨ o. Batc h pro cessing with in terv al graph compat- ibilities b et w een tasks. Discr ete Applie d Mathematics , 156(5):5 56 – 568, 2008 . [8] D. Gijswijt, V. Jost, an d M. Queyran n e. Clique p artitioning of interv al graph s with sub mo du lar costs on the cliques. Op er ations R ese ar ch , 41(3):275 –287, jul 2007. [9] F. Kuh n, T. Moscibro da, T. Nieb erg, and R. W attenhofer. F ast deterministic d istributed maximal indep enden t set computationon gro wth-b oun ded graphs. In Pierre F raigniaud, editor, DISC , v olume 3724 of L e ctur e Notes in Computer Scienc e , pages 273–28 7. Sp ringer, 2005 . [10] F. Ku hn, T. Nieb erg, T. Moscibro da, and R. W att enhofer. Lo cal approxima tion s c hemes for ad ho c and sensor net works. In DIALM-P OMC , p ages 97–103, 2005. [11] F. Kuhn and R. W attenhofer. On the complexit y of d istributed graph coloring. In Eric Rupp ert and Dahlia Malkhi, editors, P O DC , pages 7–15 . A CM, 2006. [12] K. M. Lillis, S. V. Pemmara ju , and I. A. Pirwani. T op olo gy control and geographic routing in realistic wireless net w orks. In Ev angelos Kr an akis and J arosla v Opatrn y , ed itors, ADHOC- NOW , v olume 4686 of L e ctur e N otes in Computer Scienc e , p ages 15–31. S pringer, 2007. [13] T. Nieb erg, J. Hurink, and W. Kern. Appro ximation sc hemes for wireless net wo rks. ACM T r ans actions on Algo rithms , 4(4):1–17, 2008. [14] S. P andit and S. Pe mmara ju. Fin d ing facilities fast. In ICDCN , 2009. [15] S. P andit, S. Pemmara ju, and K. V aradara jan. App ro ximation algorithms for domatic parti- tion. In RANDO M -APP ROX to app e ar , 2009 . [16] D. Pele g. Distribute d c omputing: a lo c ality-sensitive appr o ach . So ciet y for Industr ial and Applied Mathematics, Philadelphia, P A, USA, 2000. [17] S. V. P emmara ju and I. A. Pirwa ni. Energy conserv at ion via domatic p artitions. In Sergio P alazzo, Marco Cont i, and Ragh upathy Siv akumar, editors, MobiHo c , p ages 143–1 54. A CM, 2006. 20 [18] S. V. P emmara ju and I. A. Pirwani. Go o d qu alit y virtual r ealizati on of unit b all graphs. In Lars Arge, Mic hael Hoffmann, and Emo W elzl, editors, E SA , vo lume 4698 of L e ctur e Notes in Computer Scienc e , p ages 311– 322. Spr inger, 2007. [19] I. A. Pirwani and M. R. Sala v atipour . A p tas for minimum clique p artition in u nit disk graphs. CoRR , abs/0904.22 03, 2009. [20] V. Ragha v an and J. Spinrad. Robus t alg orithms for restricted d omains. J. Algorithm s , 48(1): 160–172, 2003. [21] J. Sc hneider and R. W attenhofer. A log-star distrib uted maximal indep end en t set algorithm for gro wth-b ounded graphs. In Rida A. Bazzi and Boaz Pat t-Shamir, editors, PODC , p ages 35–44 . A CM, 2008 . [22] D. Zuck erman. Linear degree extractors and the inap p ro ximabilit y of max clique and c hromatic n umb er. The ory of Computing , 3(1):103 –128, 2007. 21