Improved approximation for 3-dimensional matching via bounded pathwidth local search
aa r X i v : . [ c s . D S ] A ug Improved approximation for 3-dimensional matchingvia bounded pathwidth local search ∗ Marek Cygan † August 21, 2013
Abstract
One of the most natural optimization problems is the k -Set Packing problem, where givena family of sets of size at most k one should select a maximum size subfamily of pairwisedisjoint sets. A special case of 3- Set Packing is the well known 3 -Dimensional Matching problem, which is a maximum hypermatching problem in 3-uniform tripartite hypergraphs. Bothproblems belong to the Karp’s list of 21 NP-complete problems. The best known polynomialtime approximation ratio for k -Set Packing is ( k + ǫ ) / . ǫ )-approximation for 3 -Dimensional Matching .Those results are obtained by a simple local search algorithm, that uses constant size swaps.The main result of this paper is a new approach to local search for k -Set Packing whereonly a special type of swaps is considered, which we call swaps of bounded pathwidth. We showthat for a fixed value of k one can search the space of r -size swaps of constant pathwidth in c r poly( |F| ) time. Moreover we present an analysis proving that a local search maximum withrespect to O (log |F| )-size swaps of constant pathwidth yields a polynomial time ( k + 1 + ǫ ) / k -Set Packing .In particular we improve the approximation ratio for 3 -Dimensional Matching from 3 / ǫ to 4 / ǫ . In the
Set Packing problem, also known as
Hypergraph Matching , we are given a family
F ⊆ U of subsets of U , and the goal is to find a maximum size subfamily of F of pairwisedisjoint sets. Set Packing is a fundamental problem in combinatorial optimization with variousapplications. A simple reduction from
Independent Set (where |F | = | V | ) combined with thehardness result of H˚astad [16] makes the Set Packing problem hard to approximate. When eachset of
Set Packing is of size at most k the problem is denoted as k -Set Packing . k -Set Packing Input:
A family
F ⊆ U of sets of size at most k . Goal:
Find a maximum size subfamily of F of pairwise disjoint sets. k -Set Packing is a generalization of Independent Set in bounded degree graphs, as wellas k - Dimensional Matching and is related to plethora of other problems (see [7] for a list ∗ The preliminary version of this paper was presented at the 54th Annual IEEE Symposium on Foundations ofComputer Science (FOCS’13). The author is partially supported by Foundation for Polish Science grant HOMINGPLUS/2012-6/2. † Institute of Informatics, University of Warsaw, Poland, [email protected] .
1f connections between k -Set Packing and other combinatorial optimization problems). In 3 -Dimensional Matching the universe U is partitioned into U = X ⊎ Y ⊎ Z and F is a subset of X × Y × Z .Both 3 -Dimensional Matching and Set Packing are well studied problems, belonging toKarp’s list of 21 NP-hard problems [21]. A simple greedy algorithm returning any inclusionwisemaximal subfamily of disjoint subsets of F gives a k -approximation for k -Set Packing . One canconsider a local search routine, where as long as it is possible we remove one set from our currentfeasible solution and add two new sets. We say that such an algorithm uses size 2 swaps, as twonew sets are involved. It is known that a local search maximum with respect to size 2 swaps is a( k + 1) / k -Set Packing . If, instead of using swaps of size 2 we use swaps ofsize r for bigger values of r , then the approximation ratio approaches k/
2, and that is exactly the( k/ ǫ )-approximation algorithm by Hurkens and Schrijver [19].Despite significant interest (see Section 1.2) for over 20 years no improved polynomial time ap-proximation algorithm was obtained for k -Set Packing , even for the special case of 3 -DimensionalMatching . Meanwhile Halld´orsson [15] has shown that a local search maximum with respect to O (log |F | ) size swaps gives a ( k + 2) / k + 1 + ǫ ) / O (log |F | ) size swaps takes quasipolynomial time. Based on the work of Halld´orsson [15] a natural path to transforming a quasipolynomial timeapproximation into a polynomial time approximation would be by designing a c r poly( |F | ) timealgorithm, where c is a constant. This is exactly the framework of parameterized complexity , wherethe swap size is a natural parameter. Unfortunately, we show that this is most likely impossible,i.e. there is no such algorithm with f ( r )poly( |F | ) running time, unless W[1]=FPT, where f is somecomputable function, even for k = 3. We would like to note that W[1] =FPT is a widely believedassumption, in particular if W[1]=FPT, then the Exponential Time Hypothesis of [20] fails. Theorem 1.1.
Unless
F P T = W [1] , there is no f ( r )poly( |F | ) time algorithm, that given a family F ⊆ U of sets of size and its disjoint subfamily F ⊆ F either finds a bigger disjoint family F ⊆ F or verifies that there is no disjoint family F ⊆ F such that |F \ F | + |F \ F | ≤ r , Therefore trying to find a c r poly( |F | ) time algorithm which searches the whole r -size swapsspace is not the proper path. For this reason we introduce a notion of swaps (also called improvingsets) of bounded pathwidth (see Section 3.1). Intuitively a size r swap is of bounded pathwidth, ifthe bipartite graph where vertices represent sets that are added and removed, and edges correspondto non-empty intersections, is of constant pathwidth. Using the color-coding technique of Alon etal. [1] we show that one can search the space of swaps of size at most r of bounded pathwidth in c r poly( |F | ) time, for a constant c . As the currently best known analysis of local search maximumwith respect to logarithmic size swaps of [9] relies on swaps of unbounded pathwidth, we need todevelop a different proof strategy, and the core part of it is contained in Lemma 3.8. The algorithmand its analysis complete the main result of this paper, that is a polynomial time ( k + 1 + ǫ ) / k and ǫ . Theorem 1.2.
For any ǫ > and any integer k ≥ there is a polynomial time ( k + 1 + ǫ ) / -approximation algorithm for k -Set Packing . Similar arguments were also used by Berman and F¨urer [5] for the independent set problem in bounded degreegraphs. For further information about parameterized complexity we defer the reader to monographs [10, 13, 26].
2e believe that the usage of parameterized tools such as color-coding, pathwidth and W[1]-hardness in the setting of this work is interesting on its own, as to the best of our knowledge suchtools have not been previously used in local search based approximation algorithms.
Even though there was no improvement in terms of polynomial time approximation of k -SetPacking (and 3 -Dimensional Matching ) since the work of Hurkens and Schrijver [19], bothproblems are well studied.One can also consider weighted variant of k -Set Packing , where we want to select a max-imum weight disjoint subfamily of F . Arkin and Hassin [2] gave a ( k − ǫ )-approximationalgorithm, Chandra and Halld´orsson [8] improved it to (2 k + 2 + ǫ ) / k + 1 + ǫ ) / k -Set Packing problem Chan and Lau [7] presented astrengthened LP relaxation, which has integrality gap ( k + 1) / k -Set Packing is hard to approximatewithin a factor of O ( k/ log k ). Concerning small values of k , Berman and Karpinski [6] obtaineda 98 / − ǫ hardness for 3 -Dimensional Matching , while Hazan et al. [18] obtained 54 / − ǫ ,30 / − ǫ , and 23 / − ǫ hardness for 4, 5 and 6- Dimensional Matching respectively (note thata hardness result for k -Dimensional Matching directly gives a hardness for k -Set Packing ).Recently Sviridenko and Ward [28] have independently obtained a ( k + 2) / k -Set Packing . They observed that the analysis of Halld´orsson [15] can be com-bined with a clever application of the color coding technique. However to the best of our under-standing it is not possible to obtain ( k + 1 + ǫ ) / k -Set Packing using thetools of [28], and in particular Sviridenko and Ward do not improve the approximation ratio for3 -Dimensional Matching . The main difference between this article and [28] is in handling setsof the optimum solution, that intersect exactly one set in a local maximum. We start with preliminaries in Section 2, where we recall standard graph notation together withthe definition of pathwidth and path decompositions.Section 3 contains the main result of this paper, that is the ( k +1+ ǫ ) / k -SetPacking . First, we introduce the notion of improving set of bounded pathwidth in Section 3.1. InSection 3.2 we apply the color coding technique to obtain a polynomial time algorithm searchingan improving set of logarithmic size of bounded pathwidth. In Section 3.3 we analyse a local searchmaximum with respect to bounded pathwidth improving sets of logarithmic size. The heart of ouranalysis is contained in an abstract combinatorial Lemma 3.8 which is later applied in the proof ofLemma 3.11.The proof of Theorem 1.1 is given in Section 4. Finally, in Section 5 we conclude with potentialfuture research directions. We use standard graph notation. For an undirected graph G by V ( G ) and E ( G ) we denote the set ofits vertices and edges respectively. By N G ( v ) = { u : uv ∈ E ( G ) } we denote the open neighborhood3f a vertex v , while the closed neighborhood is defined as N G [ v ] = N G ( v ) ∪ { v } . Similarly, for asubset of vertices X we have N G [ X ] = S v ∈ X N G [ v ] and N G ( X ) = N G [ X ] \ X .By a disjoint family of sets we denote a family, where each pair of sets is pairwise disjoint. Fora positive integer r by [ r ] we denote the set { , . . . , r } . Pathwidth and path decompositions A path decomposition of a graph G = ( V, E ) is asequence P = ( B i ) qi =1 , where each set B i is a subset of vertices B i ⊆ V (called a bag ) suchthat S ≤ i ≤ q B i = V and the following properties hold:(i) For each edge uv ∈ E ( G ) there is a bag B i in P such that u, v ∈ B i .(ii) If v ∈ B i ∩ B j then v ∈ B ℓ for each min( i, j ) ≤ ℓ ≤ max( i, j ).The width of P is the size of the largest bag minus one, and the pathwidth of a graph G isthe minimum width over all possible path decompositions of G . Since our focus here is on pathdecompositions we only mention in passing that the related notion of treewidth can be definedsimilarly, except for letting the bags of the decomposition form a tree instead of a path.In order to make the description easier to follow, it is common to use path decompositions thatadhere to some simplifying properties. The most commonly used notion is that of a nice pathdecompositions, introduced by Kloks [22]; the main idea is that adjacent nodes can be assumed tohave bags differing by at most one vertex. Definition 2.1 ( nice path decomposition). A nice path decomposition is a path decomposition P = ( B i ) qi =1 , where each bag is of one of the following types: • First (leftmost) bag : the bag B is empty, B = ∅ . • Introduce bag : an internal bag B i of P with predecessor B i − such that B i = B i − ∪ { v } for some v / ∈ B i − . This bag is said to introduce v . • Forget bag : an internal bag B i of P with predecessor B i − for which B i = B i − \ { v } forsome v ∈ B i − . This bag is said to forget v . • Last (rightmost) bag : the bag associated with the largest index, i.e. q , is empty, B q = ∅ .It is easy to verify that any given path decomposition can be transformed in polynomial timeinto a nice path decomposition without increasing its width. In this section we present the main result of the paper, i.e. the ( k + 1 + ǫ ) / k -Set Packing , proving Theorem 1.2. We start with introducing the notion ofimproving set of bounded pathwidth in Section 3.1. Next, in Section 3.2 we apply the color codingtechnique to obtain a polynomial time algorithm searching an improving set of logarithmic size ofbounded pathwidth. In Section 3.3 we analyse a local search maximum with respect to boundedpathwidth improving sets of logarithmic size. The heart of our analysis is contained in an abstractcombinatorial Lemma 3.8 which is later applied in the proof of Lemma 3.11.4 .1 Bounded pathwidth improving set Let us assume that an instance
F ⊆ U of k -Set Packing is given. Moreover by F ⊆ F wedenote some disjoint subfamily of F , which we can think of as a current feasible solution of alocal search algorithm. In what follows we define a conflict graph , which is a bipartite undirectedgraph with two independent sets of vertices being F and F \ F , where an edge reflects non-emptyintersection. Definition 3.1 ( conflict graph). For a disjoint family F ⊆ F by a conflict graph G F wedenote an undirected bipartite graph with vertex set F and edge set { S S : S ∈ F , S ∈ ( F \ F ) , S ∩ S = ∅} .Next, we define an improving set X ⊆ F \ F , which can be used to increase the cardinality of F , and then we introduce a notion of an improving set of bounded pathwidth , which will be crucialin both the algorithm and the analysis of its approximation ratio. Definition 3.2 ( improving set). For a disjoint family F ⊆ F a set X ⊆ F \ F is called an improving set , if the following conditions hold: • all sets of X are pairwise disjoint, • | N G F ( X ) | < | X | , i.e. the number of sets of F having a common element with at least oneset of X is strictly smaller than | X | .Observe, that if we have an improving set X , then ( F \ N G F ( X )) ∪ X is a disjoint subfamilyof F of size greater than |F | , hence the name improving set. Definition 3.3 ( improving set of bounded pathwidth). An improving set X with respect to F ⊆ F has pathwidth at most pw , if the subgraph of the conflict graph G F induced by N G F [ X ]has pathwidth at most pw . To find an improving set of bounded pathwidth we use the color coding technique of Alon et al. [1],which is by now a well-established tool in parameterized complexity used for finding a set consistingof disjoint objects. We use two random colorings c F : F → [ r − c U : U → [ rk ], where c U ensuresthat the sets of X are disjoint, while c F is used not to consider the same set of F twice. Lemma 3.4.
There is an algorithm, that given a disjoint family F ⊆ F , and two coloring functions c F : F → [ r − , c U : U → [ rk ] in O ( rk ) |F | O ( pw ) time determines, whether there exists animproving set X ⊆ F \ F of size at most r of pathwidth at most pw , such that c F is injective on N G F ( X ) and c U is injective on S S ∈ X S .Proof. For the sake of notation by adding dummy distinct elements we ensure that each set of F has size exactly k . Define an auxiliary directed graph D = ( V D , A forget ∪ A introduce ), where eachvertex is characterized by a subset of set colors [ r − rk ], and a subsetof F of size at most pw + 1, i.e. V H = { v ( C F , C U , B ) : C F ⊆ [ r − , C U ⊆ [ rk ] ,B ⊆ F , | B | ≤ pw + 1 } . Note that this graph has O (2 r ( k +1) |F | pw +1 ) vertices.5ince there will be no possibility of confusion, to make the proof easier to follow by N [ X ] for X ⊆ F we denote N G F [ X ], i.e. we omit the subscript G F . The idea behind the construction isthat each vertex of V H describes a potential prefix of a sequence of bags in a path decompositionof N [ X ] for some X ⊆ F \ F . The set B encodes the set of vertices of N [ X ] in the current bagand ensures the bounded pathwidth property. Instead of storing all the sets of X that have alreadyappeared in the sequence of bags, we store only the colors of the elements of S S ∈ X S (encodedby C U ), as it is enough to maintain the disjointness of sets of X and keep track of the cardinalityof X - due to the assumption that each set of is size exactly k . Similarly instead of storing all thesets of N [ X ] that were already introduced, we only store their colors (encoded by C F ).To the set A introduce we add the following arcs. For s = v ( C F , C U , B ) ∈ V D , S ∈ F such that | B | ≤ pw : • if S ∈ F \ F , c U ( S ) ∩ C U = ∅ , c F is injective on N ( S ) and c F ( N ( S ) \ B ) ∩ C F = ∅ , thenadd to A introduce an arc ( s, v ( C F , C U ∪ c U ( S ) , B ∪ { S } )) • if S ∈ F , c F ( S ) C F and for each S ′ ∈ B \ F either S ∈ N ( S ′ ), or c F ( S ) c F ( N ( S ′ )),then add to A introduce an arc ( s, v ( C F ∪ { c F ( S ) } , C U , B ∪ { S } ))To the set A forget we add the following arcs. For s = v ( C F , C U , B ) ∈ V D , S ∈ B add to A forget an arc ( s, v ( C F , C U , B \ { S } )) if one of the following conditions holds: • S ∈ F , • S
6∈ F and c F ( N ( S )) ⊆ C F . Claim 3.5.
There exists a path in the graph D from the vertex v ( ∅ , ∅ , ∅ ) to a vertex v ( C F , C U , ∅ ) ∈ V D for | C F | < | C U | /k iff there exists an improving set X of size at most r of pathwidth at most pw , such that c F is injective on N ( X ) and c U is injective on S S ∈ X S .Proof. Assume that there is a path s , . . . , s q in H , where s i = ( C i F , C iU , B i ), s = ( ∅ , ∅ , ∅ ) , | C q F | < | C qU | /k and B q = ∅ . Let X = S ≤ i ≤ q B i \ F . By construction of D , we have | X | = | C qU | /k ≤ r .By the definition of A introduce and A forget since B q = ∅ , at the time a vertex v ∈ X appears forthe first time in some B i we ensure that all its neighbors in G F are either in B i or are coloredby c F with colors not yet in C i F . Moreover at the time v ∈ X is forgotten, i.e. removed fromsome B i , we ensure that all of its neighbors in G F have been already added to bags. Therefore N [ X ] ⊆ S ≤ i ≤ q B i and for each edge e of G [ N [ X ]] the endpoints of e appear in some bag B i .Since no set of F is added twice, due to the coloring c F , no set of F \ F is added twice, dueto the coloring c U , ( B i ∩ N [ X ]) qi =1 is a path decomposition of N [ X ] of width at most pw . Finally | N ( X ) | ≤ | C q F | < | C qU | /k = | X | . Hence X is an improving set of size at most r and of pathwidthat most pw .In the other direction, let X be an improving set of size at most r such that c F is injective on N ( X ), c U is injective on S S ∈ X S , and let P = ( B i ) qi =1 be a nice path decomposition of N [ X ] ofwidth at most pw . For 1 ≤ i ≤ q define s i ∈ V D as s i = v ( c F ( B ′ i ∩ F ) , c U ( S S ∈ B ′ i \F S ) , B i ), where B ′ i = S ≤ j ≤ i B i . Observe that s = ( ∅ , ∅ , ∅ ), s q = ( C F , C U , ∅ ) for | C F | = | N ( X ) | < | X | = | C U | /k and moreover if B i +1 is an introduce bag, then ( s i , s i +1 ) ∈ A introduce while if B i +1 is a forget bag,then ( s i , s i +1 ) ∈ A forget . Consequently there is a path from s to s q in the graph D .By the above claim it is enough to run a standard graph search algorithm, to check whetherthere exists a path from the vertex v ( ∅ , ∅ , ∅ ) to v ( C F , C U , ∅ ) where | C F | < | C U | /k , which finishesthe proof of Lemma 3.4. 6 heorem 3.6. There is an algorithm, that given a disjoint family F ⊆ F , in O ( rk ) |F | O ( pw ) timedetermines, whether there exists an improving set X ⊆ F \ F of size at most r of pathwidth atmost pw .Proof. Observe, that if we take c F : F → [ r −
1] where the color of each set is chosen uniformlyand independently at random, then for an improving set X of size at most r the function c F isinjective on N G F ( X ) with probability at least( r − / ( r − r − ≥ (( r − /e ) r − / ( r − r − = e − ( r − . Similarly, if we assign a color of [ rk ] to each element of U , then with probability at least e − rk the function c U : U → [ rk ] is injective on S S ∈ X S . Therefore invoking Lemma 3.4 with randomcolorings c F , c U at least e r − rk times would yield a constant error probability.To obtain a deterministic algorithm we can use the, by now standard, technique of splitters.An ( n, a, b )-splitter is a family H of functions [ n ] → [ b ], such that for any W ⊆ [ n ] of size at most a there exists f ∈ H that is injective on W . What we need is a small family of ( n, a, a )-splitters. Theorem 3.7 ([25]) . There exists an ( n, a, a ) -splitter of size e a a O (log a ) log n that can be constructedin O ( e a a O (log a ) n log n ) time. Therefore instead of using random colorings c F , c U we can use Theorem 3.7 to construct( |F | , r − , r −
1) and ( | U | , rk, rk ) splitters, leading to a deterministic algorithm, which finishes theproof of Theorem 3.6. In this subsection we analyze a local search maximum, with respect to logarithmic size improvingsets of constant pathwidth. It is well known that an undirected graph of average degree at least 2+ ǫ contains a cycle of length at most c ǫ log n , where the constant c ǫ depends on ǫ . This observationwas the base for the quasipolynomial time algorithms of [9,15]. Here, however we need to generalizethis result extensively, as the analysis of [9] relies on improving sets of unbounded pathwidth.Throughout this subsection we operate on multigraphs, as there might be several parallel edgesin a graph, however there will be no self-loops. Lemma 3.8.
Let H = ( V, E ) be an n -vertex undirected multigraph of minimum degree at least .Assume that each edge e ∈ E is associated with a subset of an alphabet w e ⊆ Σ of size at most γ ,where γ ≥ . If each element c ∈ Σ appears in at most γ sets w e , i.e. ∀ c ∈ Σ |{ e : e ∈ E, c ∈ w e }| ≤ γ ,then there exists a tree T = ( V , E ) , which is a subgraph of H , and a vertex r ∈ V , such that: • | V | ≤ / n + 2) ; • there exist two edges e , e ∈ E \ E , e = e which have both endpoints in V ; • T is a tree with at most leaves; • for each pair of edges e , e ∈ E such that w e ∩ w e = ∅ we have | dist T ( r , e ) − dist T ( r , e ) | ≤ β , where β = ⌈ log / (12 γ ) ⌉ , and dist T ( r , uv ) = min(dist T ( r , u ) , dist T ( r , v )) .Proof. First we deal with some corner cases.(i) If in H there are three parallel edges e a , e b , e c between vertices u and v , then as T we take( { u, v } , { e a } ) and set e = e b , e = e c . 7ii) If in H there are three vertices u, v, w , two parallel edges e a , e b between u and v as well as twoparallel edges e c , e d between v and w , than as T we take ( { u, v, w } , { e a , e c } ) and set e = e b , e = e d .(iii) In the last corner case let us assume that for each vertex v of H there are some two paralleledges e a , e b ∈ E ( H ) incident to v . Let uv ∈ E ( H ) be any edge of H for which there is noparallel edge in H - such an edge exists, as otherwise ( i ) or ( ii ) would hold. Let u ′ be avertex such that in H there are two parallel edges e a , e b between u and u ′ , similarly let v ′ bea vertex such that in H there are two parallel edges e c , e d between v and v ′ . Observe that u ′ = v ′ as otherwise case (ii) would hold. In that case T = ( { u, u ′ , v, v ′ } , { e a , uv, e c } ), e = e b and e = e d .Assuming that none of ( i ), ( ii ), ( iii ) holds, there is a vertex r in H , which is adjacent to at leastthree distinct vertices v , v , v .We are going to construct a sequence of logarithmic number of trees T , T , . . . rooted at r ,which are subgraphs of H satisfying two invariants: • (exponential growth) for any 1 ≤ j ≤ i the number of vertices in T i at distance exactly j from r is exactly ⌊ / j ⌋ , and there are no vertices at distance more than i , • ( Σ -nearness) for any two edges e , e of T i if w e ∩ w e = ∅ , then | dist T i ( r, e ) − dist T i ( r, e ) | ≤ β .We will show, that having constructed a tree T i for some i ≥ T i +1 satisfying the two invariants, or find a tree T with edges e , e required by the claim of the lemma.Let T = ( { r, v , v , v } , { rv , rv , rv } ) and note that it satisfies the two invariants. Assume,that T i (for some i ≥
1) was the most recently constructed tree, and we want to construct T i +1 . Let V ′ be the vertices of T i at distance exactly i from the root r . By the exponential growth invariantwe have | V ′ | = ⌊ / i ⌋ . Let E ′ ⊆ E be the set of edges of H incident to V ′ , but not contained in E ( T i ). As each vertex in H is of degree at least three, we have | E ′ | ≥ | V ′ | ≥ ⌊ / j ⌋ . (1)Let E banned = { e ∈ E ′ : ∃ e ′ ∈ E ( T i − β ) w e ∩ w e ′ = ∅} , i.e. the set of edges having a non-empty intersection with w e ′ , where e ′ is not contained in the last β levels of T i . Observe that for i ≤ β the set E banned is empty. When extending the tree T i tomaintain the Σ-nearness invariant, we use only edges of E ′ \ E banned .Let V ′′ = S uv ∈ E ′ \ E banned { u, v } \ V ( T i ). We consider two cases: either | V ′′ | ≥ ⌊ / i +1 ⌋ ornot. In the former case we will show that one can construct a tree T i +1 satisfying both exponentialgrowth and Σ-nearness invariants. In the latter case we will show that the required tree T andedges e , e exist.If | V ′′ | ≥ ⌊ / i +1 ⌋ , then we select exactly ⌊ / i +1 ⌋ vertices out of V ′′ and extend the tree T i to T i +1 by adding one more layer of vertices (at distance i + 1 from r ), connected to vertices of V ′ by edges of E ′ \ E banned . Clearly the exponential growth invariant is satisfied for T i +1 . Furthermore,since T i satisfied the Σ-nearness invariant and by the definition of E banned the tree T i +1 also satisfiesthe Σ-nearness invariant.In the remaining part of the proof we assume | V ′′ | < ⌊ / i +1 ⌋ (2)8igure 1: Edges of the tree T are gray, while edges e and e are dashed.Figure 2: Creating the tree T assuming | E ′′′ | ≤ | E ′′ | −
2. Notation as in Figure 1.and show the required tree T with edges e , e . If at least two edges of E ′ have both endpoints in V ( T i ), denote those edges uv, u ′ v ′ ∈ E ′ , then as T we take the subtree of T i induced by verticeson the paths between { u, v, u ′ , v ′ } and their least common ancestor r and set e = uv , e = u ′ v ′ (see Figure 1). Therefore let E ′′ ⊆ E ′ be the subset of edges having exactly one endpoint in V ( T i )(that is in V ′ ). By (1) we infer that | E ′′ | ≥ | E ′ | − ≥ | V ′ | − . (3)Let E ′′′ be a maximum size subset of E ′′ , such that no two edges of E ′′′ have a common endpointin V \ V ( T i ). Observe that if | E ′′′ | ≤ | E ′′ | −
2, then either: • there exists three edges e a , e b , e c ∈ E ′′ having a common endpoint in V \ V ( T i ), or • there exist four edges e a , e b , e c , e d ∈ E ′′ , such that e a , e b have a common endpoint in V \ V ( T i )and e c , e d have a common endpoint in V \ V ( T i ).In both cases we can extend the tree T i by one or two edges to construct T and set e = e b , e = e c (see Figure 2).Consequently we have | E ′′′ | ≥ | E ′′ | −
1, which together with (3) gives | E ′′′ | ≥ | V ′ | − . (4)In the last part of the proof we use the following claim. Claim 3.9. | E ′′′ \ E banned | ≥ ⌊ / i +1 ⌋ Proof.
Recall that if i ≤ β , the set E banned is empty. Hence by inequality (4) in that case | E ′′′ \ E banned | = | E ′′′ | ≥ ⌊ / i ⌋−
2. A direct check shows that for each 1 ≤ i ≤ ⌊ / i ⌋− ≥ ⌊ / i +1 ⌋ , which proves the claim in the case i ≤ < i ≤ β we have | E ′′′ \ E banned | ≥ ⌊ / i ⌋ − ≥ / i − − ≥ / i +1 . Finally for i > β we upper bound the size of E banned | E banned | ≤ i − β X j =1 γ / j ≤ γ i − β − X j =0 (3 / j ≤ γ ((3 / i − β − ≤ (3 / i − . The first inequality follows from the assumption, that each set w e is of size at most γ and eachelement of Σ is contained in at most γ sets w e , hence each of T i contributes at most γ edges to E banned . The last inequality follows from the choice of β and the assumption γ ≥
1. Therefore | E ′′′ \ E banned | ≥ | E ′′′ | − | E banned |≥ ⌊ / i ⌋ − − ( (3 / i − ≥ / i +1 . Observe that by the definition of E ′′′ we have | V ′′ | ≥ | E ′′′ \ E banned | , but then Claim 3.9contradicts inequality (2). Corollary 3.10.
Let H = ( V, E ) be an undirected multigraph with n vertices and of minimumdegree at least . Assume that each edge e ∈ V is associated with a subset of an alphabet w e ⊆ Σ ofsize at most γ , for some γ ≥ , such that each element of Σ appears in at most γ sets w e . Thereexists a subgraph H = ( V , E ) of H , and a path decomposition ( B i ) qi =1 of H of width at most β + 3) , where β = ⌈ log / (12 γ ) ⌉ and(a) | E | = | V | + 1 ,(b) | V | ≤ / n + 2) ,(c) for each pair of edges e , e ∈ E , such that w e ∩ w e = ∅ there exists a bag B i for some ≤ i ≤ q , such that all of the endpoints of both e and e are contained in B i ,(d) for each edge uv ∈ E the set of indices { i : u, v ∈ B i } is an interval.Proof. First, we use Lemma 3.8 to obtain T = ( V , E ), r ∈ V , such that | V | ≤ / n +2), where for each pair of edges e , e ∈ E such that w e ∩ w e = ∅ we have | dist T ( r , e ) − dist T ( r , e ) | ≤ β . Let e , e ∈ E \ E be two edges with both endpoints in V . Define H =( V , E ∪ { e , e } ), clearly H is a subgraph of H and the number of edges is the number of verticesplus one. Therefore properties ( a ) and ( b ) are satisfied and it remains to show that there exists apath decomposition of H of width at most 4( β + 3), satisfying ( c ) and ( d ).Let D i be the set of vertices of V at distance exactly i from r in T . Consider a sequence( B i ) qi =0 , where q = 4(log / n + 2), and B i = S max(0 ,i − β − ≤ j ≤ i D i ∪ e ∪ e . It is straightforwardto check that this is in fact a path decomposition of H , and since T has at most 4 leaves, this10mplies that the size of each D i is upper bounded by 4, and hence the path decomposition is ofwidth at most 4( β + 3).Observe that property ( c ) required by the corollary follows from the last property of Lemma 3.8,because all of the endpoints of edges e , e ∈ E , such that w e ∩ w e = ∅ , are contained in B max(dist T ( r ,e )+1 , dist T ( r ,e )+1) . To prove property ( d ) let e = uv be an arbitrary edge of E anddefine I u = { i : u ∈ B i } and I v = { i : v ∈ B i } . As we already know that ( B i ) qi =0 is a pathdecomposition it follows that both sets I u , I v form an interval, hence I u ∩ I v is also an interval,which proves ( d ). Lemma 3.11.
Fix an arbitrary ǫ > . There are constants c , c (depending on k and ǫ ), suchthat for any disjoint family F ⊆ F , for which there is no improving set of size at most c log n ofpathwidth at most c we have | OP T | ≤ (( k + 1) / ǫ ) |F | , where OP T ⊆ F is a maximum sizedisjoint subfamily of F .Proof. Let C = F ∩ OP T and denote A = F \ C , B = OP T \ C . Let G be the subgraph of G F induced by A ∪ B . We are going to construct a sequence of at most 1 /ǫ subgraphs of G ,namely G i = G [ A i ∪ B i ] for i ≥
1, where A i ⊆ A , B i ⊆ B , satisfying two invariants:(a) in G i there is no subset X ⊆ B i of size at most 2( k + 1) /ǫ − i , such that | N G i ( X ) | < | X | ,(b) | A \ A i | = | B \ B i | .Observe G trivially satisfies ( b ) and in order to make G satisfy ( a ) it is enough to set c and c so that c ≥ k + 1) /ǫ ,c ≥ k + 1) /ǫ , as there is no improving set of size at most 2( k + 1) /ǫ and pathwidth of an improving set of size x is at most 2 x . Consider subsequent values of i starting from 0. Split the vertices of B i into groups B i , B i , B i , consisting of vertices of B i of degree exactly one, exactly two and at least three in G i ,respectively. Observe that because of ( a ) there is no isolated vertex of B i in G i and moreover notwo vertices of B i have a common neighbour in G i . Consider the following two cases: • | B i | ≥ ǫ | OP T | : in this case we construct a graph G i +1 = G [ A i +1 ∪ B i +1 ], where A i +1 = A i \ N G i ( B i ) and B i +1 = B i ∪ B i = B i \ B i . The invariant ( a ) is satisfied, as any set X ⊆ B i +1 of size at most 2( k + 1) /ǫ − i − such that | N G i +1 ( X ) | < | X | would imply existenceof a set X ′ = X ∪ ( N G i ( N G i ( X )) ∩ B i ) of size at most ( k + 1) · | X | ≤ k + 1) /ǫ − i , such that | N G i ( X ′ ) | < | X ′ | (see Figure 3). • | B i | < ǫ | OP T | : We are going to use the following claim, which we prove later. Claim 3.12. | B i | ≤ (1 + ǫ ) | A i | As each vertex of A i is of degree at most k in G i , the number of edges of G i is at most k | A i | .At the same time the number of edges of G i is at least | B i | + 2 | B i | + 3 | B i | , therefore | B i | + 2 | B i | + 3 | B i | ≤ k | A i | . X ′ \ X N G i +1 ( X )Figure 3: Lifting an improving set X in G i +1 to an improving set X ′ in G i . Gray vertices belongto G i but not to G i +1 .Note that summing the inequalities: | B i | ≤ ǫ | A i || B i | ≤ ǫ | A i || B i | ≤ (1 + ǫ ) | A i || B i | + 2 | B i | + 3 | B i | ≤ k | A i | we obtain | B i | ≤ (( k + 1) / ǫ ) | A i | . However | OP T \ B i | = | C | + | B \ B i | = | C | + | A \ A i | = |F \ A i | , where the second equalityfollows from invariant ( b ), hence | OP T | ≤ (( k + 1) / ǫ ) |F | .In the second case we have proved the thesis, while the first case can appear only 1 /ǫ number oftimes, as in each step we remove at least ǫ | OP T | vertices from B i . Therefore to finish the proof ofLemma 3.11 it suffices to prove Claim 3.12. Proof of Claim 3.12.
Assume the contrary. Construct a multigraph H = ( A i , E H ), where E H = { e x = uv : x ∈ B i , N G i ( x ) = { u, v }} . Set Σ = F and for each edge e x = uv ∈ E H , set as w e x the set of all vertices of G at distance at most 2 /ǫ from x . Observe that since G is of maximumdegree at most k , we have | w e x | ≤ k /ǫ . For the same reason each vertex of G appears in at most2 k /ǫ sets w e x .In order to use Corollary 3.10 we need to reduce the graph H , in a way ensuring all its verticesare of degree at least 3. However we know, that the graph H is of average degree at least 2 + 2 ǫ ,since | E H | / | A i | = | B i | / | A i | ≥ ǫ . Let H ′ = H . As long as there exist an isolated vertex, or avertex of degree one in H ′ remove it. Note that such a reduction rule does not decrease the averagedegree of H ′ . Similarly if H ′ contains a path v , v , . . . , v ℓ , v ℓ +1 , where all vertices v j for 1 ≤ j ≤ ℓ are of degree exactly 2 and ℓ ≥ /ǫ , then remove all the vertices v j for 1 ≤ j ≤ ℓ from H ′ . Asthis operation removes ℓ vertices, but only ℓ + 1 edges, and ℓ ≥ /ǫ , the average degree does notdecrease. Finally, we construct H ′′ from H ′ by simultaneously considering all the maximal paths v , v , . . . , v ℓ , v ℓ +1 , with all internal vertices of degree two, and contracting each of such paths to asingle edge e ′ = v v ℓ +1 and setting w e ′ = S ≤ j ≤ ℓ w v j v j +1 . Observe that for each edge e of H ′′ thesize of w e is upper bounded by 2 k /ǫ (1 /ǫ + 1), as a contracted path consist of at most ⌊ /ǫ + 1 ⌋ edges. 12 i = XY i − \ Y i N G i ( Y i ) a b c a b cH Figure 4: The right graph is H = ( V , E ) provided by Corollary 3.10. The left side depicts theset X corresponding to E , as well as lifting the set Y i = X to Y i − . Gray vertices belong to G i − but not to G i . The dashed path on the left between a and b in H ′ is contracted into an edge of H ′′ on the right.As H ′′ is of minimum degree at least 3, we apply Corollary 3.10 to it, where γ = 2 k /ǫ (1 /ǫ + 1).Let H = ( V , E ) and P = ( B i ) qi =1 be as defined in Corollary 3.10. Let X ⊆ B i be the set of allthe vertices of B i corresponding to the edges of E , including the vertices of B i that correspondto edges of H ′ that were contracted into some edge of E (see Figure 4). As | E | > | V | we have | N G i ( X ) | < | X | . Clearly X is of size at most | E | (1 /ǫ + 1) ≤ (4(log / |F | + 2) + 1)(1 /ǫ + 1), thatis logarithmic in |F | , as ǫ is a constant. It remains to show that we can lift X to an improving setof bounded pathwidth, while increasing its size only by a constant factor.Let Y i = X . For j = i − , . . . , Y j = Y j +1 ∪ ( N G j ( N G j ( Y j )) ∩ B j ) (see Figure 4). Observethat at each step the size of Y j increases by a factor of at most k + 1, hence | Y | ≤ | Y i | ( k + 1) i andmoreover Y is an improving set w.r.t. F . Since Y is of size logarithmic in |F | it remains to showthat N G F [ Y ] is of constant pathwidth.Create a sequence of subsets P ′ = ( B ′ i ) qi =1 , by taking as B ′ i the set ( S e = uv ∈ E ,u,v ∈ B i w e ∩ N G F [ Y ]). The size of each B ′ i is at most ( w + 1) γ , where w is the width of P , hence it re-mains to show that P ′ is indeed a path decomposition. Each vertex of N G F [ Y ] is within distanceat most 2 /ǫ from some vertex of X , hence each vertex of N G F [ Y ] is contained in some set w e for e ∈ E . Similarly each edge of G F [ N G F [ Y ]] is within distance at most 2 /ǫ from some vertex of X ,so it has both its endpoints in some set w e for e ∈ E . Since P is a path decomposition each edge e ∈ E has both its endpoints in some bag B i , therefore S ≤ i ≤ q B ′ i = N G F [ Y ] and each edge of N G F [ Y ] has both its endpoints in some bag B ′ i . Property (d) of Corollary 3.10 implies that each w e contributes to B ′ i for values of i forming an interval I e . Moreover if for two edges e , e ∈ E the intersection w e ∩ w e is non-empty, then by property (c) of Corollary 3.10 we know that theintervals I e and I e have non-empty intersection. This ensures that each vertex v of N G F [ Y ]appears in a set of bags B ′ i forming an interval in the sequence P ′ , as each pair of intervals in { I e : v ∈ w e } has non-empty intersection.Therefore Y is an improving set of logarithmic size and of constant pathwidth, which is acontradiction. Consequently | B i | ≤ (1 + ǫ ) | A i | , which finishes the proof of Claim 3.12.Lemma 3.11 together with the algorithm for searching improving sets of bounded pathwidthfrom Theorem 3.6 gives a polynomial time ( k +1+ ǫ ) / k -Set Packing for any constant k , proving Theorem 1.2. In particular there is a (4 / ǫ )-approximation for the3 -Dimensional Matching problem. 13 Local search hardness
In this section we are going to show, that there is no algorithm verifying for a given F ⊆ F ,whether there exists an improving set (see Definition 3.2) of size at most r in f ( r )poly( |F | ) time,even when k = 3. In fact we show a stronger hardness result, ruling out existence of an algorithm,that either finds a bigger disjoint family F (without any restriction on its distance from F ), orverifies that there is no improving set of size at most r . That is exactly the notion of permissive parameterized local search introduced by Marx and Schlotter in [24] (for more information aboutparameterized local search see [11, 14, 23]).In our reduction, we use a standard W[1]-hard problem [12], namely Multicolored Clique parameterized by the clique size.
Multicolored Clique
Input:
An undirected graph G = ( V, E ), a positive integer k , and a color function c : V →{ , . . . , k − } . Question:
Does the graph G contain a clique of size k , where each vertex is of different color? Theorem 4.1.
There is a constant α > , such that given an instance I = ( G, k, c ) of Multi-colored Clique one can in polynomial time construct an instance
F ⊆ U of -Set Packing ,together with a disjoint subfamily F ⊆ F of size | U | / − , such that: • If I is a YES-instance, then there exists a family F ⊆ F of disjoint | U | / sets, such that |F \ F | + |F \ F | ≤ αk , • if there exists a disjoint subfamily F ⊆ F of size | U | / , then I is a YES-instance.Proof. We start with a definition of a simple gadget, that will be used a couple of times in theconstruction.
Definition 4.2.
For a positive integer h ≥ x an ( x, h )-amplifier is a family F x ⊆ U x of sets of size 3, where U x = { x , . . . , x · h − } , and F x = {{ x i , x i , x i +1 } : 1 ≤ i < h } Let I = ( G = ( V, E ) , k, c ) be an instance of Multicolored Clique . W.l.o.g. we may assumethat k = 4 h , where h is a positive integer, since otherwise we may add universal vertices (adjacentto all other vertices). We start with constructing an ( x, h )-amplifier, which will be called the topamplifier , and ( v, h )-amplifier for each v ∈ V , called vertex amplifiers . As the universe U we take U = U x ∪ ( [ v ∈ V U v ) ∪ { v ′ , v ′′ : v ∈ V } ∪ { s ( i,j ) : 0 ≤ i < j < k } ∪ { ℓ i : 1 ≤ i ≤ k } . To the family F we add all the sets of F x and F v for v ∈ V , as well as:(i) sets { v , v ′ , v ′′ } for v ∈ V ,(ii) sets { x k + i , v ′ , v ′′ } for 0 ≤ i < k for v ∈ c − ( i ),(iii) sets { u k + c ( v ) , v k + c ( u ) , s ( c ( u ) ,c ( v )) } for uv ∈ E , c ( u ) < c ( v ),(iv) sets { v k + c ( v ) , ℓ c ( v ) − , ℓ c ( v ) } for v ∈ V , 14v) sets { ℓ i − , ℓ i − , ℓ i } for 1 ≤ i ≤ ⌊ k/ ⌋ (note that 2 k = 2 · h ≡ s ( i,j ) in lexicographic order of pairs ( i, j ), take subsequent triples ofelements and add them to the family F , that is add sets { s (0 , , s (0 , , s (0 , } , . . . , { s ( k − ,k − , s ( k − ,k − , s ( k − ,k − } (note that (cid:0) k (cid:1) ≡ k − ≡ F of size | U | / − • add to F sets { x i , x i , x i +1 } ∈ F x for 1 ≤ i < k such that ⌊ log i ⌋ is odd. • add to F sets { v i , v i , v i +1 } ∈ F v for v ∈ V and 1 ≤ i < k , such that ⌊ log i ⌋ is odd. • add to F all the sets from points (i), (v), (vi) of the construction of F .Note that the size of F equals | U | / −
1, as the only elements which are not covered by F are x , ℓ k − and ℓ k . Claim 4.3. If I is a YES-instance, then there exists a disjoint family F ⊆ F of size | U | / , suchthat |F \ F | + |F \ F | = O ( k ) .Proof. Let K ⊆ V be a solution to I , that is a multicolored clique of size k . Construct a disjointfamily F as follows:(a) add to F sets { x i , x i , x i +1 } ∈ F x for each 1 ≤ i < k , such that ⌊ log i ⌋ is even,(b) add to F sets { v i , v i , v i +1 } ∈ F x for v ∈ K and 1 ≤ i < k , such that ⌊ log i ⌋ is even,(c) add to F sets { v i , v i , v i +1 } ∈ F x for v ∈ V \ K and 1 ≤ i < k , such that ⌊ log i ⌋ is odd,(d) for 0 ≤ i < k add to F the set { x k + i , v ′ , v ′′ } , where v is the unique vertex of K of color i ,(e) add to F sets { v , v ′ , v ′′ } for v ∈ V \ K ,(f) add to F sets { u k + c ( u ) , v k + c ( v ) , s c ( u ) ,c ( v ) } for u, v ∈ K , c ( u ) < c ( v ),(g) add to F sets { v k + c ( v ) , ℓ c ( v ) − , ℓ c ( v ) } for v ∈ K .A direct check shows that the above family is disjoint and covers all the elements of U , hence |F | = | U | /
3. Note that in the above construction of F in each of the points (a), (d), (g) we addto F only O ( k ) sets, while in points (b), (f) we add to F O ( k ) sets, whereas in points (c) and(e) we add to F sets that are present in F . Therefore the number of sets of F which are notpresent in F is upper bounded by a linear function in k . Claim 4.4.
If there exists a disjoint family F of size | U | / , then I is a YES-instance.Proof. Let F ⊆ F be any disjoint family of size | U | /
3. Since the element x can be covered onlyby the set { x , x , x } , the family F contains all the sets { x i , x i , x i +1 } ∈ F x for 1 ≤ i < k ,where ⌊ log i ⌋ is even, and consequently elements x k + i for 0 ≤ i < k are not covered by sets of F x .Therefore elements x k + i are covered by sets from point (ii) of the construction of F , hence for each0 ≤ i < k in F there is exactly one set { v , v , v } ∈ F for v ∈ c − ( i ), and let K be the set ofthose k multicolored vertices. 15e want to show that K is a clique. As for each v ∈ K we have { v , v , v } ∈ F , the family F contains all the sets { v i , v i , v i +1 } for 1 ≤ i < k where ⌊ log i ⌋ is even. Consequently elements v k + i for 0 ≤ i < k , i = c ( v ) are covered by sets from point (iii) of the construction of F . Considerany pair 0 ≤ i < j < k . Denote as u the unique vertex of K ∩ c − ( i ) and let { u k + j , v k + i , s ( i,j ) } bethe set of F covering u k + j , where v ∈ c − ( j ). This implies that v k + i is not covered by a set of the( v, h )-amplifier, hence v is covered by the ( v, h )-amplifier, i.e. by { v , v , v } . Therefore v ∈ K and the vertices of colors i and j of K are adjacent. Since i and j were selected arbitrarily, K is aclique.The proof of Theorem 4.1 follows from Claim 4.3 and Claim 4.4.Theorem 4.1, together with the well-known fact that Multicolored Clique is W[1]-hard [12]implies Theorem 1.1.
One can try to continue the research direction of Chan and Lau [7], who presented a strengtheningof the standard LP relaxation, proving integrality gap of ( k + 1) / k + c ) / c .Finally, we believe that it is worth looking into other problems, where local search algorithmswere applied successfully, such as k -Median [3] or Restricted Max-Min Fair Allocation [27].A potential goal would be to design improved approximation local search algorithms using non-constant size swaps in the spirit of the framework of this paper.
Acknowledgements
We would like to thank Marcin Mucha for helpful discussions.
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