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Efficient Algorithms for the Max K -Vertex Cover Problem

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Efficient Algorithms for the Max K -Vertex Cover Problem Efficient Algorithms for the max k -vertex cover Problem Federico Della Croce, Vangelis Paschos To cite this version: Federico Della Croce, Vangelis Paschos. Efficient Algorithms for the max k -vertex cover Problem. 7th International Conference on Theoretical Computer Science (TCS), Sep 2012, Amsterdam, Netherlands. pp.295-309, 10.1007/978-3-642-33475-7_21. hal-01511883 HAL Id: hal-01511883 https://hal.archives-ouvertes.fr/hal-01511883 Submitted on 4 Jul 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Efficient algorithms for the max k-vertex cover problem⋆ Federico Della Croce1 and Vangelis Th. Paschos2,3 1D.A.I., Politecnico di Torino, Italy, [email protected] 2PSL Research University, Université Paris-Dauphine, LAMSADE, CNRS, UMR 7243, France, [email protected] 3Institut Universitaire de France Abstract. We first devise moderately exponential exact algorithms for max k-vertex cover, with time-complexity exponential in n but with polynomial space-complexity by developing a branch and reduce method based upon the measure-and-conquer technique. We then prove that, there exists an exact algorithm for max k-vertex cover with com- plexity bounded above by the maximum among ck and γτ , for some γ < 2, where τ is the cardinality of a minimum vertex cover of G (note that max k-vertex cover ∈/ FPT with respect to parameter k unless FPT = W[1]), using polynomial space. We finally study approxima- tion of max k-vertex cover by moderately exponential algorithms. The general goal of the issue of moderately exponential approxima- tion is to catch-up on polynomial inapproximability, by providing al- gorithms achieving, with worst-case running times importantly smaller than those needed for exact computation, approximation ratios unachiev- able in polynomial time. 1 Introduction In the max k-vertex cover problem a graph G(V, E) with V = n vertices 1,...,n and E edges (i, j) is given together with an integer value| | k<n. The goal is to find| a| subset K V with cardinality k, that is K = k, such that the total number of edges covered⊂ by K is maximized. In its| | decision version, max k-vertex cover can be defined as follows: “given G, k and ℓ, does G contain k vertices that cover at least ℓ edges?”. max k-vertex cover is NP- hard (it contains the minimum vertex cover problem as particular case), but it is polynomially approximable within approximation ratio 3/4, while it cannot be solved by a polynomial time approximation schema unless P = NP. The inter- ested reader can be referred to [19,30] for more information about approximation issues for this problem. In the literature, we often find this problem under the name partial vertex cover problem. It is mainly studied from a parameterized complexity point of ⋆ Research supported by the French Agency for Research under the program TODO, ANR-09-EMER-010 and by a Lagrange fellowship of the Fondazione CRT, Torino, Italy 2 Federico Della Croce and Vangelis Th. Paschos view (see [17] for information on fixed-parameter (in)tractability). A problem is fixed-parameter tractable with respect to a parameter t, if it can be solved (to optimality) with time-complexity O(f(t)p(n)) where f is a function that depends on the parameter t, and p is a polynomial on the size n of the instance. In what follows, when dealing with fixed parameter tractability of max k-vertex cover, we shall use notation max k-vertex cover(t) to denote that we speak about fixed parameter tractability with respect to parameter t. Parameterized complexity issues for max k-vertex cover are first studied in [3] where it is proved that partial vertex cover is fixed-parameter tractable with respect to parameter ℓ, next in [28] where it is proved that it is W[1]-hard with respect to parameter k (another proof of the same result can be found in [9]) and finally in [31] where the fixed-parameter tractability results of [3] are further improved. Let us also quote the paper by [24], where it is proved that in apex-minor-free graphs graphs, partial vertex cover can be solved with complexity that is subexponential in k. The seminal Courcelle’s Theorem [13] (see also [21,20] as well as [37] for a comprehensive study around this theorem) assures that decision problems de- fined on graphs that are expressible in terms of monadic second-order logic formulæ are fixed parameter tractable when the treewidth1 of the the input- graph G, denoted by w, is used as parameter. Courcelle’s Theorem can be also extended to a broad class of optimization problems [1]. As max k-vertex cover belongs to this class, it is fixed parameter tractable with respect to w. In most of cases, “rough” application of this theorem, involves very large functions f(w) (see definition of fixed-parameter tractability given above). In [34], it is proved that given a nice tree decomposition, there exists a fixed- parameter algorithm (based upon dynamic programming), with respect to param- eter w that solves max k-vertex cover in time O(2wk(w2 + k) I ), where I is the number of nodes of the nice tree decomposition and in exponential· | | space| .| In other words, max k-vertex cover(w) FPT, but the fixed-parameter algorithm of [34] uses exponential space. Let∈ us note that in any graph G, de- noting by τ the size of a minimum vertex cover of G, it holds that w 6 τ. So, max k-vertex cover(τ) FPT too, but through the use of exponential space (recall that, as adopted above,∈ max k-vertex cover(τ) denotes the max k- vertex cover problem parameterized by the size τ of a minimum vertex cover. Very frequently, a serious problem about fixed-parameter tractability with respect to w is that it takes too much time to compute the “nice tree decomposi- n tion” that also derives the value of w. More precisely, this takes time O∗(1.7549 ) (notation O∗( ) ignores polynomial factors) by making use of exponential space · 1 A tree decomposition of a graph G(V,E) is a pair (X,T ) where T is a tree on vertex set V (T ) the vertices of which we call nodes and X = ({Xi : i ∈ V (T )}) is a collection of subsets of V such that: (i) ∪i∈V (T )Xi = V , (ii) for each edge (v,w) ∈ E, there exist an i ∈ V (T ) such that {v,w} ∈ Xi, and (iii) for each v ∈ V , the set of nodes {i : v ∈ Xi} forms a subtree of T . The width of a tree decomposition ({Xi : i ∈ V (T )},T ) equals maxi∈V (T ){|Xi|− 1}. The treewidth of a graph G is the minimum width over all tree decompositions of G. Efficient algorithms for the max k-vertex cover problem 3 n and time O∗(2.6151 ) by making use of polynomial space [25]. Note that the problem of deciding if the treewidth of a graph is at most w is fixed-parameter 3 tractable and takes time O(2O(w )n) [33]. Dealing with solution of max k-vertex cover by exact algorithms with running times (exponential) functions of n, let us note that a trivial optimal algo- max -vertex cover n k rithm for k takes time O∗( k )= O∗(n ), and polynomial n space, producing all the subsets of V of size k. This turns to a worst-case O∗(2 ) n 6 n n time (since k 2 with equality for k = 2 ). An improvement of this bound is ω k/3 +O(1) presented in [9], where an exact algorithm with complexity O∗(n ⌈ ⌉ ) was ωt proposed based upon a generalization of the O∗(n ) algorithm of [35] for finding 0.792k a 3t-clique in a graph, where ω =2.376. This induces a complexity O∗(n ), but exponential space is needed. As far as we know, no exact algorithm with n running time O∗(γ ), for some γ < 2, is known for max k-vertex cover. In this paper, we first devise an exact branch and reduce algorithm based upon the measure-and-conquer paradigm by [22] (Section 2) requiring running ∆−1 ∆ n time O∗(2 +1 ), where ∆ denotes the maximum degree of G, and polynomial space. The algorithm is then tailored to graphs with maximum degree 3 in- n ducing a running time O∗(1.3339 ) (Section 4). In Section 3, we devise a fixed parameter algorithm, with respect to parameter τ where, as mentioned above, τ τ is the cardinality of a minimum vertex cover of G that works in time O∗(2 ) and needs only polynomial space. By elaborating a bit more this result we then τ show that the time-complexity of this algorithm is indeed either O∗(γ ) for some k γ < 2 or O∗(c ), for some c > 2. In other words, this algorithm either works in time better than 2τ or it is fixed parameter with respect to the size k of the desired cover. Finally, we show that the technique used for proving that max k-vertex cover(τ) FPT, can be used to prove inclusion in the same class of many other well-known∈ combinatorial problems.
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