33 research outputs found

    Slightly Superexponential Parameterized Problems

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    A central problem in parameterized algorithms is to obtain algorithms with running time f(k) center dot n(O(1)) such that f is as slow growing a function of the parameter k as possible. In particular, a large number of basic parameterized problems admit parameterized algorithms where f (k) is single-exponential, that is, c(k) for some constant c, which makes aiming for such a running time a natural goal for other problems as well. However, there are still plenty of problems where the f(k) appearing in the best-known running time is worse than single-exponential and it remained "slightly superexponential" even after serious attempts to bring it down. A natural question to ask is whether the f (k) appearing in the running time of the best-known algorithms is optimal for any of _ these problems. In this paper, we examine parameterized problems where f(k) is k(O(k)) = 2(O(k log k)) in the best-known running time, and for a number of such problems we show that the dependence on k in the running time cannot be improved to single-exponential. More precisely we prove the following tight lower bounds, for four natural problems, arising from three different domains: (1) In the CLOSEST STRING problem, given strings S-1,..., s(t) over an alphabet Sigma of length L each, and an integer d, the question is whether there exists a string s over E of length L, such that its hamming distance from each of the strings s,, 1 <= i <= t, is at most d. The pattern matching problem CLOSEST STRING is known to be solvable in times 2(O(d log d)) center dot n(O(1)) and 2(O(d log vertical bar Sigma vertical bar)) center dot n(O(1)). We show that there are no 2(O(d log d)) center dot n(O(1)) or 2(O(d log vertical bar Sigma vertical bar)) time algorithms, unless the Exponential Time Hypothesis (ETH) fails. (2) The graph embedding problem DISTORTION, that is, deciding whether a graph G has a metric embedding into the integers with distortion at most d can be solved in time 2(O(d log d)) center dot n(O(1)). We show that there is no 2(O(w log w)) center dot n(O(1)) time algorithm, unless the ETH fails. (3) The DISJOINT PATHS problem can be solved in time 2(O(w log w)) center dot n(O(1)) on graphs of treewidth at most w. We show that there is no 2(O(w log w)) center dot n(O(1)) time algorithm, unless the ETH fails. (4) The CHROMATIC NUMBER problem can be solved in time 2(O(w log w)) center dot n(O(1)) on graphs of treewidth at most w. We show that there is no 2(O(w log w)) center dot n(O(1)) time algorithm, unless the ETH fails. To obtain our results, we first prove the lower bound for variants of basic problems: finding cliques, independent sets, and hitting sets. These artificially constrained variants form a good starting point for proving lower bounds on natural problems without any technical restrictions and could be of independent interest. Several follow-up works have already obtained tight lower bounds by using our framework, and we believe it will prove useful in obtaining even more lower bounds in the future

    Slightly Superexponential Parameterized Problems

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    Lower bounds for approximation schemes for Closest String

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    In the Closest String problem one is given a family S\mathcal S of equal-length strings over some fixed alphabet, and the task is to find a string yy that minimizes the maximum Hamming distance between yy and a string from S\mathcal S. While polynomial-time approximation schemes (PTASes) for this problem are known for a long time [Li et al., J. ACM'02], no efficient polynomial-time approximation scheme (EPTAS) has been proposed so far. In this paper, we prove that the existence of an EPTAS for Closest String is in fact unlikely, as it would imply that FPT=W[1]\mathrm{FPT}=\mathrm{W}[1], a highly unexpected collapse in the hierarchy of parameterized complexity classes. Our proof also shows that the existence of a PTAS for Closest String with running time f(ε)no(1/ε)f(\varepsilon)\cdot n^{o(1/\varepsilon)}, for any computable function ff, would contradict the Exponential Time Hypothesis

    On Exact Algorithms for Permutation CSP

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    In the Permutation Constraint Satisfaction Problem (Permutation CSP) we are given a set of variables VV and a set of constraints C, in which constraints are tuples of elements of V. The goal is to find a total ordering of the variables, π :V[1,...,V]\pi\ : V \rightarrow [1,...,|V|], which satisfies as many constraints as possible. A constraint (v1,v2,...,vk)(v_1,v_2,...,v_k) is satisfied by an ordering π\pi when π(v1)<π(v2)<...<π(vk)\pi(v_1)<\pi(v_2)<...<\pi(v_k). An instance has arity kk if all the constraints involve at most kk elements. This problem expresses a variety of permutation problems including {\sc Feedback Arc Set} and {\sc Betweenness} problems. A naive algorithm, listing all the n!n! permutations, requires 2O(nlogn)2^{O(n\log{n})} time. Interestingly, {\sc Permutation CSP} for arity 2 or 3 can be solved by Held-Karp type algorithms in time O(2n)O^*(2^n), but no algorithm is known for arity at least 4 with running time significantly better than 2O(nlogn)2^{O(n\log{n})}. In this paper we resolve the gap by showing that {\sc Arity 4 Permutation CSP} cannot be solved in time 2o(nlogn)2^{o(n\log{n})} unless ETH fails

    Multidimensional Binary Vector Assignment problem: standard, structural and above guarantee parameterizations

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    In this article we focus on the parameterized complexity of the Multidimensional Binary Vector Assignment problem (called \BVA). An input of this problem is defined by mm disjoint sets V1,V2,,VmV^1, V^2, \dots, V^m, each composed of nn binary vectors of size pp. An output is a set of nn disjoint mm-tuples of vectors, where each mm-tuple is obtained by picking one vector from each set ViV^i. To each mm-tuple we associate a pp dimensional vector by applying the bit-wise AND operation on the mm vectors of the tuple. The objective is to minimize the total number of zeros in these nn vectors. mBVA can be seen as a variant of multidimensional matching where hyperedges are implicitly locally encoded via labels attached to vertices, but was originally introduced in the context of integrated circuit manufacturing. We provide for this problem FPT algorithms and negative results (ETHETH-based results, WW[2]-hardness and a kernel lower bound) according to several parameters: the standard parameter kk i.e. the total number of zeros), as well as two parameters above some guaranteed values.Comment: 16 pages, 6 figure

    Lower Bounds for the Graph Homomorphism Problem

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    The graph homomorphism problem (HOM) asks whether the vertices of a given nn-vertex graph GG can be mapped to the vertices of a given hh-vertex graph HH such that each edge of GG is mapped to an edge of HH. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the 22-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound 2Ω(nloghloglogh)2^{\Omega\left( \frac{n \log h}{\log \log h}\right)}. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound 2O(nlogh)2^{{\mathcal O}(n\log{h})} is almost asymptotically tight. We also investigate what properties of graphs GG and HH make it difficult to solve HOM(G,H)(G,H). An easy observation is that an O(hn){\mathcal O}(h^n) upper bound can be improved to O(hvc(G)){\mathcal O}(h^{\operatorname{vc}(G)}) where vc(G)\operatorname{vc}(G) is the minimum size of a vertex cover of GG. The second lower bound hΩ(vc(G))h^{\Omega(\operatorname{vc}(G))} shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph HH, it is known that HOM(G,H)(G,H) can be solved in time (f(Δ(H)))n(f(\Delta(H)))^n and (f(tw(H)))n(f(\operatorname{tw}(H)))^n where Δ(H)\Delta(H) is the maximum degree of HH and tw(H)\operatorname{tw}(H) is the treewidth of HH. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number χ(H)\chi(H) does not exceed tw(H)\operatorname{tw}(H) and Δ(H)+1\Delta(H)+1, it is natural to ask whether similar upper bounds with respect to χ(H)\chi(H) can be obtained. We provide a negative answer to this question by establishing a lower bound (f(χ(H)))n(f(\chi(H)))^n for any function ff. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.Comment: 19 page

    Subset feedback vertex set is fixed parameter tractable

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    The classical Feedback Vertex Set problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. Feedback Vertex Set has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field. In this paper we consider a more general and difficult version of the problem, named Subset Feedback Vertex Set (SUBSET-FVS in short) where an instance comes additionally with a set S ? V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSET-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP'00, SIDMA'00]. The question whether the SUBSET-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2^k n^O(1) instances with the size of S bounded by O(k^3), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2^O(k log k) n^O(1) time algorithm solving the Subset Feedback Vertex Set problem, proving that it is indeed fixed-parameter tractable.Comment: full version of a paper presented at ICALP'1

    On Directed Feedback Vertex Set parameterized by treewidth

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    We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time 2o(tlogt)nO(1)2^{o(t\log t)}\cdot n^{\mathcal{O}(1)} on general directed graphs, where tt is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time 2O(tlogt)nO(1)2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}. On the other hand, we show that if the input digraph is planar, then the running time can be improved to 2O(t)nO(1)2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}.Comment: 20
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