arXiv subjects

Marek Cygan

Publications and source records attributed to Marek Cygan.

At least 73 records · Page 4Linked to original sources

arXiv2013

On Pairwise Spanners

Given an undirected $n$-node unweighted graph $G = (V, E)$, a spanner with stretch function $f(\cdot)$ is a subgraph $H\subseteq G$ such that, if two nodes are at distance $d$ in $G$, then they are at distance at most $f(d)$ in $H$. Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the $u$-$v$ distance only for pairs $(u,v)$ in a given set $\cP \subseteq V\times V$. Such $\cP$-spanners were studied before [Coppersmith,Elkin'05] only in the special case that $f(\cdot)$ is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same $\cP$) and of the best known spanners (with the same $f(\cdot)$). In more detail, for arbitrary $\cP$, we show that there exists a $\mathcal{P}$-spanner of size $O(n(|\cP|\log n)^{1/4})$ with $f(d)=d+4\log n$. Alternatively, for any $\eps>0$, there exists a $\cP$-spanner of size $O(n|\cP|^{1/4}\sqrt{\frac{\log n}{\eps}})$ with $f(d)=(1+\eps)d+4$. We also consider the relevant special case that there is a critical set of nodes $S\subseteq V$, and we wish to approximate either the distances within nodes in $S$ or from nodes in $S$ to any other node. We show that there exists an $(S\times S)$-spanner of size $O(n\sqrt{|S|})$ with $f(d)=d+2$, and an $(S\times V)$-spanner of size $O(n\sqrt{|S|\log n})$ with $f(d)=d+2\log n$. All the mentioned pairwise spanners can be constructed in polynomial time.

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arXiv2012

Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time

It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in terms of the weights). We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms, also for weighted and counting versions. For example, in this time we can solve the traveling salesman problem or count the number of Hamiltonian cycles. The rank-based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying "small" sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the matrix tree theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.

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arXiv2012

Fast Hamiltonicity checking via bases of perfect matchings

For an even integer t \geq 2, the Matchings Connecivity matrix H_t is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph K_t on t vertices; an entry H_t[M_1,M_2] is 1 if M_1\cup M_2 is a Hamiltonian cycle and 0 otherwise. Motivated by the computational study of the Hamiltonicity problem, we present three results on the structure of H_t: We first show that H_t has rank at most 2^{t/2-1} over GF(2) via an appropriate factorization that explicitly provides families of matchings X_t forming bases for H_t. Second, we show how to quickly change representation between such bases. Third, we notice that the sets of matchings X_t induce permutation matrices within H_t. Subsequently, we use the factorization to obtain an 1.888^n n^{O(1)} time Monte Carlo algorithm that solves the Hamiltonicity problem in directed bipartite graphs. Our algorithm as well counts the number of Hamiltonian cycles modulo two in directed bipartite or undirected graphs in the same time bound. Moreover, we use the fast basis change algorithm from the second result to present a Monte Carlo algorithm that given an undirected graph on n vertices along with a path decomposition of width at most pw decides Hamiltonicity in (2+\sqrt{2})^{pw}n^{O(1)} time. Finally, we use the third result to show that for every ε>0 this cannot be improved to (2+\sqrt{2}-ε)^{pw}n^{O(1)} time unless the Strong Exponential Time Hypothesis fails, i.e., a faster algorithm for this problem would imply the breakthrough result of a (2-ε)^n time algorithm for CNF-Sat.

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arXiv2012

Known algorithms for EDGE CLIQUE COVER are probably optimal

In the EDGE CLIQUE COVER (ECC) problem, given a graph G and an integer k, we ask whether the edges of G can be covered with k complete subgraphs of G or, equivalently, whether G admits an intersection model on k-element universe. Gramm et al. [JEA 2008] have shown a set of simple rules that reduce the number of vertices of G to 2^k, and no algorithm is known with significantly better running time bound than a brute-force search on this reduced instance. In this paper we show that the approach of Gramm et al. is essentially optimal: we present a polynomial time algorithm that reduces an arbitrary 3-CNF-SAT formula with n variables and m clauses to an equivalent ECC instance (G,k) with k = O(log n) and |V(G)| = O(n + m). Consequently, there is no 2^{2^{o(k)}}poly(n) time algorithm for the ECC problem, unless the Exponential Time Hypothesis fails. To the best of our knowledge, these are the first results for a natural, fixed-parameter tractable problem, and proving that a doubly-exponential dependency on the parameter is essentially necessary.

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arXiv2012

Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings

Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this paper, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen's theorem and of Strojohann's determinant algorithm. It allows us to give new and simple solutions to the following problems: * Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^ω) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with non-negative weights) this matches the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the other hand, no algorithm working in \tilde{O}(Wn^ω) time was previously known for undirected graphs with negative weights. Furthermore our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. * Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^ω) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge no algorithm with this running time was known for undirected graphs with negative weights. * Finding Minimum Weight Perfect Matchings -- We present an \tilde{O}(Wn^ω) time algorithm for finding minimum weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski in 2006, who presented such an algorithm but only in the case of bipartite graphs. In order to solve minimum weight perfect matching problem we develop a novel combinatorial interpretation of the dual solution which sheds new light on this problem. Such a combinatorial interpretation was not know previously, and is of independent interest.

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arXiv2012

LP Rounding for k-Centers with Non-uniform Hard Capacities

In this paper we consider a generalization of the classical k-center problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node has to travel to get to its assigned center. This problem is NP-hard, even when centers have no capacity restrictions and optimal factor 2 approximation algorithms are known. With capacities, when all centers have identical capacities, a 6 approximation is known with no better lower bounds than for the infinite capacity version. While many generalizations and variations of this problem have been studied extensively, no progress was made on the capacitated version for a general capacity function. We develop the first constant factor approximation algorithm for this problem. Our algorithm uses an LP rounding approach to solve this problem, and works for the case of non-uniform hard capacities, when multiple copies of a node may not be chosen and can be extended to the case when there is a hard bound on the number of copies of a node that may be selected. In addition we establish a lower bound on the integrality gap of 7(5) for non-uniform (uniform) hard capacities. In addition we prove that if there is a (3-eps)-factor approximation for this problem then P=NP. Finally, for non-uniform soft capacities we present a much simpler 11-approximation algorithm, which we find as one more evidence that hard capacities are much harder to deal with.

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arXiv2012

On fixed-parameter algorithms for Split Vertex Deletion

In the Split Vertex Deletion problem, given a graph G and an integer k, we ask whether one can delete k vertices from the graph G to obtain a split graph (i.e., a graph, whose vertex set can be partitioned into two sets: one inducing a clique and the second one inducing an independent set). In this paper we study fixed-parameter algorithms for Split Vertex Deletion parameterized by k: we show that, up to a factor quasipolynomial in k and polynomial in n, the Split Vertex Deletion problem can be solved in the same time as the well-studied Vertex Cover problem. Plugging the currently best fixed-parameter algorithm for Vertex Cover due to Chen et al. [TCS 2010], we obtain an algorithm that solves Split Vertex Deletion in time O(1.2738^k * k^O(log k) + n^O(1)). To achieve our goal, we prove the following structural result that may be of independent interest: for any graph G we may compute a family P of size n^O(log n) containing partitions of V(G) into two parts, such for any two disjoint subsets X_C, X_I of V(G) where G[X_C] is a clique and G[X_I] is an independent set, there is a partition in P which contains all vertices of X_C on one side and all vertices of X_I on the other.

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arXiv2012

Steiner Forest Orientation Problems

We consider connectivity problems with orientation constraints. Given a directed graph $D$ and a collection of ordered node pairs $P$ let $P[D]=\{(u,v) \in P: D {contains a} uv{-path}}$. In the {\sf Steiner Forest Orientation} problem we are given an undirected graph $G=(V,E)$ with edge-costs and a set $P \subseteq V \times V$ of ordered node pairs. The goal is to find a minimum-cost subgraph $H$ of $G$ and an orientation $D$ of $H$ such that $P[D]=P$. We give a 4-approximation algorithm for this problem. In the {\sf Maximum Pairs Orientation} problem we are given a graph $G$ and a multi-collection of ordered node pairs $P$ on $V$. The goal is to find an orientation $D$ of $G$ such that $|P[D]|$ is maximum. Generalizing the result of Arkin and Hassin [DAM'02] for $|P|=2$, we will show that for a mixed graph $G$ (that may have both directed and undirected edges), one can decide in $n^{O(|P|)}$ time whether $G$ has an orientation $D$ with $P[D]=P$ (for undirected graphs this problem admits a polynomial time algorithm for any $P$, but it is NP-complete on mixed graphs). For undirected graphs, we will show that one can decide whether $G$ admits an orientation $D$ with $|P[D]| \geq k$ in $O(n+m)+2^{O(k\cdot \log \log k)}$ time; hence this decision problem is fixed-parameter tractable, which answers an open question from Dorn et al. [AMB'11]. We also show that {\sf Maximum Pairs Orientation} admits ratio $O(\log |P|/\log\log |P|)$, which is better than the ratio $O(\log n/\log\log n)$ of Gamzu et al. [WABI'10] when $|P|<n$. Finally, we show that the following node-connectivity problem can be solved in polynomial time: given a graph $G=(V,E)$ with edge-costs, $s,t \in V$, and an integer $\ell$, find a min-cost subgraph $H$ of $G$ with an orientation $D$ such that $D$ contains $\ell$ internally-disjoint $st$-paths, and $\ell$ internally-disjoint $ts$-paths.

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arXiv2012

Scheduling partially ordered jobs faster than 2^n

In the SCHED problem we are given a set of n jobs, together with their processing times and precedence constraints. The task is to order the jobs so that their total completion time is minimized. SCHED is a special case of the Traveling Repairman Problem with precedences. A natural dynamic programming algorithm solves both these problems in 2^n n^O(1) time, and whether there exists an algorithms solving SCHED in O(c^n) time for some constant c < 2 was an open problem posted in 2004 by Woeginger. In this paper we answer this question positively.

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arXiv2012

Deterministic parameterized connected vertex cover

In the Connected Vertex Cover problem we are given an undirected graph G together with an integer k and we are to find a subset of vertices X of size at most k, such that X contains at least one end-point of each edge and moreover X induces a connected subgraph. For this problem we present a deterministic algorithm running in O(2^k n^O(1)) time and polynomial space, improving over previously best O(2.4882^k n^O(1)) deterministic algorithm and O(2^k n^O(1)) randomized algorithm. Furthermore, when usage of exponential space is allowed, we present an O(2^k k(n+m)) time algorithm that solves a more general variant with arbitrary real weights. Finally, we show that in O(2k k(n + m)) time and O(2^k k) space one can count the number of connected vertex covers of size at most k, which can not be improved to O((2 - eps)^k nO(1)) for any eps > 0 under the Strong Exponential Time Hypothesis, as shown by Cygan et al. [CCC'12].

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arXiv2012

Sitting closer to friends than enemies, revisited

Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space R^l in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into R^1 can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this particular case, answering several questions posed by Kermarrec and Thraves. First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by showing that the problem is closely related to the recognition of proper interval graphs. Second, we prove that the general case, whose polynomial-time tractability remained open, is in fact NP-complete. Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming approach gives a running time single-exponential in the number of vertices.

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arXiv2011

On group feedback vertex set parameterized by the size of the cutset

We study the parameterized complexity of a robust generalization of the classical Feedback Vertex Set problem, namely the Group Feedback Vertex Set problem; we are given a graph G with edges labeled with group elements, and the goal is to compute the smallest set of vertices that hits all cycles of G that evaluate to a non-null element of the group. This problem generalizes not only Feedback Vertex Set, but also Subset Feedback Vertex Set, Multiway Cut and Odd Cycle Transversal. Completing the results of Guillemot [Discr. Opt. 2011], we provide a fixed-parameter algorithm for the parameterization by the size of the cutset only. Our algorithm works even if the group is given as a polynomial-time oracle.

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arXiv2011

Clique cover and graph separation: New incompressibility results

The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. Although a framework for proving kernelization lower bounds has been discovered in 2008 and successfully applied multiple times over the last three years, establishing kernelization complexity of many important problems remains open. In this paper we show that, unless NP is a subset of coNP/poly and the polynomial hierarchy collapses up to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduces the size of an instance to polynomial in the parameter: - EDGE CLIQUE COVER, parameterized by the number of cliques, - DIRECTED EDGE/VERTEX MULTIWAY CUT, parameterized by the size of the cutset, even in the case of two terminals, - EDGE/VERTEX MULTICUT, parameterized by the size of the cutset, and - k-WAY CUT, parameterized by the size of the cutset. The existence of a polynomial kernelization for EDGE CLIQUE COVER was a seasoned veteran in open problem sessions. Furthermore, our results complement very recent developments in designing parameterized algorithms for cut problems by Marx and Razgon [STOC'11], Bousquet et al. [STOC'11], Kawarabayashi and Thorup [FOCS'11] and Chitnis et al. [SODA'12].

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arXiv2011

Parameterized Complexity of Firefighting Revisited

The Firefighter problem is to place firefighters on the vertices of a graph to prevent a fire with known starting point from lighting up the entire graph. In each time step, a firefighter may be permanently placed on an unburned vertex and the fire spreads to its neighborhood in the graph in so far no firefighters are protecting those vertices. The goal is to let as few vertices burn as possible. This problem is known to be NP-complete, even when restricted to bipartite graphs or to trees of maximum degree three. Initial study showed the Firefighter problem to be fixed-parameter tractable on trees in various parameterizations. We complete these results by showing that the problem is in FPT on general graphs when parameterized by the number of burned vertices, but has no polynomial kernel on trees, resolving an open problem. Conversely, we show that the problem is W[1]-hard when parameterized by the number of unburned vertices, even on bipartite graphs. For both parameterizations, we additionally give refined algorithms on trees, improving on the running times of the known algorithms.

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arXiv2011

Dominating Set is Fixed Parameter Tractable in Claw-free Graphs

We show that the dominating set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw (K(1,3)), the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2^O(k^2)n^O(1) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k. Note that this parameterization of dominating set is W[2]-hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general. The most general class of graphs for which an FPT algorithm was previously known for this parameterization of dominating set is the class of K(i,j)-free graphs, which exclude, for some fixed i,j, the complete bipartite graph K(i,j) as a subgraph. For i,i >= 2, the class of claw-free graphs and any class of K(i,j)-free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of dominating set is known to be fixed-parameter tractable. We also show that, in some sense, it is the presence of the claw that makes this parameterization of the dominating set problem hard. More precisely, we show that for any t ?>= 4, the dominating set problem parameterized by the solution size is W[2]-hard in graphs that exclude the t-claw K(1,t) as an induced subgraph. Our arguments also imply that the related connected dominating set and dominating clique problems are W[2]-hard in these graph classes. Finally, we show that for any t, the clique problem parameterized by solution size, which is W[1]-hard on general graphs, is FPT in t-claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.

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arXiv2011

Subset feedback vertex set is fixed parameter tractable

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.

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arXiv2011

On Multiway Cut parameterized above lower bounds

In this paper we consider two above lower bound parameterizations of the Node Multiway Cut problem - above the maximum separating cut and above a natural LP-relaxation - and prove them to be fixed-parameter tractable. Our results imply O*(4^k) algorithms for Vertex Cover above Maximum Matching and Almost 2-SAT as well as an O*(2^k) algorithm for Node Multiway Cut with a standard parameterization by the solution size, improving previous bounds for these problems.

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arXiv2011

Channel Assignment via Fast Zeta Transform

We show an O*((l+1)^n)-time algorithm for the channel assignment problem, where l is the maximum edge weight. This improves on the previous O*((l+2)^n)-time algorithm by Kral, as well as algorithms for important special cases, like L(2,1)-labelling. For the latter problem, our algorithm works in O*(3^n) time. The progress is achieved by applying the fast zeta transform in combination with the inclusion-exclusion principle.

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