arXiv subjects

Marek Cygan

Publications and source records attributed to Marek Cygan.

At least 55 records · Page 3Linked to original sources

arXiv2016

Tight Lower Bounds on Graph Embedding Problems

We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of $|V(H)|^{o(|V(H)|)}$-time algorithm deciding if graph $G$ is a subgraph of $H$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems. Moreover, as a consequence of our reductions conditional lower bounds follow for other related problems such as Locally Injective Homomorphism, Graph Minors, Topological Graph Minors, Minimum Distortion Embedding and Quadratic Assignment Problem.

cs.DS↗

arXiv2015

Lower bounds for the parameterized complexity of Minimum Fill-in and other completion problems

In this work, we focus on several completion problems for subclasses of chordal graphs: Minimum Fill-In, Interval Completion, Proper Interval Completion, Threshold Completion, and Trivially Perfect Completion. In these problems, the task is to add at most k edges to a given graph in order to obtain a chordal, interval, proper interval, threshold, or trivially perfect graph, respectively. We prove the following lower bounds for all these problems, as well as for the related Chain Completion problem: Assuming the Exponential Time Hypothesis, none of these problems can be solved in time 2^O(n^(1/2) / log^c n) or 2^O(k^(1/4) / log^c k) n^O(1), for some integer c. Assuming the non-existence of a subexponential-time approximation scheme for Min Bisection on d-regular graphs, for some constant d, none of these problems can be solved in time 2^o(n) or 2^o(sqrt(k)) n^O(1). For all the aforementioned completion problems, apart from Proper Interval Completion, FPT algorithms with running time of the form 2^O(sqrt(k) log k) n^O(1) are known. Thus, the second result proves that a significant improvement of any of these algorithms would lead to a surprising breakthrough in the design of approximation algorithms for Min Bisection. To prove our results, we use a reduction methodology based on combining the classic approach of starting with a sparse instance of 3-Sat, prepared using the Sparsification Lemma, with the existence of almost linear-size Probabilistically Checkable Proofs (PCPs). Apart from our main results, we also obtain lower bounds excluding the existence of subexponential algorithms for the Optimum Linear Arrangement problem, as well as improved, yet still not tight, lower bounds for Feedback Arc Set in Tournaments.

cs.CC↗

arXiv2015

Lower bounds for approximation schemes for Closest String

In the Closest String problem one is given a family $\mathcal S$ of equal-length strings over some fixed alphabet, and the task is to find a string $y$ that minimizes the maximum Hamming distance between $y$ and a string from $\mathcal S$. 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 $\mathrm{FPT}=\mathrm{W}[1]$, 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 $f(\varepsilon)\cdot n^{o(1/\varepsilon)}$, for any computable function $f$, would contradict the Exponential Time Hypothesis.

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arXiv2015

Algorithmic Complexity of Power Law Networks

It was experimentally observed that the majority of real-world networks follow power law degree distribution. The aim of this paper is to study the algorithmic complexity of such "typical" networks. The contribution of this work is twofold. First, we define a deterministic condition for checking whether a graph has a power law degree distribution and experimentally validate it on real-world networks. This definition allows us to derive interesting properties of power law networks. We observe that for exponents of the degree distribution in the range $[1,2]$ such networks exhibit double power law phenomenon that was observed for several real-world networks. Our observation indicates that this phenomenon could be explained by just pure graph theoretical properties. The second aim of our work is to give a novel theoretical explanation why many algorithms run faster on real-world data than what is predicted by algorithmic worst-case analysis. We show how to exploit the power law degree distribution to design faster algorithms for a number of classical P-time problems including transitive closure, maximum matching, determinant, PageRank and matrix inverse. Moreover, we deal with the problems of counting triangles and finding maximum clique. Previously, it has been only shown that these problems can be solved very efficiently on power law graphs when these graphs are random, e.g., drawn at random from some distribution. However, it is unclear how to relate such a theoretical analysis to real-world graphs, which are fixed. Instead of that, we show that the randomness assumption can be replaced with a simple condition on the degrees of adjacent vertices, which can be used to obtain similar results. As a result, in some range of power law exponents, we are able to solve the maximum clique problem in polynomial time, although in general power law networks the problem is NP-complete.

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arXiv2015

The Hardness of Subgraph Isomorphism

Subgraph Isomorphism is a very basic graph problem, where given two graphs $G$ and $H$ one is to check whether $G$ is a subgraph of $H$. Despite its simple definition, the Subgraph Isomorphism problem turns out to be very broad, as it generalizes problems such as Clique, $r$-Coloring, Hamiltonicity, Set Packing and Bandwidth. However, for all of the mentioned problems $2^{\mathcal{O}(n)}$ time algorithms exist, so a natural and frequently asked question in the past was whether there exists a $2^{\mathcal{O}(n)}$ time algorithm for Subgraph Isomorphism. In the monograph of Fomin and Kratsch [Springer'10] this question is highlighted as an open problem, among few others. Our main result is a reduction from 3-SAT, producing a subexponential number of sublinear instances of the Subgraph Isomorphism problem. In particular, our reduction implies a $2^{Ω(n \sqrt{\log n})}$ lower bound for Subgraph Isomorphism under the Exponential Time Hypothesis. This shows that there exist classes of graphs that are strictly harder to embed than cliques or Hamiltonian cycles. The core of our reduction consists of two steps. First, we preprocess and pack variables and clauses of a 3-SAT formula into groups of logarithmic size. However, the grouping is not arbitrary, since as a result we obtain only a limited interaction between the groups. In the second step, we overcome the technical hardness of encoding evaluations as permutations by a simple, yet fruitful scheme of guessing the sizes of preimages of an arbitrary mapping, reducing the case of arbitrary mapping to bijections. In fact, when applying this step to a recent independent result of Fomin et al.[arXiv:1502.05447 (2015)], who showed hardness of Graph Homomorphism, we can transfer their hardness result to Subgraph Isomorphism, implying a nearly tight lower bound of $2^{Ω(n \log n / \log \log n)}$.

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arXiv2015

Polynomial kernelization for removing induced claws and diamonds

A graph is called (claw,diamond)-free if it contains neither a claw (a $K_{1,3}$) nor a diamond (a $K_4$ with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ unless the Exponential Time Hypothesis fail

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arXiv2015

Kernelization lower bound for Permutation Pattern Matching

A permutation $π$ contains a permutation $σ$ as a pattern if it contains a subsequence of length $|σ|$ whose elements are in the same relative order as in the permutation $σ$. This notion plays a major role in enumerative combinatorics. We prove that the problem does not have a polynomial kernel (under the widely believed complexity assumption $\mbox{NP} \not\subseteq \mbox{co-NP}/\mbox{poly}$) by introducing a new polynomial reduction from the clique problem to permutation pattern matching.

cs.DS↗

arXiv2014

Directed Subset Feedback Vertex Set is Fixed-Parameter Tractable

Given a graph $G$ and an integer $k$, the Feedback Vertex Set (FVS) problem asks if there is a vertex set $T$ of size at most $k$ that hits all cycles in the graph. The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is FPT by giving a $4^{k}k!n^{O(1)}$ time algorithm. In the subset versions of this problems, we are given an additional subset $S$ of vertices (resp., edges) and we want to hit all cycles passing through a vertex of $S$ (resp. an edge of $S$). Recently, the Subset Feedback Vertex Set in undirected graphs was shown to be FPT by Cygan et al. (ICALP '11) and independently by Kakimura et al. (SODA '12). We generalize the result of Chen et al. (STOC '08) by showing that Subset Feedback Vertex Set in directed graphs can be solved in time $2^{O(k^3)}n^{O(1)}$. By our result, we complete the picture for feedback vertex set problems and their subset versions in undirected and directed graphs. Besides proving the fixed-parameter tractability of Directed Subset Feedback Vertex Set, we reformulate the random sampling of important separators technique in an abstract way that can be used for a general family of transversal problems. Moreover, we modify the probability distribution used in the technique to achieve better running time; in particular, this gives an improvement from $2^{2^{O(k)}}$ to $2^{O(k^2)}$ in the parameter dependence of the Directed Multiway Cut algorithm of Chitnis et al. (SODA '12).

cs.DS↗

arXiv2014

Hitting forbidden subgraphs in graphs of bounded treewidth

We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph $H$ and an input graph $G$, the task is to find a set $X \subseteq V(G)$ of minimum size that hits all subgraphs of $G$ isomorphic to $H$. In the colorful variant of the problem, each vertex of $G$ is precolored with some color from $V(H)$ and we require to hit only $H$-subgraphs with matching colors. Standard techniques shows that for every fixed $H$, the problem is fixed-parameter tractable parameterized by the treewidth of $G$; however, it is not clear how exactly the running time should depend on treewidth. For the colorful variant, we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of $G$ is tightly governed by $μ(H)$, the maximum size of a minimal vertex separator in $H$. That is, we show for every fixed $H$ that, on a graph of treewidth $t$, the colorful problem can be solved in time $2^{\mathcal{O}(t^{μ(H)})}\cdot|V(G)|$, but cannot be solved in time $2^{o(t^{μ(H)})}\cdot |V(G)|^{O(1)}$, assuming the Exponential Time Hypothesis (ETH). Furthermore, we give some preliminary results showing that, in the absence of colors, the parameterized complexity landscape of H-Subgraph Hitting is much richer.

cs.DS↗

arXiv2014

Approximating Upper Degree-Constrained Partial Orientations

In the Upper Degree-Constrained Partial Orientation problem we are given an undirected graph $G=(V,E)$, together with two degree constraint functions $d^-,d^+ : V \to \mathbb{N}$. The goal is to orient as many edges as possible, in such a way that for each vertex $v \in V$ the number of arcs entering $v$ is at most $d^-(v)$, whereas the number of arcs leaving $v$ is at most $d^+(v)$. This problem was introduced by Gabow [SODA'06], who proved it to be MAXSNP-hard (and thus APX-hard). In the same paper Gabow presented an LP-based iterative rounding $4/3$-approximation algorithm. Since the problem in question is a special case of the classic 3-Dimensional Matching, which in turn is a special case of the $k$-Set Packing problem, it is reasonable to ask whether recent improvements in approximation algorithms for the latter two problems [Cygan, FOCS'13; Sviridenko & Ward, ICALP'13] allow for an improved approximation for Upper Degree-Constrained Partial Orientation. We follow this line of reasoning and present a polynomial-time local search algorithm with approximation ratio $5/4+\varepsilon$. Our algorithm uses a combination of two types of rules: improving sets of bounded pathwidth from the recent $4/3+\varepsilon$-approximation algorithm for 3-Set Packing [Cygan, FOCS'13], and a simple rule tailor-made for the setting of partial orientations. In particular, we exploit the fact that one can check in polynomial time whether it is possible to orient all the edges of a given graph [Gyárfás & Frank, Combinatorics'76].

cs.DS↗

arXiv2014

On Problems as Hard as CNFSAT

The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-Sat. In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2^n), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o(2^n), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every epsilon<1, there is a (large) integer k such that that k-CNF-Sat cannot be computed in time 2^{epsilon n}. In this paper, we show that, for every epsilon < 1, the problems Hitting Set, Set Splitting, and NAE-Sat cannot be computed in time O(2^{epsilon n}) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for Set Cover, and prove that, under this assumption, the fastest known algorithms for Steinter Tree, Connected Vertex Cover, Set Partitioning, and the pseudo-polynomial time algorithm for Subset Sum cannot be significantly improved. Finally, we justify our assumption about the hardness of Set Cover by showing that the parity of the number of set covers cannot be computed in time O(2^{epsilon n}) for any epsilon<1 unless SETH fails.

cs.DS↗

arXiv2014

Minimum Bisection is fixed parameter tractable

In the classic Minimum Bisection problem we are given as input a graph $G$ and an integer $k$. The task is to determine whether there is a partition of $V(G)$ into two parts $A$ and $B$ such that $||A|-|B|| \leq 1$ and there are at most $k$ edges with one endpoint in $A$ and the other in $B$. In this paper we give an algorithm for Minimum Bisection with running time $O(2^{O(k^{3})}n^3 \log^3 n)$. This is the first fixed parameter tractable algorithm for Minimum Bisection. At the core of our algorithm lies a new decomposition theorem that states that every graph $G$ can be decomposed by small separators into parts where each part is "highly connected" in the following sense: any cut of bounded size can separate only a limited number of vertices from each part of the decomposition. Our techniques generalize to the weighted setting, where we seek for a bisection of minimum weight among solutions that contain at most $k$ edges.

cs.DS↗

arXiv2014

Constant Factor Approximation for Capacitated k-Center with Outliers

The $k$-center problem is a classic facility location problem, where given an edge-weighted graph $G = (V,E)$ one is to find a subset of $k$ vertices $S$, such that each vertex in $V$ is "close" to some vertex in $S$. The approximation status of this basic problem is well understood, as a simple 2-approximation algorithm is known to be tight. Consequently different extensions were studied. In the capacitated version of the problem each vertex is assigned a capacity, which is a strict upper bound on the number of clients a facility can serve, when located at this vertex. A constant factor approximation for the capacitated $k$-center was obtained last year by Cygan, Hajiaghayi and Khuller [FOCS'12], which was recently improved to a 9-approximation by An, Bhaskara and Svensson [arXiv'13]. In a different generalization of the problem some clients (denoted as outliers) may be disregarded. Here we are additionally given an integer $p$ and the goal is to serve exactly $p$ clients, which the algorithm is free to choose. In 2001 Charikar et al. [SODA'01] presented a 3-approximation for the $k$-center problem with outliers. In this paper we consider a common generalization of the two extensions previously studied separately, i.e. we work with the capacitated $k$-center with outliers. We present the first constant factor approximation algorithm with approximation ratio of 25 even for the case of non-uniform hard capacities.

cs.DS↗

arXiv2013

Improved approximation for 3-dimensional matching via bounded pathwidth local search

One of the most natural optimization problems is the k-Set Packing problem, where given a family of sets of size at most k one should select a maximum size subfamily of pairwise disjoint sets. A special case of 3-Set Packing is the well known 3-Dimensional Matching problem. Both problems belong to the Karp`s list of 21 NP-complete problems. The best known polynomial time approximation ratio for k-Set Packing is (k + eps)/2 and goes back to the work of Hurkens and Schrijver [SIDMA`89], which gives (1.5 + eps)-approximation for 3-Dimensional Matching. Those results are obtained by a simple local search algorithm, that uses constant size swaps. The main result of the paper is a new approach to local search for k-Set Packing where only a special type of swaps is considered, which we call swaps of bounded pathwidth. We show that 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 with respect to O(log |F|)-size swaps of constant pathwidth yields a polynomial time (k + 1 + eps)/3-approximation algorithm, improving the best known approximation ratio for k-Set Packing. In particular we improve the approximation ratio for 3-Dimensional Matching from 3/2 + eps to 4/3 + eps.

cs.DS↗

arXiv2013

Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

In this paper we consider kernelization for problems on d-degenerate graphs, i.e. graphs such that any subgraph contains a vertex of degree at most $d$. This graph class generalizes many classes of graphs for which effective kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and H-topological-minor free graphs. We show that for several natural problems on d-degenerate graphs the best known kernelization upper bounds are essentially tight.

cs.DS↗

arXiv2013

Fast branching algorithm for Cluster Vertex Deletion

In the family of clustering problems, we are given a set of objects (vertices of the graph), together with some observed pairwise similarities (edges). The goal is to identify clusters of similar objects by slightly modifying the graph to obtain a cluster graph (disjoint union of cliques). Hueffner et al. [Theory Comput. Syst. 2010] initiated the parameterized study of Cluster Vertex Deletion, where the allowed modification is vertex deletion, and presented an elegant O(2^k * k^9 + n * m)-time fixed-parameter algorithm, parameterized by the solution size. In our work, we pick up this line of research and present an O(1.9102^k * (n + m))-time branching algorithm.

cs.DS↗

arXiv2013

The planar directed k-Vertex-Disjoint Paths problem is fixed-parameter tractable

Given a graph G and k pairs of vertices (s_1,t_1), ..., (s_k,t_k), the k-Vertex-Disjoint Paths problem asks for pairwise vertex-disjoint paths P_1, ..., P_k such that P_i goes from s_i to t_i. Schrijver [SICOMP'94] proved that the k-Vertex-Disjoint Paths problem on planar directed graphs can be solved in time n^{O(k)}. We give an algorithm with running time 2^{2^{O(k^2)}} n^{O(1)} for the problem, that is, we show the fixed-parameter tractability of the problem.

cs.DM↗

arXiv2013

Faster exponential-time algorithms in graphs of bounded average degree

We first show that the Traveling Salesman Problem in an n-vertex graph with average degree bounded by d can be solved in O*(2^{(1-\eps_d)n}) time and exponential space for a constant \eps_d depending only on d, where the O*-notation suppresses factors polynomial in the input size. Thus, we generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of bounded degree. Then, we move to the problem of counting perfect matchings in a graph. We first present a simple algorithm for counting perfect matchings in an n-vertex graph in O*(2^{n/2}) time and polynomial space; our algorithm matches the complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on inclusion-exclusion principle instead of algebraic transformations. Building upon this result, we show that the number of perfect matchings in an n-vertex graph with average degree bounded by d can be computed in O*(2^{(1-\eps_{2d})n/2}) time and exponential space, where \eps_{2d} is the constant obtained by us for the Traveling Salesman Problem in graphs of average degree at most 2d. Moreover we obtain a simple algorithm that counts the number of perfect matchings in an n-vertex bipartite graph of average degree at most d in O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of Izumi and Wadayama [FOCS 2012].

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