34 research outputs found

    Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering

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    We prove the following theorem. Given a planar graph GG and an integer kk, it is possible in polynomial time to randomly sample a subset AA of vertices of GG with the following properties: (i) AA induces a subgraph of GG of treewidth O(klogk)\mathcal{O}(\sqrt{k}\log k), and (ii) for every connected subgraph HH of GG on at most kk vertices, the probability that AA covers the whole vertex set of HH is at least (2O(klog2k)nO(1))1(2^{\mathcal{O}(\sqrt{k}\log^2 k)}\cdot n^{\mathcal{O}(1)})^{-1}, where nn is the number of vertices of GG. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound 2O(klog2k)nO(1)2^{\mathcal{O}(\sqrt{k} \log^2 k)} n^{\mathcal{O}(1)}. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph, examples of such problems include Directed kk-Path, Weighted kk-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface

    FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges

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    We study the ?-Fixed Cardinality Graph Partitioning (?-FCGP) problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph G, two numbers k,p and 0 ? ? ? 1, the question is whether there is a set S ? V of size k with a specified coverage function cov_?(S) at least p (or at most p for the minimization version). The coverage function cov_?(?) counts edges with exactly one endpoint in S with weight ? and edges with both endpoints in S with weight 1 - ?. ?-FCGP generalizes a number of fundamental graph problems such as Densest k-Subgraph, Max k-Vertex Cover, and Max (k,n-k)-Cut. A natural question in the study of ?-FCGP is whether the algorithmic results known for its special cases, like Max k-Vertex Cover, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for Max k-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greed vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for ? > 0 and the subexponential-time algorithms for the problem on apex-minor free graphs for maximization with ? > 1/3 and minimization with ? < 1/3

    Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs

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    We give algorithms with running time 2^{O({sqrt{k}log{k}})} n^{O(1)} for the following problems. Given an n-vertex unit disk graph G and an integer k, decide whether G contains (i) a path on exactly/at least k vertices, (ii) a cycle on exactly k vertices, (iii) a cycle on at least k vertices, (iv) a feedback vertex set of size at most k, and (v) a set of k pairwise vertex disjoint cycles. For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previously best known parameterized algorithms that run in time 2^{O(k^{0.75}log{k})} n^{O(1)}. Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to k^{O(1)} and there exists a solution that crosses every separator at most O(sqrt{k}) times. The running times of our algorithms are optimal up to the log{k} factor in the exponent, assuming the Exponential Time Hypothesis

    The bidimensionality theory and its algorithmic applications

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 201-219).Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k x k grid graph (and similar graphs) grows with k, typically as Q(k²), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to efficient-often subexponential-fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth, in the sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we show that such a bound is always at most linear. The bidimensionality theory unifies and improves several previous results.(cont.) The theory is based on algorithmic and combinatorial extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating a parallel theory of graph contractions. The foundation of this work is the topological theory of drawings of graphs on surfaces and our results regarding the relation (the linearity) of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a fixed graph H as a minor. In this thesis, we also develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L₁ (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an O[sq. root( log n)] approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be [theta][sq. root(log n)]. We also prove various approximate max-flow/min-vertex- cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n). These results have a number of applications. We exhibit an O[sq. root (log n)] pseudo-approximation for finding balanced vertex separators in general graphs.(cont.) Furthermore, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k[sq. root(log k)]), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor, we give a constant-factor approximation for the treewidth; this via the bidimensionality theory can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.by MohammadTaghi Hajiaghayi.Ph.D

    Subexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H-Minor-Free Graphs

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    We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on HH-minor free graphs. In particular, we obtain the following results (where kk is the solution-size parameter). 1. 2O(klogk)nO(1)2^{O(\sqrt{k}\log k)} \cdot n^{O(1)} time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a 2O(klog4k)nO(1)2^{O(\sqrt{k}\log^4 k)} \cdot n^{O(1)} time algorithm for Edge Multiway Cut and a 2O(rklogk)nO(1)2^{O(r \sqrt{k} \log k)} \cdot n^{O(1)} time algorithm for Vertex Multiway Cut, where rr is the number of terminals to be separated; 3. a 2O((r+k)log4(rk))nO(1)2^{O((r+\sqrt{k})\log^4 (rk))} \cdot n^{O(1)} time algorithm for Edge Multicut and a 2O((rk+r)log(rk))nO(1)2^{O((\sqrt{rk}+r) \log (rk))} \cdot n^{O(1)} time algorithm for Vertex Multicut, where rr is the number of terminal pairs to be separated; 4. a 2O(klogglog4k)nO(1)2^{O(\sqrt{k} \log g \log^4 k)} \cdot n^{O(1)} time algorithm for Group Feedback Edge Set and a 2O(gklog(gk))nO(1)2^{O(g \sqrt{k}\log(gk))} \cdot n^{O(1)} time algorithm for Group Feedback Vertex Set, where gg is the size of the group. 5. In addition, our approach also gives nO(k)n^{O(\sqrt{k})} time algorithms for all above problems with the exception of nO(r+k)n^{O(r+\sqrt{k})} time for Edge/Vertex Multicut and (ng)O(k)(ng)^{O(\sqrt{k})} time for Group Feedback Edge/Vertex Set. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an hh-almost-embeddable graph for any fixed constant hh. In particular we show the following. Let GG be an hh-almost-embeddable graph for a constant hh. Then for every pNp\in\mathbb{N}, there exist disjoint sets Z1,,ZpV(G)Z_1,\dots,Z_p \subseteq V(G) such that for every i{1,,p}i \in \{1,\dots,p\} and every ZZiZ'\subseteq Z_i, the treewidth of G/(Zi\Z)G/(Z_i\backslash Z') is O(p+Z)O(p+|Z'|). Here G/(Zi\Z)G/(Z_i\backslash Z') is the graph obtained from GG by contracting edges with both endpoints in Zi\ZZ_i \backslash Z'.Comment: A preliminary version appears in SODA'2

    Parameterized approximation algorithms for bidirected steiner network problems

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    The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E) and a set {D}subseteq V x V of k demand pairs. The aim is to compute the cheapest network N subseteq G for which there is an s -> t path for each (s,t)in {D}. It is known that this problem is notoriously hard as there is no k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parameterizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k. For the bi-DSN_Planar problem, the aim is to compute a planar optimum solution N subseteq G in a bidirected graph G, i.e. for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSN_Planar, unless FPT=W[1]. Additionally we study several generalizations of bi-DSN_Planar and obtain upper and lower bounds on obtainable runtimes parameterized by k. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network N subseteq G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists when parameterized by k [Chitnis et al., IPEC 2013]. We show a tight inapproximability result: under Gap-ETH there is no (2-{epsilon})-approximation algorithm parameterized by k (for any epsilon>0). To the best of our knowledge, this is the first example of a W[1]-hard problem admitting a non-trivial parameterized approximation factor which is also known to be tight! Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k
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