167 research outputs found

    Subexponential parameterized algorithms for graphs of polynomial growth

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    We show that for a number of parameterized problems for which only 2O(k)nO(1)2^{O(k)} n^{O(1)} time algorithms are known on general graphs, subexponential parameterized algorithms with running time 2O(k111+δlog2k)nO(1)2^{O(k^{1-\frac{1}{1+\delta}} \log^2 k)} n^{O(1)} are possible for graphs of polynomial growth with growth rate (degree) δ\delta, that is, if we assume that every ball of radius rr contains only O(rδ)O(r^\delta) vertices. The algorithms use the technique of low-treewidth pattern covering, introduced by Fomin et al. [FOCS 2016] for planar graphs; here we show how this strategy can be made to work for graphs with polynomial growth. Formally, we prove that, given a graph GG of polynomial growth with growth rate δ\delta and an integer kk, one can in randomized polynomial time find a subset AV(G)A \subseteq V(G) such that on one hand the treewidth of G[A]G[A] is O(k111+δlogk)O(k^{1-\frac{1}{1+\delta}} \log k), and on the other hand for every set XV(G)X \subseteq V(G) of size at most kk, the probability that XAX \subseteq A is 2O(k111+δlog2k)2^{-O(k^{1-\frac{1}{1+\delta}} \log^2 k)}. Together with standard dynamic programming techniques on graphs of bounded treewidth, this statement gives subexponential parameterized algorithms for a number of subgraph search problems, such as Long Path or Steiner Tree, in graphs of polynomial growth. We complement the algorithm with an almost tight lower bound for Long Path: unless the Exponential Time Hypothesis fails, no parameterized algorithm with running time 2k11δεnO(1)2^{k^{1-\frac{1}{\delta}-\varepsilon}}n^{O(1)} is possible for any ε>0\varepsilon > 0 and an integer δ3\delta \geq 3

    Subexponential Parameterized Algorithms for Graphs of Polynomial Growth

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    We show that for a number of parameterized problems for which only 2^{O(k)} n^{O(1)} time algorithms are known on general graphs, subexponential parameterized algorithms with running time 2^{O(k^{1-1/(1+d)} log^2 k)} n^{O(1)} are possible for graphs of polynomial growth with growth rate (degree) d, that is, if we assume that every ball of radius r contains only O(r^d) vertices. The algorithms use the technique of low-treewidth pattern covering, introduced by Fomin et al. [FOCS 2016] for planar graphs; here we show how this strategy can be made to work for graphs of polynomial growth. Formally, we prove that, given a graph G of polynomial growth with growth rate d and an integer k, one can in randomized polynomial time find a subset A of V(G) such that on one hand the treewidth of G[A] is O(k^{1-1/(1+d)} log k), and on the other hand for every set X of vertices of size at most k, the probability that X is a subset of A is 2^{-O(k^{1-1/(1+d)} log^2 k)}. Together with standard dynamic programming techniques on graphs of bounded treewidth, this statement gives subexponential parameterized algorithms for a number of subgraph search problems, such as Long Path or Steiner Tree, in graphs of polynomial growth. We complement the algorithm with an almost tight lower bound for Long Path: unless the Exponential Time Hypothesis fails, no parameterized algorithm with running time 2^{k^{1-1/d-epsilon}}n^{O(1)} is possible for any positive epsilon and any integer d >= 3

    Dynamic Programming for Graphs on Surfaces

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    We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2^{O(k log k)} n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called "surface cut decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2^{O(k)} n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.Comment: 28 pages, 3 figure

    H-free graphs, Independent Sets, and subexponential-time algorithms

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    It is an old open question in algorithmic graph theory to determine the complexity of the Maximum Independent Set problem on Pt-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t ≤ 5 [Lokshtanov et al., SODA 2014, pp. 570-581, 2014]. Here we study the existence of subexponential-time algorithms for the problem: by generalizing an earlier result of Randerath and Schiermeyer for t = 5 [Discrete Appl. Math., 158 (2010), pp. 1041-1044], we show that for any t ≥ 5, there is an algorithm for Maximum Independent Set on Pt-free graphs whose running time is subexponential in the number of vertices. Scattered Set is the generalization of Maximum Independent Set where the vertices of the solution are required to be at distance at least d from each other. We give a complete characterization of those graphs H for which d-Scattered Set on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus number of edges): If every component of H is a path, then d-Scattered Set on H-free graphs with n vertices and m edges can be solved in time 2(n+m) 1-O(1/|V (H)|), even if d is part of the input. Otherwise, assuming ETH, there is no 2o(n+m)-time algorithm for d-Scattered Set for any fixed d ≥ 3 on H-free graphs with n-vertices and m-edges. © 2016 Gábor Bacsó, Dániel Marx, and Zsolt Tuza

    Hyperbolic intersection graphs and (quasi)-polynomial time

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    We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in dd-dimensional hyperbolic space, which we denote by Hd\mathbb{H}^d. Using a new separator theorem, we show that unit ball graphs in Hd\mathbb{H}^d enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in 2O(n11/(d1))2^{O(n^{1-1/(d-1)})} time for any fixed d3d\geq 3, while the same problems need 2O(n11/d)2^{O(n^{1-1/d})} time in Rd\mathbb{R}^d. We also show that these algorithms in Hd\mathbb{H}^d are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in H2\mathbb{H}^2, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial (nO(logn)n^{O(\log n)}) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and 33-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in H2\mathbb{H}^2 have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require 2Ω(n)2^{\Omega(\sqrt{n})} time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching nΩ(logn)n^{\Omega(\log n)} lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.Comment: Short version appears in SODA 202

    Fast counting with tensor networks

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    We introduce tensor network contraction algorithms for counting satisfying assignments of constraint satisfaction problems (#CSPs). We represent each arbitrary #CSP formula as a tensor network, whose full contraction yields the number of satisfying assignments of that formula, and use graph theoretical methods to determine favorable orders of contraction. We employ our heuristics for the solution of #P-hard counting boolean satisfiability (#SAT) problems, namely monotone #1-in-3SAT and #Cubic-Vertex-Cover, and find that they outperform state-of-the-art solvers by a significant margin.Comment: v2: added results for monotone #1-in-3SAT; published versio

    When Maximum Stable Set Can Be Solved in FPT Time

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    Maximum Independent Set (MIS for short) is in general graphs the paradigmatic W[1]-hard problem. In stark contrast, polynomial-time algorithms are known when the inputs are restricted to structured graph classes such as, for instance, perfect graphs (which includes bipartite graphs, chordal graphs, co-graphs, etc.) or claw-free graphs. In this paper, we introduce some variants of co-graphs with parameterized noise, that is, graphs that can be made into disjoint unions or complete sums by the removal of a certain number of vertices and the addition/deletion of a certain number of edges per incident vertex, both controlled by the parameter. We give a series of FPT Turing-reductions on these classes and use them to make some progress on the parameterized complexity of MIS in H-free graphs. We show that for every fixed t >=slant 1, MIS is FPT in P(1,t,t,t)-free graphs, where P(1,t,t,t) is the graph obtained by substituting all the vertices of a four-vertex path but one end of the path by cliques of size t. We also provide randomized FPT algorithms in dart-free graphs and in cricket-free graphs. This settles the FPT/W[1]-hard dichotomy for five-vertex graphs H

    Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

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    The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time

    TSP in a Simple Polygon

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    We study the Traveling Salesman Problem inside a simple polygon. In this problem, which we call tsp in a simple polygon, we wish to compute a shortest tour that visits a given set S of n sites inside a simple polygon P with m edges while staying inside the polygon. This natural problem has, to the best of our knowledge, not been studied so far from a theoretical perspective. It can be solved exactly in poly(n,m) + 2^O(?nlog n) time, using an algorithm by Marx, Pilipczuk, and Pilipczuk (FOCS 2018) for subset tsp as a subroutine. We present a much simpler algorithm that solves tsp in a simple polygon directly and that has the same running time

    Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction

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    In this work, we initiate the complexity study of Biclique Contraction and Balanced Biclique Contraction. In these problems, given as input a graph G and an integer k, the objective is to determine whether one can contract at most k edges in G to obtain a biclique and a balanced biclique, respectively. We first prove that these problems are NP-complete even when the input graph is bipartite. Next, we study the parameterized complexity of these problems and show that they admit single exponential-time FPT algorithms when parameterized by the number k of edge contractions. Then, we show that Balanced Biclique Contraction admits a quadratic vertex kernel while Biclique Contraction does not admit any polynomial compression (or kernel) under standard complexity-theoretic assumptions
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