179 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

    Full text link
    Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on nn-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth dd and using kk labels, we can solve - Independent Set in time 2O(dk)nO(1)2^{O(dk)}\cdot n^{O(1)} using O(dk2logn)O(dk^2\log n) space; - Max Cut in time nO(dk)n^{O(dk)} using O(dklogn)O(dk\log n) space; and - Dominating Set in time 2O(dk)nO(1)2^{O(dk)}\cdot n^{O(1)} using nO(1)n^{O(1)} space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with dd if one wishes to keep the space complexity polynomial.Comment: Conference version to appear at the European Symposium on Algorithms (ESA 2023

    Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

    Get PDF
    Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, - Independent Set can be solved in time 2^(dk) ⋅ n^(1) using (dk²log n) space; - Max Cut can be solved in time n^(dk) using (dk log n) space; and - Dominating Set can be solved in time 2^(dk) ⋅ n^(1) using n^(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    The Complexity of Some Geometric Proof Systems

    Get PDF
    In this Thesis we investigate proof systems based on Integer Linear Programming. These methods inspect the solution space of an unsatisfiable propositional formula and prove that this space contains no integral points. We begin by proving some size and depth lower bounds for a recent proof system, Stabbing Planes, and along the way introduce some novel methods for doing so. We then turn to the complexity of propositional contradictions generated uniformly from first order sentences, in Stabbing Planes and Sum-Of-Squares. We finish by investigating the complexity-theoretic impact of the choice of method of generating these propositional contradictions in Sherali-Adams

    Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

    Get PDF
    Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth d and using k labels, we can solve - Independent Set in time 2O(dk)⋅nO(1) using O(dk2logn) space; - Max Cut in time nO(dk) using O(dklogn) space; and - Dominating Set in time 2O(dk)⋅nO(1) using nO(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial

    Distinct Difference Configurations in Groups

    Get PDF

    Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

    Get PDF

    Counting Subgraphs in Somewhere Dense Graphs

    Get PDF
    We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time f(H)?|G|^O(1) for some computable function f. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes ? as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis: - Counting k-matchings in a graph G ? ? is fixed-parameter tractable if and only if ? is nowhere dense. - Counting k-independent sets in a graph G ? ? is fixed-parameter tractable if and only if ? is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if ? is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19). At the same time our proofs are much simpler: using structural characterisations of somewhere dense graphs, we show that a colourful version of a recent breakthrough technique for analysing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting
    corecore