67 research outputs found

    Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space

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    A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(?^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems. In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal

    ETH Tight Algorithms for Geometric Intersection Graphs: Now in Polynomial Space

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    De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexponential algorithms on geometric graphs with tight (up to ETH) running times. This framework is based on dynamic programming on graphs of weighted treewidth resulting in algorithms that use super-polynomial space. We introduce the notion of weighted treedepth and use it to refine the framework of de Berg et al. for obtaining polynomial space (with tight running times) on geometric graphs. As a result, we prove that for any fixed dimension d ≥ 2 on intersection graphs of similarly-sized fat objects many well-known graph problems including Independent Set, r-Dominating Set for constant r, Cycle Cover, Hamiltonian Cycle, Hamiltonian Path, Steiner Tree, Connected Vertex Cover, Feedback Vertex Set, and (Connected) Odd Cycle Transversal are solvable in time 2^(n^{1-1/d}) and within polynomial space.publishedVersio

    Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

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    For many algorithmic problems on graphs of treewidth tt, a standard dynamic programming approach gives an algorithm with time and space complexity 2O(t)nO(1)2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}. It turns out that when one considers the more restrictive parameter treedepth, it is often the case that a variation of this technique can be used to reduce the space complexity to polynomial, while retaining time complexity of the form 2O(d)nO(1)2^{\mathcal{O}(d)}\cdot n^{\mathcal{O}(1)}, where dd is the treedepth. This transfer of methodology is, however, far from automatic. For instance, for problems with connectivity constraints, standard dynamic programming techniques give algorithms with time and space complexity 2O(tlogt)nO(1)2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)} on graphs of treewidth tt, but it is not clear how to convert them into time-efficient polynomial space algorithms for graphs of low treedepth. Cygan et al. (FOCS'11) introduced the Cut&Count technique and showed that a certain class of problems with connectivity constraints can be solved in time and space complexity 2O(t)nO(1)2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}. Recently, Hegerfeld and Kratsch (STACS'20) showed that, for some of those problems, the Cut&Count technique can be also applied in the setting of treedepth, and it gives algorithms with running time 2O(d)nO(1)2^{\mathcal{O}(d)}\cdot n^{\mathcal{O}(1)} and polynomial space usage. However, a number of important problems eluded such a treatment, with the most prominent examples being Hamiltonian Cycle and Longest Path. In this paper we clarify the situation by showing that Hamiltonian Cycle, Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit 5dnO(1)5^d\cdot n^{\mathcal{O}(1)}-time and polynomial space algorithms on graphs of treedepth dd. The algorithms are randomized Monte Carlo with only false negatives.Comment: Presented at WG2020. 20 pages, 2 figure

    Computing Treedepth in Polynomial Space and Linear FPT Time

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    Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

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    Isolation Schemes for Problems on Decomposable Graphs

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    The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time. In this paper, we study a setting that is more typical for NP\mathsf{NP}-complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use - O(tlogn+log2n)O(t\log n + \log^2{n}) random bits on graphs of treewidth at most tt; - O(n)O(\sqrt{n}) random bits on planar or HH-minor free graphs; and - O(n)O(n)-random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed HH we obtain an algorithm for detecting a Hamiltonian cycle in an HH-minor-free graph that runs in deterministic time 2O(n)2^{O(\sqrt{n})} and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most dd that use O(d)O(d) random bits and assign polynomially-bounded weights. We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved

    Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth

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    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

    On space efficiency of algorithms working on structural decompositions of graphs

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    Dynamic programming on path and tree decompositions of graphs is a technique that is ubiquitous in the field of parameterized and exponential-time algorithms. However, one of its drawbacks is that the space usage is exponential in the decomposition's width. Following the work of Allender et al. [Theory of Computing, '14], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely related to a conjecture that the Longest Common Subsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we complete the complexity landscape sketched for pathwidth and treewidth by Allender et al. by considering the parameter tree-depth. We prove that computations on tree-depth decompositions correspond to a model of non-deterministic machines that work in polynomial time and logarithmic space, with access to an auxiliary stack of maximum height equal to the decomposition's depth. Together with the results of Allender et al., this describes a hierarchy of complexity classes for polynomial-time non-deterministic machines with different restrictions on the access to working space, which mirrors the classic relations between treewidth, pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new version is augmented with a space-efficient algorithm for Dominating Set using the Chinese remainder theore

    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
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