46 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

    Narrow sieves for parameterized paths and packings

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    We present randomized algorithms for some well-studied, hard combinatorial problems: the k-path problem, the p-packing of q-sets problem, and the q-dimensional p-matching problem. Our algorithms solve these problems with high probability in time exponential only in the parameter (k, p, q) and using polynomial space; the constant bases of the exponentials are significantly smaller than in previous works. For example, for the k-path problem the improvement is from 2 to 1.66. We also show how to detect if a d-regular graph admits an edge coloring with dd colors in time within a polynomial factor of O(2^{(d-1)n/2}). Our techniques build upon and generalize some recently published ideas by I. Koutis (ICALP 2009), R. Williams (IPL 2009), and A. Bj\"orklund (STACS 2010, FOCS 2010)

    A Proof Checking View of Parameterized Complexity

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    The PCP Theorem is one of the most stunning results in computational complexity theory, a culmination of a series of results regarding proof checking it exposes some deep structure of computational problems. As a surprising side-effect, it also gives strong non-approximability results. In this paper we initiate the study of proof checking within the scope of Parameterized Complexity. In particular we adapt and extend the PCP[n log log n, n log log n] result of Feige et al. to several parameterized classes, and discuss some corollaries

    A Discharging Method: Improved Kernels for Edge Triangle Packing and Covering

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    \textsc{Edge Triangle Packing} and \textsc{Edge Triangle Covering} are dual problems extensively studied in the field of parameterized complexity. Given a graph GG and an integer kk, \textsc{Edge Triangle Packing} seeks to determine whether there exists a set of at least kk edge-disjoint triangles in GG, while \textsc{Edge Triangle Covering} aims to find out whether there exists a set of at most kk edges that intersects all triangles in GG. Previous research has shown that \textsc{Edge Triangle Packing} has a kernel of (3+ϵ)k(3+\epsilon)k vertices, while \textsc{Edge Triangle Covering} has a kernel of 6k6k vertices. In this paper, we show that the two problems allow kernels of 3k3k vertices, improving all previous results. A significant contribution of our work is the utilization of a novel discharging method for analyzing kernel size, which exhibits potential for analyzing other kernel algorithms

    Equal-Subset-Sum Faster Than the Meet-in-the-Middle

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    In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A,B subseteq S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O^*(3^(n/2)) <= O^*(1.7321^n) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O^*(1.7088^n) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O^*(3^n) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O^*(2.6817^n) time and polynomial space

    ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs

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    We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2^{?(?k)}(n+m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2^{o(?k)}(n+m)^?(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2^{?(?k)}(n+m)^?(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2^{?(?klog k)}(n+m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width ?(?k)

    Counting Problems in Parameterized Complexity

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    This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs. While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way

    A fixed-parameter tractable algorithm for elimination distance to bounded degree graphs

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    In the literature on parameterized graph problems, there has been an increased effort in recent years aimed at exploring novel notions of graph edit-distance that are more powerful than the size of a modulator to a specific graph class. In this line of research, Bulian and Dawar [Algorithmica, 75 (2016), pp. 363--382] introduced the notion of elimination distance and showed that deciding whether a given graph has elimination distance at most kk to any minor-closed class of graphs is fixed-parameter tractable parameterized by kk [Algorithmica, 79 (2017), pp. 139--158]. They showed that graph isomorphism parameterized by the elimination distance to bounded degree graphs is fixed-parameter tractable and asked whether determining the elimination distance to the class of bounded degree graphs is fixed-parameter tractable. Recently, Lindermayr, Siebertz, and Vigny [MFCS 2020, LIPIcs Leibniz Int. Proc. Inform. 170, Wadern Germany, 2020, 65] obtained a fixed-parameter algorithm for this problem in the special case where the input is restricted to K5K_5-minor free graphs. In this paper, we answer the question of Bulian and Dawar in the affirmative for general graphs. In fact, we give a more general result capturing elimination distance to any graph class characterized by a finite set of graphs as forbidden induced subgraphs
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