2,278 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    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

    Towards Efficient Explainability of Schedulability Properties in Real-Time Systems

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    The notion of efficient explainability was recently introduced in the context of hard-real-time scheduling: a claim that a real-time system is schedulable (i.e., that it will always meet all deadlines during run-time) is defined to be efficiently explainable if there is a proof of such schedulability that can be verified by a polynomial-time algorithm. We further explore this notion by (i) classifying a variety of common schedulability analysis problems according to whether they are efficiently explainable or not; and (ii) developing strategies for dealing with those determined to not be efficiently schedulable, primarily by identifying practically meaningful sub-problems that are efficiently explainable

    Overlapping and Robust Edge-Colored Clustering in Hypergraphs

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    A recent trend in data mining has explored (hyper)graph clustering algorithms for data with categorical relationship types. Such algorithms have applications in the analysis of social, co-authorship, and protein interaction networks, to name a few. Many such applications naturally have some overlap between clusters, a nuance which is missing from current combinatorial models. Additionally, existing models lack a mechanism for handling noise in datasets. We address these concerns by generalizing Edge-Colored Clustering, a recent framework for categorical clustering of hypergraphs. Our generalizations allow for a budgeted number of either (a) overlapping cluster assignments or (b) node deletions. For each new model we present a greedy algorithm which approximately minimizes an edge mistake objective, as well as bicriteria approximations where the second approximation factor is on the budget. Additionally, we address the parameterized complexity of each problem, providing FPT algorithms and hardness results

    Clones over Finite Sets and Minor Conditions

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    Achieving a classification of all clones of operations over a finite set is one of the goals at the heart of universal algebra. In 1921 Post provided a full description of the lattice of all clones over a two-element set. However, over the following years, it has been shown that a similar classification seems arduously reachable even if we only focus on clones over three-element sets: in 1959 Janov and Mučnik proved that there exists a continuum of clones over a k-element set for every k > 2. Subsequent research in universal algebra therefore focused on understanding particular aspects of clone lattices over finite domains. Remarkable results in this direction are the description of maximal and minimal clones. One might still hope to classify all operation clones on finite domains up to some equivalence relation so that equivalent clones share many of the properties that are of interest in universal algebra. In a recent turn of events, a weakening of the notion of clone homomorphism was introduced: a minor-preserving map from a clone C to D is a map which preserves arities and composition with projections. The minor-equivalence relation on clones over finite sets gained importance both in universal algebra and in computer science: minor-equivalent clones satisfy the same set identities of the form f(x_1,...,x_n) = g(y_1,...,y_m), also known as minor-identities. Moreover, it was proved that the complexity of the CSP of a finite structure A only depends on the set of minor-identities satisfied by the polymorphism clone of A. Throughout this dissertation we focus on the poset that arises by considering clones over a three-element set with the following order: we write C ≤_{m} D if there exist a minor-preserving map from C to D. It has been proved that ≤_{m} is a preorder; we call the poset arising from ≤_{m} the pp-constructability poset. We initiate a systematic study of the pp-constructability poset. To this end, we distinguish two cases that are qualitatively distinct: when considering clones over a finite set A, one can either set a boundary on the cardinality of A, or not. We denote by P_n the pp-constructability poset restricted to clones over a set A such that |A|=n and by P_{fin} we denote the whole pp-constructability poset, i.e., we only require A to be finite. First, we prove that P_{fin} is a semilattice and that it has no atoms. Moreover, we provide a complete description of P_2 and describe a significant part of P_3: we prove that P_3 has exactly three submaximal elements and present a full description of the ideal generated by one of these submaximal elements. As a byproduct, we prove that there are only countably many clones of self-dual operations over {0,1,2} up to minor-equivalence

    Gap Preserving Reductions Between Reconfiguration Problems

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    Combinatorial reconfiguration is a growing research field studying problems on the transformability between a pair of solutions for a search problem. For example, in SAT Reconfiguration, for a Boolean formula ? and two satisfying truth assignments ?_? and ?_? for ?, we are asked to determine whether there is a sequence of satisfying truth assignments for ? starting from ?_? and ending with ?_?, each resulting from the previous one by flipping a single variable assignment. We consider the approximability of optimization variants of reconfiguration problems; e.g., Maxmin SAT Reconfiguration requires to maximize the minimum fraction of satisfied clauses of ? during transformation from ?_? to ?_?. Solving such optimization variants approximately, we may be able to obtain a reasonable reconfiguration sequence comprising almost-satisfying truth assignments. In this study, we prove a series of gap-preserving reductions to give evidence that a host of reconfiguration problems are PSPACE-hard to approximate, under some plausible assumption. Our starting point is a new working hypothesis called the Reconfiguration Inapproximability Hypothesis (RIH), which asserts that a gap version of Maxmin CSP Reconfiguration is PSPACE-hard. This hypothesis may be thought of as a reconfiguration analogue of the PCP theorem. Our main result is PSPACE-hardness of approximating Maxmin 3-SAT Reconfiguration of bounded occurrence under RIH. The crux of its proof is a gap-preserving reduction from Maxmin Binary CSP Reconfiguration to itself of bounded degree. Because a simple application of the degree reduction technique using expander graphs due to Papadimitriou and Yannakakis (J. Comput. Syst. Sci., 1991) does not preserve the perfect completeness, we modify the alphabet as if each vertex could take a pair of values simultaneously. To accomplish the soundness requirement, we further apply an explicit family of near-Ramanujan graphs and the expander mixing lemma. As an application of the main result, we demonstrate that under RIH, optimization variants of popular reconfiguration problems are PSPACE-hard to approximate, including Nondeterministic Constraint Logic due to Hearn and Demaine (Theor. Comput. Sci., 2005), Independent Set Reconfiguration, Clique Reconfiguration, and Vertex Cover Reconfiguration

    FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii

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    Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into kk clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints. In particular, we give a (15+ϵ)(15+\epsilon)-approximation algorithm that runs in 20(k2logk)n32^{0(k^2\log k)}\cdot n^3 time. When capacities are uniform, we obtain the following improved approximation bounds: A (4 + ϵ\epsilon)-approximation with running time 2O(klog(k/ϵ))n32^{O(k\log(k/\epsilon))}n^3, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 + ϵ\epsilon)-approximation with running time 2O(k/ϵ2log(k/ϵ))dn32^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3 and a (1+ϵ)(1+\epsilon)-approximation with running time 2O(kdlog((k/ϵ)))n32^{O(kd\log ((k/\epsilon)))}n^{3} in the Euclidean space; and a (1 + ϵ\epsilon)-approximation in the Euclidean space with running time 2O(k/ϵ2log(k/ϵ))dn32^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3 if we are allowed to violate the capacities by (1 + ϵ\epsilon)-factor. We complement this result by showing that there is no (1 + ϵ\epsilon)-approximation algorithm running in time f(k)nO(1)f(k)\cdot n^{O(1)}, if any capacity violation is not allowed.Comment: Full version of a paper accepted to SoCG 202

    Parameterized Max Min Feedback Vertex Set

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    Constrained Submodular Maximization via New Bounds for DR-Submodular Functions

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    Submodular maximization under various constraints is a fundamental problem studied continuously, in both computer science and operations research, since the late 19701970's. A central technique in this field is to approximately optimize the multilinear extension of the submodular objective, and then round the solution. The use of this technique requires a solver able to approximately maximize multilinear extensions. Following a long line of work, Buchbinder and Feldman (2019) described such a solver guaranteeing 0.3850.385-approximation for down-closed constraints, while Oveis Gharan and Vondr\'ak (2011) showed that no solver can guarantee better than 0.4780.478-approximation. In this paper, we present a solver guaranteeing 0.4010.401-approximation, which significantly reduces the gap between the best known solver and the inapproximability result. The design and analysis of our solver are based on a novel bound that we prove for DR-submodular functions. This bound improves over a previous bound due to Feldman et al. (2011) that is used by essentially all state-of-the-art results for constrained maximization of general submodular/DR-submodular functions. Hence, we believe that our new bound is likely to find many additional applications in related problems, and to be a key component for further improvement.Comment: 48 page

    FPT Approximations for Capacitated/Fair Clustering with Outliers

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    Clustering problems such as kk-Median, and kk-Means, are motivated from applications such as location planning, unsupervised learning among others. In such applications, it is important to find the clustering of points that is not ``skewed'' in terms of the number of points, i.e., no cluster should contain too many points. This is modeled by capacity constraints on the sizes of clusters. In an orthogonal direction, another important consideration in clustering is how to handle the presence of outliers in the data. Indeed, these clustering problems have been generalized in the literature to separately handle capacity constraints and outliers. To the best of our knowledge, there has been very little work on studying the approximability of clustering problems that can simultaneously handle both capacities and outliers. We initiate the study of the Capacitated kk-Median with Outliers (CkkMO) problem. Here, we want to cluster all except mm outlier points into at most kk clusters, such that (i) the clusters respect the capacity constraints, and (ii) the cost of clustering, defined as the sum of distances of each non-outlier point to its assigned cluster-center, is minimized. We design the first constant-factor approximation algorithms for CkkMO. In particular, our algorithm returns a (3+\epsilon)-approximation for CkkMO in general metric spaces, and a (1+\epsilon)-approximation in Euclidean spaces of constant dimension, that runs in time in time f(k,m,ϵ)ImO(1)f(k, m, \epsilon) \cdot |I_m|^{O(1)}, where Im|I_m| denotes the input size. We can also extend these results to a broader class of problems, including Capacitated k-Means/k-Facility Location with Outliers, and Size-Balanced Fair Clustering problems with Outliers. For each of these problems, we obtain an approximation ratio that matches the best known guarantee of the corresponding outlier-free problem.Comment: Abstract shortened to meet arxiv requirement
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