160 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Parameterized Graph Modification Beyond the Natural Parameter

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    Parameterized Complexity of Equality MinCSP

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    We study the parameterized complexity of MinCSP for so-called equality languages, i.e., for finite languages over an infinite domain such as ?, where the relations are defined via first-order formulas whose only predicate is =. This is an important class of languages that forms the starting point of all study of infinite-domain CSPs under the commonly used approach pioneered by Bodirsky, i.e., languages defined as reducts of finitely bounded homogeneous structures. Moreover, MinCSP over equality languages forms a natural class of optimisation problems in its own right, covering such problems as Edge Multicut, Steiner Multicut and (under singleton expansion) Edge Multiway Cut. We classify MinCSP(?) for every finite equality language ?, under the natural parameter, as either FPT, W[1]-hard but admitting a constant-factor FPT-approximation, or not admitting a constant-factor FPT-approximation unless FPT=W[2]. In particular, we describe an FPT case that slightly generalises Multicut, and show a constant-factor FPT-approximation for Disjunctive Multicut, the generalisation of Multicut where the "cut requests" come as disjunctions over O(1) individual cut requests s_i ? t_i. We also consider singleton expansions of equality languages, enriching an equality language with the capability for assignment constraints (x = i) for either a finite or infinitely many constants i, and fully characterize the complexity of the resulting MinCSP

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Improved Hardness of Approximating k-Clique under ETH

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    In this paper, we prove that assuming the exponential time hypothesis (ETH), there is no f(k)nko(1/loglogk)f(k)\cdot n^{k^{o(1/\log\log k)}}-time algorithm that can decide whether an nn-vertex graph contains a clique of size kk or contains no clique of size k/2k/2, and no FPT algorithm can decide whether an input graph has a clique of size kk or no clique of size k/f(k)k/f(k), where f(k)f(k) is some function in k1o(1)k^{1-o(1)}. Our results significantly improve the previous works [Lin21, LRSW22]. The crux of our proof is a framework to construct gap-producing reductions for the kk-Clique problem. More precisely, we show that given an error-correcting code C:Σ1kΣ2kC:\Sigma_1^k\to\Sigma_2^{k'} that is locally testable and smooth locally decodable in the parallel setting, one can construct a reduction which on input a graph GG outputs a graph GG' in (k)O(1)nO(logΣ2/logΣ1)(k')^{O(1)}\cdot n^{O(\log|\Sigma_2|/\log|\Sigma_1|)} time such that: \bullet If GG has a clique of size kk, then GG' has a clique of size KK, where K=(k)O(1)K = (k')^{O(1)}. \bullet If GG has no clique of size kk, then GG' has no clique of size (1ε)K(1-\varepsilon)\cdot K for some constant ε(0,1)\varepsilon\in(0,1). We then construct such a code with k=kΘ(loglogk)k'=k^{\Theta(\log\log k)} and Σ2=Σ1k0.54|\Sigma_2|=|\Sigma_1|^{k^{0.54}}, establishing the hardness results above. Our code generalizes the derivative code [WY07] into the case with a super constant order of derivatives.Comment: 48 page

    Parameterized Graph Modification Beyond the Natural Parameter

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    Fully dynamic approximation schemes on planar and apex-minor-free graphs

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    The classic technique of Baker [J. ACM '94] is the most fundamental approach for designing approximation schemes on planar, or more generally topologically-constrained graphs, and it has been applied in a myriad of different variants and settings throughout the last 30 years. In this work we propose a dynamic variant of Baker's technique, where instead of finding an approximate solution in a given static graph, the task is to design a data structure for maintaining an approximate solution in a fully dynamic graph, that is, a graph that is changing over time by edge deletions and edge insertions. Specifically, we address the two most basic problems -- Maximum Weight Independent Set and Minimum Weight Dominating Set -- and we prove the following: for a fully dynamic nn-vertex planar graph GG, one can: * maintain a (1ε)(1-\varepsilon)-approximation of the maximum weight of an independent set in GG with amortized update time f(ε)no(1)f(\varepsilon)\cdot n^{o(1)}; and, * under the additional assumption that the maximum degree of the graph is bounded at all times by a constant, also maintain a (1+ε)(1+\varepsilon)-approximation of the minimum weight of a dominating set in GG with amortized update time f(ε)no(1)f(\varepsilon)\cdot n^{o(1)}. In both cases, f(ε)f(\varepsilon) is doubly-exponential in poly(1/ε)\mathrm{poly}(1/\varepsilon) and the data structure can be initialized in time f(ε)n1+o(1)f(\varepsilon)\cdot n^{1+o(1)}. All our results in fact hold in the larger generality of any graph class that excludes a fixed apex-graph as a minor.Comment: 37 pages, accepted to SODA '2

    Parameterized Complexity Classification for Interval Constraints

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    Constraint satisfaction problems form a nicely behaved class of problems that lends itself to complexity classification results. From the point of view of parameterized complexity, a natural task is to classify the parameterized complexity of MinCSP problems parameterized by the number of unsatisfied constraints. In other words, we ask whether we can delete at most kk constraints, where kk is the parameter, to get a satisfiable instance. In this work, we take a step towards classifying the parameterized complexity for an important infinite-domain CSP: Allen's interval algebra (IA). This CSP has closed intervals with rational endpoints as domain values and employs a set AA of 13 basic comparison relations such as ``precedes'' or ``during'' for relating intervals. IA is a highly influential and well-studied formalism within AI and qualitative reasoning that has numerous applications in, for instance, planning, natural language processing and molecular biology. We provide an FPT vs. W[1]-hard dichotomy for MinCSP(Γ)(\Gamma) for all ΓA\Gamma \subseteq A. IA is sometimes extended with unions of the relations in AA or first-order definable relations over AA, but extending our results to these cases would require first solving the parameterized complexity of Directed Symmetric Multicut, which is a notorious open problem. Already in this limited setting, we uncover connections to new variants of graph cut and separation problems. This includes hardness proofs for simultaneous cuts or feedback arc set problems in directed graphs, as well as new tractable cases with algorithms based on the recently introduced flow augmentation technique. Given the intractability of MinCSP(A)(A) in general, we then consider (parameterized) approximation algorithms and present a factor-22 fpt-approximation algorithm

    Baby PIH: Parameterized Inapproximability of Min CSP

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    The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only (1ε)(1-\varepsilon)-satisfiable (where the parameter is the number of variables) for some constant 0<ε<10<\varepsilon<1. We consider a minimization version of CSPs (Min-CSP), where one may assign rr values to each variable, and the goal is to ensure that every constraint is satisfied by some choice among the r×rr \times r pairs of values assigned to its variables (call such a CSP instance rr-list-satisfiable). We prove the following strong parameterized inapproximability for Min CSP: For every r1r \ge 1, it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even rr-list-satisfiable. We refer to this statement as "Baby PIH", following the recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the combinatorial arguments underlying the Baby PCP theorem, overcoming some basic obstacles that arise in the parameterized setting. Furthermore, our reduction runs in time polynomially bounded in both the number of variables and the alphabet size, and thus implies the Baby PCP theorem as well

    On streaming approximation algorithms for constraint satisfaction problems

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    In this thesis, we explore streaming algorithms for approximating constraint satisfaction problems (CSPs). The setup is roughly the following: A computer has limited memory space, sees a long "stream" of local constraints on a set of variables, and tries to estimate how many of the constraints may be simultaneously satisfied. The past ten years have seen a number of works in this area, and this thesis includes both expository material and novel contributions. Throughout, we emphasize connections to the broader theories of CSPs, approximability, and streaming models, and highlight interesting open problems. The first part of our thesis is expository: We present aspects of previous works that completely characterize the approximability of specific CSPs like Max-Cut and Max-Dicut with n\sqrt{n}-space streaming algorithm (on nn-variable instances), while characterizing the approximability of all CSPs in n\sqrt n space in the special case of "composable" (i.e., sketching) algorithms, and of a particular subclass of CSPs with linear-space streaming algorithms. In the second part of the thesis, we present two of our own joint works. We begin with a work with Madhu Sudan and Santhoshini Velusamy in which we prove linear-space streaming approximation-resistance for all ordering CSPs (OCSPs), which are "CSP-like" problems maximizing over sets of permutations. Next, we present joint work with Joanna Boyland, Michael Hwang, Tarun Prasad, and Santhoshini Velusamy in which we investigate the n\sqrt n-space streaming approximability of symmetric Boolean CSPs with negations. We give explicit n\sqrt n-space sketching approximability ratios for several families of CSPs, including Max-kkAND; develop simpler optimal sketching approximation algorithms for threshold predicates; and show that previous lower bounds fail to characterize the n\sqrt n-space streaming approximability of Max-33AND.Comment: Harvard College senior thesis; 119 pages plus references; abstract shortened for arXiv; formatted with Dissertate template (feel free to copy!); exposits papers arXiv:2105.01782 (APPROX 2021) and arXiv:2112.06319 (APPROX 2022
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