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Register a new userExtended Formulation for CSP that is Compact for Instances of Bounded Treewidth
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Added by
Martin Koutecky
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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In this paper we provide an extended formulation for the class of constraint satisfaction problems and prove that its size is polynomial for instances whose constraint graph has bounded treewidth. This implies new upper bounds on extension complexity of several important NP-hard problems on graphs of bounded treewidth.
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We show that CSP is fixed-parameter tractable when parameterized by the treewidth of a backdoor into any tractable CSP problem over a finite constraint language. This result combines the two prominent approaches for achieving tractability for CSP: (i) by structural restrictions on the interaction between the variables and the constraints and (ii) by language restrictions on the relations that can be used inside the constraints. Apart from defining the notion of backdoor-treewidth and showing how backdoors of small treewidth can be used to efficiently solve CSP, our main technical contribution is a fixed-parameter algorithm that finds a backdoor of small treewidth.
We present an algorithm for strongly refuting smoothed instances of all Boolean CSPs. The smoothed model is a hybrid between worst and average-case input models, where the input is an arbitrary instance of the CSP with only the negation patterns of the literals re-randomized with some small probability. For an $n$-variable smoothed instance of a $k$-arity CSP, our algorithm runs in $n^{O(ell)}$ time, and succeeds with high probability in bounding the optimum fraction of satisfiable constraints away from $1$, provided that the number of constraints is at least $tilde{O}(n) (frac{n}{ell})^{frac{k}{2} - 1}$. This matches, up to polylogarithmic factors in $n$, the trade-off between running time and the number of constraints of the state-of-the-art algorithms for refuting fully random instances of CSPs [RRS17]. We also make a surprising new connection between our algorithm and even covers in hypergraphs, which we use to positively resolve Feiges 2008 conjecture, an extremal combinatorics conjecture on the existence of even covers in sufficiently dense hypergraphs that generalizes the well-known Moore bound for the girth of graphs. As a corollary, we show that polynomial-size refutation witnesses exist for arbitrary smoothed CSP instances with number of constraints a polynomial factor below the spectral threshold of $n^{k/2}$, extending the celebrated result for random 3-SAT of Feige, Kim and Ofek [FKO06].
We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems are essential to data analysis and used for example in road networks and data visualization. A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set, and it offers significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings. Our main result is a near-linear time construction of a coreset for k-Median in a general graph $G$, with size $O_{epsilon, k}(mathrm{tw}(G))$ where $mathrm{tw}(G)$ is the treewidth of $G$, and we complement the construction with a nearly-tight size lower bound. The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of $O(mathrm{tw}(G))$ on the shattering dimension under any point weights. We validate our coreset on real-world road networks, and our scalable algorithm constructs tiny coresets with high accuracy, which translates to a massive speedup of existing approximation algorithms such as local search for graph k-Median.
We show that any quantum circuit of treewidth $t$, built from $r$-qubit gates, requires at least $Omega(frac{n^{2}}{2^{O(rcdot t)}cdot log^4 n})$ gates to compute the element distinctness function. Our result generalizes a near-quadratic lower bound for quantum formula size obtained by Roychowdhury and Vatan [SIAM J. on Computing, 2001]. The proof of our lower bound follows by an extension of Nev{c}iporuks method to the context of quantum circuits of constant treewidth. This extension is made via a combination of techniques from structural graph theory, tensor-network theory, and the connected-component counting method, which is a classic tool in algebraic geometry.
The Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-coding that runs in time $O(n^{tw(H)+1})$ [Alon, Yuster, Zwick95], where $n$ is the number of vertices of the host graph $G$. While there are pattern graphs known for which Subgraph Isomorphism can be solved in an improved running time of $O(n^{tw(H)+1-varepsilon})$ or even faster (e.g. for $k$-cliques), it is not known whether such improvements are possible for all patterns. The only known lower bound rules out time $n^{o(tw(H) / log(tw(H)))}$ for any class of patterns of unbounded treewidth assuming the Exponential Time Hypothesis [Marx07]. In this paper, we demonstrate the existence of maximally hard pattern graphs $H$ that require time $n^{tw(H)+1-o(1)}$. Specifically, under the Strong Exponential Time Hypothesis (SETH), a standard assumption from fine-grained complexity theory, we prove the following asymptotic statement for large treewidth $t$: For any $varepsilon > 0$ there exists $t ge 3$ and a pattern graph $H$ of treewidth $t$ such that Subgraph Isomorphism on pattern $H$ has no algorithm running in time $O(n^{t+1-varepsilon})$. Under the more recent 3-uniform Hyperclique hypothesis, we even obtain tight lower bounds for each specific treewidth $t ge 3$: For any $t ge 3$ there exists a pattern graph $H$ of treewidth $t$ such that for any $varepsilon>0$ Subgraph Isomorphism on pattern $H$ has no algorithm running in time $O(n^{t+1-varepsilon})$. In addition to these main results, we explore (1) colored and uncolored problem variants (and why they are equivalent for most cases), (2) Subgraph Isomorphism for $tw < 3$, (3) Subgraph Isomorphism parameterized by pathwidth, and (4) a weighted problem variant.
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