1,288 research outputs found

    On Coloring Resilient Graphs

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    We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we study rr-resiliently kk-colorable graphs, which are those kk-colorable graphs that remain kk-colorable even after the addition of any rr new edges. We prove lower bounds on the NP-hardness of coloring resiliently colorable graphs, and provide an algorithm that colors sufficiently resilient graphs. We also analyze the corresponding notion of resilience for kk-SAT. This notion of resilience suggests an array of open questions for graph coloring and other combinatorial problems.Comment: Appearing in MFCS 201

    Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

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    The study of graph products is a major research topic and typically concerns the term f(G∗H)f(G*H), e.g., to show that f(G∗H)=f(G)f(H)f(G*H)=f(G)f(H). In this paper, we study graph products in a non-standard form f(R[G∗H]f(R[G*H] where RR is a "reduction", a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. (1) A tight n1−ϵn^{1-\epsilon}-approximation hardness for the minimum consistent deterministic finite automaton (DFA) problem, where nn is the sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming NP≠RPNP\neq RP (the weakest possible assumption). (2) A tight n1/2−ϵn^{1/2-\epsilon} hardness for the edge-disjoint paths (EDP) problem on directed acyclic graphs (DAGs), where nn denotes the number of vertices. (3) A tight hardness of packing vertex-disjoint kk-cycles for large kk. (4) An alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst.Sci. 2008]

    Decentralized Constraint Satisfaction

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    We show that several important resource allocation problems in wireless networks fit within the common framework of Constraint Satisfaction Problems (CSPs). Inspired by the requirements of these applications, where variables are located at distinct network devices that may not be able to communicate but may interfere, we define natural criteria that a CSP solver must possess in order to be practical. We term these algorithms decentralized CSP solvers. The best known CSP solvers were designed for centralized problems and do not meet these criteria. We introduce a stochastic decentralized CSP solver and prove that it will find a solution in almost surely finite time, should one exist, also showing it has many practically desirable properties. We benchmark the algorithm's performance on a well-studied class of CSPs, random k-SAT, illustrating that the time the algorithm takes to find a satisfying assignment is competitive with stochastic centralized solvers on problems with order a thousand variables despite its decentralized nature. We demonstrate the solver's practical utility for the problems that motivated its introduction by using it to find a non-interfering channel allocation for a network formed from data from downtown Manhattan

    On Local Regret

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    Online learning aims to perform nearly as well as the best hypothesis in hindsight. For some hypothesis classes, though, even finding the best hypothesis offline is challenging. In such offline cases, local search techniques are often employed and only local optimality guaranteed. For online decision-making with such hypothesis classes, we introduce local regret, a generalization of regret that aims to perform nearly as well as only nearby hypotheses. We then present a general algorithm to minimize local regret with arbitrary locality graphs. We also show how the graph structure can be exploited to drastically speed learning. These algorithms are then demonstrated on a diverse set of online problems: online disjunct learning, online Max-SAT, and online decision tree learning.Comment: This is the longer version of the same-titled paper appearing in the Proceedings of the Twenty-Ninth International Conference on Machine Learning (ICML), 201

    Clustering of solutions in the random satisfiability problem

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    Using elementary rigorous methods we prove the existence of a clustered phase in the random KK-SAT problem, for K≥8K\geq 8. In this phase the solutions are grouped into clusters which are far away from each other. The results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks. It can be generalized to other systems of both physical and computational interest.Comment: 4 pages, 1 figur

    Learning-Based Constraint Satisfaction With Sensing Restrictions

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    In this paper we consider graph-coloring problems, an important subset of general constraint satisfaction problems that arise in wireless resource allocation. We constructively establish the existence of fully decentralized learning-based algorithms that are able to find a proper coloring even in the presence of strong sensing restrictions, in particular sensing asymmetry of the type encountered when hidden terminals are present. Our main analytic contribution is to establish sufficient conditions on the sensing behaviour to ensure that the solvers find satisfying assignments with probability one. These conditions take the form of connectivity requirements on the induced sensing graph. These requirements are mild, and we demonstrate that they are commonly satisfied in wireless allocation tasks. We argue that our results are of considerable practical importance in view of the prevalence of both communication and sensing restrictions in wireless resource allocation problems. The class of algorithms analysed here requires no message-passing whatsoever between wireless devices, and we show that they continue to perform well even when devices are only able to carry out constrained sensing of the surrounding radio environment

    Breaking Instance-Independent Symmetries In Exact Graph Coloring

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    Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

    On Derandomizing Local Distributed Algorithms

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    The gap between the known randomized and deterministic local distributed algorithms underlies arguably the most fundamental and central open question in distributed graph algorithms. In this paper, we develop a generic and clean recipe for derandomizing LOCAL algorithms. We also exhibit how this simple recipe leads to significant improvements on a number of problem. Two main results are: - An improved distributed hypergraph maximal matching algorithm, improving on Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for edge-coloring, maximum matching approximation, and low out-degree edge orientation. The first gives an improved algorithm for Open Problem 11.4 of the book of Barenboim and Elkin, and the last gives the first positive resolution of their Open Problem 11.10. - An improved distributed algorithm for the Lov\'{a}sz Local Lemma, which gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads to improved distributed algorithms for problems such as defective coloring and kk-SAT.Comment: 37 page
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