11,348 research outputs found

    Dynamic Algorithms for Graph Coloring

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    We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for (Δ+1)(\Delta+1)- vertex coloring and (2Δ1)(2\Delta-1)-edge coloring in a graph with maximum degree Δ\Delta. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results. (1) We present a randomized algorithm which maintains a (Δ+1)(\Delta+1)-vertex coloring with O(logΔ)O(\log \Delta) expected amortized update time. (2) We present a deterministic algorithm which maintains a (1+o(1))Δ(1+o(1))\Delta-vertex coloring with O(polylogΔ)O(\text{poly} \log \Delta) amortized update time. (3) We present a simple, deterministic algorithm which maintains a (2Δ1)(2\Delta-1)-edge coloring with O(logΔ)O(\log \Delta) worst-case update time. This improves the recent O(Δ)O(\Delta)-edge coloring algorithm with O~(Δ)\tilde{O}(\sqrt{\Delta}) worst-case update time by Barenboim and Maimon.Comment: To appear in SODA 201

    Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm

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    Many practical problems in almost all scientific and technological disciplines have been classified as computationally hard (NP-hard or even NP-complete). In life sciences, combinatorial optimization problems frequently arise in molecular biology, e.g., genome sequencing; global alignment of multiple genomes; identifying siblings or discovery of dysregulated pathways.In almost all of these problems, there is the need for proving a hypothesis about certain property of an object that can be present only when it adopts some particular admissible structure (an NP-certificate) or be absent (no admissible structure), however, none of the standard approaches can discard the hypothesis when no solution can be found, since none can provide a proof that there is no admissible structure. This article presents an algorithm that introduces a novel type of solution method to "efficiently" solve the graph 3-coloring problem; an NP-complete problem. The proposed method provides certificates (proofs) in both cases: present or absent, so it is possible to accept or reject the hypothesis on the basis of a rigorous proof. It provides exact solutions and is polynomial-time (i.e., efficient) however parametric. The only requirement is sufficient computational power, which is controlled by the parameter αN\alpha\in\mathbb{N}. Nevertheless, here it is proved that the probability of requiring a value of α>k\alpha>k to obtain a solution for a random graph decreases exponentially: P(α>k)2(k+1)P(\alpha>k) \leq 2^{-(k+1)}, making tractable almost all problem instances. Thorough experimental analyses were performed. The algorithm was tested on random graphs, planar graphs and 4-regular planar graphs. The obtained experimental results are in accordance with the theoretical expected results.Comment: Working pape

    Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

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    We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank rr. Our main result is a deterministic algorithm to generate a matching which is an O(r)O(r)-approximation to the maximum weight matching, running in O~(rlogΔ+log2Δ+logn)\tilde O(r \log \Delta + \log^2 \Delta + \log^* n) rounds. (Here, the O~()\tilde O() notations hides polyloglog Δ\text{polyloglog } \Delta and polylog r\text{polylog } r factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris & Kuhn (2017). As a main application, we obtain nearly-optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a (1+ϵ)(1+\epsilon) approximation algorithm using O~(logΔ/ϵ3+polylog(1/ϵ,loglogn))\tilde O(\log \Delta / \epsilon^3 + \text{polylog}(1/\epsilon, \log \log n)) randomized time and O~(log2Δ/ϵ4+logn/ϵ)\tilde O(\log^2 \Delta / \epsilon^4 + \log^*n / \epsilon) deterministic time. The second application is a faster algorithm for hypergraph maximal matching, a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of local graph algorithms. This gives an algorithm for (2Δ1)(2 \Delta - 1)-edge-list coloring in O~(log2Δlogn)\tilde O(\log^2 \Delta \log n) rounds deterministically or O~((loglogn)3)\tilde O( (\log \log n)^3 ) rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most (1+ϵ)λ\lceil (1+\epsilon) \lambda \rceil for a graph of arboricity λ\lambda; for fixed ϵ\epsilon this runs in O~(log6n)\tilde O(\log^6 n) rounds deterministically or O~(log3n)\tilde O(\log^3 n ) rounds randomly
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