3,528 research outputs found

    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 α∈N\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

    Grundy Coloring & Friends, Half-Graphs, Bicliques

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    The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order ?, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering ?, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f(k)n^{2^{k-1}}-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS \u2717]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K_{t,t}-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest

    Complexity of Grundy coloring and its variants

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    The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least kk. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time O∗(2.443n)O^*(2.443^n) and WEAK GRUNDY COLORING in time O∗(2.716n)O^*(2.716^n) on graphs of order nn. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time O∗(2O(wk))O^*(2^{O(wk)}) for graphs of treewidth ww (where kk is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time O∗(2o(wlog⁡w))O^*(2^{o(w\log w)}). We also describe an O∗(22O(k))O^*(2^{2^{O(k)}}) algorithm for WEAK GRUNDY COLORING, which is therefore \fpt for the parameter kk. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for GRUNDY COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with the two other problems, we show that CONNECTED GRUNDY COLORING is \np-complete already for k=7k=7 colors.Comment: 24 pages, 7 figures. This version contains some new results and improvements. A short paper based on version v2 appeared in COCOON'1

    Fast Distributed Approximation for Max-Cut

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    Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the LOCAL\mathcal{LOCAL} or CONGEST\mathcal{CONGEST} models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any Ï”>0\epsilon > 0, we develop randomized approximation algorithms achieving a ratio of (1−ϔ)(1-\epsilon) to the optimum for Max-Cut on bipartite graphs in the CONGEST\mathcal{CONGEST} model, and on general graphs in the LOCAL\mathcal{LOCAL} model. We further present efficient deterministic algorithms, including a 1/31/3-approximation for Max-Dicut in our models, thus improving the best known (randomized) ratio of 1/41/4. Our algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest

    Leveraging Physical Layer Capabilites: Distributed Scheduling in Interference Networks with Local Views

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    In most wireless networks, nodes have only limited local information about the state of the network, which includes connectivity and channel state information. With limited local information about the network, each node's knowledge is mismatched; therefore, they must make distributed decisions. In this paper, we pose the following question - if every node has network state information only about a small neighborhood, how and when should nodes choose to transmit? While link scheduling answers the above question for point-to-point physical layers which are designed for an interference-avoidance paradigm, we look for answers in cases when interference can be embraced by advanced PHY layer design, as suggested by results in network information theory. To make progress on this challenging problem, we propose a constructive distributed algorithm that achieves rates higher than link scheduling based on interference avoidance, especially if each node knows more than one hop of network state information. We compare our new aggressive algorithm to a conservative algorithm we have presented in [1]. Both algorithms schedule sub-networks such that each sub-network can employ advanced interference-embracing coding schemes to achieve higher rates. Our innovation is in the identification, selection and scheduling of sub-networks, especially when sub-networks are larger than a single link.Comment: 14 pages, Submitted to IEEE/ACM Transactions on Networking, October 201
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