30,655 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 Ξ±>k\alpha>k to obtain a solution for a random graph decreases exponentially: P(Ξ±>k)≀2βˆ’(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

    Injective colorings of sparse graphs

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    Let mad(G)mad(G) denote the maximum average degree (over all subgraphs) of GG and let Ο‡i(G)\chi_i(G) denote the injective chromatic number of GG. We prove that if mad(G)≀5/2mad(G) \leq 5/2, then Ο‡i(G)≀Δ(G)+1\chi_i(G)\leq\Delta(G) + 1; and if mad(G)<42/19mad(G) < 42/19, then Ο‡i(G)=Ξ”(G)\chi_i(G)=\Delta(G). Suppose that GG is a planar graph with girth g(G)g(G) and Ξ”(G)β‰₯4\Delta(G)\geq 4. We prove that if g(G)β‰₯9g(G)\geq 9, then Ο‡i(G)≀Δ(G)+1\chi_i(G)\leq\Delta(G)+1; similarly, if g(G)β‰₯13g(G)\geq 13, then Ο‡i(G)=Ξ”(G)\chi_i(G)=\Delta(G).Comment: 10 page
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