Constructive generation of very hard 3-colorability instances

Mizuno, Kazunori; Nishihara, Seiichi

doi:10.1016/j.dam.2006.07.015

Cited by 38 publications

(16 citation statements)

References 14 publications

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“…Nevertheless, there are some lines of research suggesting special distributions of graph instances on which purported NP-complete problem solvers should be evaluated in order to appropriately determine their performances (e.g., [17], [52], [53]). …”

Section: Resultsmentioning

confidence: 99%

“…Hence, an interesting theoretical analysis that should follow is to study the behavior of on 4-critical graphs since in this class, there is no subgraph with chromatic number four, and hence, finding unavoidable vertex contractions may be hard (e.g., see Ref. [53] for a good initial development of this idea). Hence, a classification of 4-critical graphs on the basis of can lead to very significant results.…”

Section: Discussionmentioning

confidence: 99%

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

Martı́n

2013

PLoS ONE

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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 if and only if 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 . Nevertheless, here it is proved that the probability of requiring a value of to obtain a solution for a random graph decreases exponentially: , 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.

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Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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

Martı́n

2013

PLoS ONE

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“…When testing these graphs for 3-colorability, Mizuno and Nishihara (2008) experimented with numerous algorithms and software platforms, but always found exponential growth in the runtime for larger and larger instances. Based on these experimental observations, the authors propose these graphs as "hard" examples of 3-colorability.…”

Section: Graph 3-coloring Instancesmentioning

confidence: 99%

The Cunningham-Geelen Method in Practice: Branch-Decompositions and Integer Programming

Margulies

Hicks

2013

INFORMS Journal on Computing

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I n 2007, W. H. Cunningham and J. Geelen describe an algorithm for solving max c Twhere A ∈ m×n ≥0 , b ∈ m , and c ∈ n , which utilizes a branch-decomposition of the matrix A and techniques from dynamic programming. In this paper, we report on the first implementation of the CG algorithm and compare our results with the commercial integer programming software Gurobi. Using branch-decomposition trees produced by heuristics and optimal trees produced by algorithms developed in our previous studies, we test both a memory-intensive and low-memory version of the CG algorithm on problem instances such as graph 3-coloring, set partition, market split, and knapsack. We isolate a class of set partition instances where the CG algorithm runs twice as fast as Gurobi, and demonstrate that certain infeasible market split and knapsack instances with width ≤ 6 range from running twice as fast as Gurobi, to running in a matter of minutes versus a matter of hours.

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“…Corollary 1.9 is consistent with their findings. Indeed, in all the graphs they present in Figure 3 (see [12]), the removal of any edge leaves a 3-colorable graph, and each of these graphs has an edge that does not lie on 3 or 4-cycle. This implies that when = GF (2), N (G) > 1 for such graphs G, so computationally determining that they are not 3-colorable is not immediate under the NulLA paradigm.…”

Section: Introductionmentioning

confidence: 99%

Low Degree Nullstellensatz Certificates for 3-Colorability

Lowenstein

Omar

2016

Electron. J. Combin.

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In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field . In this article, we give a new direct combinatorial characterization of graphs that can be determined to be non-3colorable in the first iteration of this algorithm when = GF (2). This greatly simplifies the work of De Loera et. al, as we express the combinatorial characterization directly in terms of the graphs themselves without introducing superfluous directed graphs. Furthermore, for all graphs on at most 12 vertices, we determine at which iteration NulLA detects a graph is not 3-colorable when = GF (2).

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Constructive generation of very hard 3-colorability instances

Cited by 38 publications

References 14 publications

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

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

The Cunningham-Geelen Method in Practice: Branch-Decompositions and Integer Programming

Low Degree Nullstellensatz Certificates for 3-Colorability

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