Integration of AI and OR Techniques in Constraint Programming
Integration of AI and OR Techniques in Constraint Programming
复制标题
约束规划中 AI 和 OR 技术的集成
DOI：
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发表时间：
2016
期刊：
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通讯作者：
M. Lombardi
中科院分区：
文献类型：
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作者：
Domenico Salvagnin;M. Lombardi
图着色问题：拉姆齐数R(4，3，3)=30 Michael Coish，Michael Frank，Avraham Itzhakov，and Miller 1 of the Negev，Ben-Gurion University of the Negev，Beersheba，以色列2计算科学学院，格拉斯哥大学，格拉斯哥，苏格兰，拉姆齐数是出了名的难图着色问题。一个独立的R1.。.；Rk；n？Ramsey着色是对完全图Kn的k个颜色进行着色的图，对于每个1i k，它不包含颜色为i的单色完全子图Kri。所有这样的着色的集合被表示为R(？)。.；rk；n？拉姆齐数Rçr1；。.；Rk？是最小的n[0，使得不存在唯一的R_1；。.；rk；n？存在着色。拉姆齐数R；3；3？通常被表示为未知的拉姆齐数，最有可能“很快”被发现。然而，50多年来，它的确切价值一直不得而知。本文提出了一种基于抽象和对称破缺的方法，并用它计算了Rin4；3；3？1⁄4 30的值。以前已知的是30R in4；3；3？31[4]。1966年，Kalbfleisch[2]证明了R in4；3；3？30，Piwakowski[3]在1997年证明了R in4；3；3？32，一年后Piwakowski和Radziszowski[4]证明了R in4；3；3？31。我们演示了我们的方法论如何应用于计算证明R？1⁄4 30。我们的方法涉及到应用嵌入技术来得出结论：如果存在α4；3；3；30？Ramsey染色，则它一定是H13；8；8i正则的。若要确定是否存在H13；8；8i正则图4；3；3；30？Ramsey着色需要首先计算先前未知的集合R in3；3；3；13？，它被证明具有78,892的大小。为了做到这一点，我们证明了现有的结合SAT求解和对称破缺的对称破缺技术[1]适用于较小的实例，但不适用于R-3；3；3；13？取而代之的是，我们使用了一种称为度矩阵的新抽象。在确定了Rα3；3；3；13？之后，我们将其应用于嵌入方法中，从而得到了本文的主要结果：不存在α4；3；3；30？Ramsey染色，因此Rα4；3；3？1⁄4 30.由以色列科学基金会支持，拨款82/13。计算资源由IBM共享大学奖(以色列)提供。
s of Fast Tracked Journal Papers Breaking Symmetries in Graph Coloring Problems with Degree Matrices: The Ramsey Number R(4, 3, 3) = 30 Michael Codish, Michael Frank, Avraham Itzhakov, and Alice Miller 1 Department of Computer Science, Ben-Gurion University of the Negev, Beersheba, Israel 2 School of Computing Science, University of Glasgow, Glasgow, Scotland Ramsey numbers are notoriously hard graph coloring problems. An ðr1; . . .; rk; nÞ Ramsey coloring is a graph coloring in k colors of the complete graph Kn that does not contain a monochromatic complete sub-graph Kri in color i for each 1 i k. The set of all such colorings is denoted Rðr1; . . .; rk; nÞ. The Ramsey number Rðr1; . . .; rkÞ is the least n[ 0 such that no ðr1; . . .; rk; nÞ coloring exists. The Ramsey number Rð4; 3; 3Þ is often presented as the unknown Ramsey number with the best chance of being found “soon”. Yet, its precise value has remained unknown for more than 50 years. This paper presents a methodology based on abstraction and symmetry breaking that is demonstrated by using it to compute the value Rð4; 3; 3Þ 1⁄4 30. It was previously known that 30 Rð4; 3; 3Þ 31 [4]. Kalbfleisch [2] proved in 1966 that Rð4; 3; 3Þ 30, Piwakowski [3] proved in 1997 that Rð4; 3; 3Þ 32, and one year later Piwakowski and Radziszowski [4] proved that Rð4; 3; 3Þ 31. We demonstrate how our methodology applies to computationally prove that Rð4; 3; 3Þ 1⁄4 30. Our approach involves applying an embedding technique to conclude that if a ð4; 3; 3; 30Þ Ramsey coloring exists then it must be h13; 8; 8i regular. To determine if there exists a h13; 8; 8i regular ð4; 3; 3; 30Þ Ramsey coloring required first computing the previously unknown set Rð3; 3; 3; 13Þ, which was shown to have size 78,892. To do this we demonstrate that an existing symmetry breaking technique combining SAT solving with symmetry breaking [1] works for smaller instances but not for Rð3; 3; 3; 13Þ. Instead we use a new abstraction referred to as degree matrices. Having determined Rð3; 3; 3; 13Þ we then use it within the embedding approach to achieve the major result of this paper: that there is no ð4; 3; 3; 30Þ Ramsey coloring, and so Rð4; 3; 3Þ 1⁄4 30. Supported by the Israel Science Foundation, grant 82/13. Computational resources provided by an IBM Shared University Award (Israel).