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A completely positive formulation of the graph isomorphism problem and its positive semidefinite relaxation
Journal of Combinatorial Optimization ( IF 1 ) Pub Date : 2020-06-06 , DOI: 10.1007/s10878-020-00598-w
Pawan Aurora , Shashank K. Mehta

Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) show that two graphs \(G_1,G_2\), on n vertices each, are isomorphic if and only if the feasible region of a certain linear program, LP-GI, intersects with the Quadratic Assignment Problem (QAP)-polytope in \({\mathbb {R}}^{(n^4+n^2)/2}\). The linear program LP-GI in Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) is obtained by relaxing an integer linear program whose feasible points correspond to the isomorphisms between \(G_1,G_2\). In this paper we take an analogous approach with the linear programs replaced with conic programs. A completely positive description of the QAP-polytope was obtained in Povh and Rendl (Discrete Optim 6(3):231–241, 2009). By adding the graph conditions to this description we get a completely positive formulation of the graph isomorphism problem. However, analogous to integer linear programs, it is NP-hard to optimize over the cone of completely positive matrices. So we relax this formulation by replacing the cone of completely positive matrices with the cone of positive semidefinite matrices. We observe that the resulting SDP is the Lovász Theta function (Lovász in IEEE Trans Inf Theory 25(1):1–7, 1979) of a graph product of \(G_1,G_2\) and can be efficiently computed. We provide a natural heuristic that uses the SDP to solve the graph isomorphism problem. We run our heuristic on several pairs of non-isomorphic strongly regular graphs and find the results to be encouraging. Further, by adding the non-negativity constraints to the SDP, we obtain a doubly non-negative formulation, DNN-GI. We show that if the set of optimal points in DNN-GI contains a point of rank at most 3, then the given pair of graphs must be isomorphic.

中文翻译:

图同构问题的一个完全正公式及其正半定松弛

Aurora和Mehta(J Comb Optim 36(3):965-1006,2018)表明,当且仅当某个线性程序的可行区域内,两个图\(G_1,G_2 \)分别n个顶点上是同构的, LP-GI与\({\ mathbb {R}} ^ {(n ^ 4 + n ^ 2)/ 2} \)中的二次分配问题(QAP)多位点相交。Aurora和Mehta中的线性程序LP-GI(J Comb Optim 36(3):965–1006,2018)是通过放宽一个整数线性程序而获得的,该线性程序的可行点对应于\(G_1,G_2 \)之间的同构。在本文中,我们采用类似的方法,将线性程序替换为圆锥程序。在Povh和Rendl中获得了对QAP多聚体的完全肯定的描述(Discrete Optim 6(3):231–241,2009)。通过将图形条件添加到此描述中,我们得到了图形同构问题的完全肯定的公式。但是,类似于整数线性程序,要在完全正矩阵的圆锥上进行优化是NP难的。因此,我们通过用正半定矩阵的锥代替完全正矩阵的锥来放宽此公式。我们观察到,生成的SDP是\(G_1,G_2 \)图积的LovászTheta函数(IEEE Trans Inf Theory 25(1):1–7,1979中的Lovász 并可以有效地进行计算。我们提供了一种自然的启发式方法,该方法使用SDP解决了图形同构问题。我们在几对非同构的强正则图上运行启发式方法,发现结果令人鼓舞。此外,通过将非负约束添加到SDP中,我们获得了双重非负公式DNN-GI。我们表明,如果DNN-GI中的最佳点集包含最多3个等级点,则给定的图对必须是同构的。
更新日期:2020-06-06
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