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A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem

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Abstract

The minimum cut problem, MC, and the special case of the vertex separator problem, consists in partitioning the set of nodes of a graph G into k subsets of given sizes in order to minimize the number of edges cut after removing the k-th set. Previous work on approximate solutions uses, in increasing strength and expense: eigenvalue, semidefinite programming, SDP, and doubly nonnegative, DNN, bounding techniques. In this paper, we derive strengthened SDP and DNN relaxations, and we propose a scalable algorithmic approach for efficiently evaluating, theoretically verifiable, both upper and lower bounds. Our stronger relaxations are based on a new gangster set, and we demonstrate how facial reduction, FR, fits in well to allow for regularized relaxations. Moreover, the FR appears to be perfectly well suited for a natural splitting of variables, and thus for the application of splitting methods. Here, we adopt the strictly contractive Peaceman-Rachford splitting method, sPRSM. Further, we bring useful redundant constraints back into the subproblems, and show empirically that this accelerates sPRSM.In addition, we employ new strategies for obtaining lower bounds and upper bounds of the optimal value of MC from approximate iterates of the sPRSM thus aiding in early termination of the algorithm. We compare our approach with others in the literature on random datasets and vertex separator problems. This illustrates the efficiency and robustness of our proposed method.

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Notes

  1. There are several ways of constructing such a matrix \({\widetilde{V}}\). Another way is presented in (2.45), below.

  2. The name gangster refers to shooting holes in the matrix, a term coined originally by Philippe Toint.

  3. The singularity degree is essentially the minimum number of FR steps needed to find the so-called minimal face, the smallest face containing the feasible set. The singularity degree depends on the data of the problem, i.e., the linear constraints, \({{\mathcal {A}}} (X) = b\), and the semidefinite cone. For the original SDP relaxation before FR, it can be shown that the singularity degree is one, i.e., one can use the lifted linear equality constraints to find an exposing vector and use it to construct the matrix \({\widehat{V}}\).

  4. Robinson regularity: strict feasibility holds and the linear constraints are onto, [23].

  5. There is a misprint error in [29, Lemma 4.1]: the variable Z in item (c) should be Y.

  6. Strict feasibility holds and the linear constraints are onto, [23].

  7. This strengthens [18, Lemma 3.2].

  8. Note that the inner maximization forces \(Y={\widehat{V}} R{\widehat{V}}^T\).

  9. Note that the Lagrangian is linear in R, Y and linear in Z. Moreover, both constraint sets \({{\mathcal {R}}} ,{{\mathcal {Y}}} \) are convex and compact. Therefore, the result also follows from the classical Von Neumann-Fan minmax theorem.

  10. Note that if \(Y^{\text {out}}\) is rank-1 and feasible, then the first two methods in Item 1a and Item 1b yield exact solutions to MC. This motivates the use of eigenvector information.

  11. MATLAB: \(r = \min (\hbox {sum}(\lambda /(n+1)>0.1)+1,n+1)\);

  12. The DNN relaxation in [19] imposes the additional nonnegativity constraints \({\widehat{V}} Z{\widehat{V}}^T\ge 0\) onto their \({\mathbf{SDP}} _{\text {final}}\) relaxation.

  13. Note that our data are integral and we round up the lower bound, therefore the gap is integer valued. Thus, finding a zero duality gap is reasonable. Moreover, the lower bounds are nonnegative.

  14. In this case and the next, we have \(m_i=1\) for some i. One typical application is the side chain positioning problem that involves rotamer selections in protein folding. Some rotamer sets typically often only one rotamer, see [5, 13].

  15. As pointed out by one reviewer, in this case, we have \(n= k\), and the objective function in (1.1) is the number of nonzero entries that remain in the upper triangular part of \(X^TAX\) after removing the last row and column. Thus, an optimal solution \(X\in {{\mathcal {M}}}_e\) is a permutation matrix that puts the vertex with the highest degree as vertex n. Here, we include these instances to test the effectiveness of our new gangster constraints.

  16. In MATLAB: A = abs(sprandsym(sum(m),densityA))\(>0\); A = A - diag(diag(A)); Note that densityA is different from the density of the graph p defined in Table 1.

  17. These results use extra cutting planes, and therefore they obtain stronger lower bounds on \({{\,{\mathrm{{cut}}}\,}}(m)\).

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, China

    Xinxin Li

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, The People’s Republic of China

    Ting Kei Pong

  3. Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Canada

    Hao Sun & Henry Wolkowicz

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  1. Xinxin Li
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  2. Ting Kei Pong
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Correspondence to Henry Wolkowicz.

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This paper is partially based on the Master′s thesis of Hao Sun [27]. The authors Hao Sun and Henry Wolkowicz thank the Natural Sciences and Engineering Research Council of Canada for their support. Xinxin Li′s research was supported by the National Natural Science Foundation of China (No. 11601183, No. 61872162), Natural Science Foundation for Young Scientist of Jilin Province (No. 20180520212JH) and the China Scholarship Council (No. 201806175127). Ting Kei Pong′s research was supported partly by Hong Kong Research Grants Council PolyU153004/18p.

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Li, X., Pong, T.K., Sun, H. et al. A strictly contractive Peaceman-Rachford splitting method for the doubly nonnegative relaxation of the minimum cut problem. Comput Optim Appl 78, 853–891 (2021). https://doi.org/10.1007/s10589-020-00261-4

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