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INFORMS journal on computing

Exact Facetial Odd-Cycle Separation for Maximum Cut and Binary Quadratic Optimization

Michael Jünger, Sven Mallach

Published Feb 11, 2021• 6 citations
Publisher

Abstract

The exact solution of the NP-hard (nondeterministic polynomial-time hard) maximum cut problem is important in many applications across, for example, physics, chemistry, neuroscience, and circuit layout—which is also due to its equivalence to the unconstrained binary quadratic optimization problem. Leading solution methods are based on linear or semidefinite programming and require the separation of the so-called odd-cycle inequalities. In their groundbreaking research, F. Barahona and A. R. Mahjoub have given an informal description of a polynomial-time algorithm for this problem. As pointed out recently, however, additional effort is necessary to guarantee that the inequalities obtained correspond to facets of the cut polytope. In this paper, we shed more light on a so enhanced separation procedure and investigate experimentally how it performs in comparison with an ideal setting where one could even employ the sparsest, most violated, or geometrically most promising facet-defining odd-cycle inequalities. Summary of Contribution: This paper aims at a better capability to solve binary quadratic optimization or maximum cut problems and their various applications using integer programming techniques. To this end, the paper describes enhancements to a well-known algorithm for the central separation problem arising in this context; it is demonstrated experimentally that these enhancements are worthwhile from a computational point of view. The linear relaxations of the aforementioned problems are typically solved using fewer iterations and cutting planes than with a nonenhanced approach. It is also shown that the enhanced procedure is only slightly inferior to an ideal, enumerative, and, in practice, intractable global cutting-plane selection.

DOI
10.1287/ijoc.2020.1008

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The exact solution of the NP-hard (nondeterministic polynomial-time hard) maximum cut problem is important in many applications across, for example, physics, chemistry, neuroscience, and circuit layout—which is also due to its equivalence to the unconstrained binary quadratic optimization problem. Leading solution methods are based on linear or semidefinite programming and require the separation of the so-called odd-cycle inequalities. In their groundbreaking research, F. Barahona and A. R. Mahjoub have given an informal description of a polynomial-time algorithm for this problem. As pointed out recently, however, additional effort is necessary to guarantee that the inequalities obtained correspond to facets of the cut polytope. In this paper, we shed more light on a so enhanced separation procedure and investigate experimentally how it performs in comparison with an ideal setting where one could even employ the sparsest, most violated, or geometrically most promising facet-defining odd-cycle inequalities. Summary of Contribution: This paper aims at a better capability to solve binary quadratic optimization or maximum cut problems and their various applications using integer programming techniques. To this end, the paper describes enhancements to a well-known algorithm for the central separation problem arising in this context; it is demonstrated experimentally that these enhancements are worthwhile from a computational point of view. The linear relaxations of the aforementioned problems are typically solved using fewer iterations and cutting planes than with a nonenhanced approach. It is also shown that the enhanced procedure is only slightly inferior to an ideal, enumerative, and, in practice, intractable global cutting-plane selection.
Abstract The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the equivalent quadratic unconstrained binary optimization problem have experienced much interest in recent years. This article aims to advance the state of the art in the exact solution of both problems—by using mathematical programming techniques. The main focus lies on sparse problem instances, although also dense ones can be solved. We enhance several algorithmic components such as reduction techniques and cutting-plane separation algorithms, and combine them in an exact branch-and-cut solver. Furthermore, we provide a parallel implementation. The new solver is shown to significantly outperform existing state-of-the-art software for sparse maximum-cut and quadratic unconstrained binary optimization instances. Furthermore, we improve the best known bounds for several instances from the 7th DIMACS Challenge and the QPLIB, and solve some of them (for the first time) to optimality.
The maximum-cut problem is one of the fundamental problems in combinatorial optimization. With the advent of quantum computers, both the maximum-cut and the equivalent quadratic unconstrained binary optimization problem have experienced much interest in recent years. This article aims to advance the state of the art in the exact solution of both problems -- by using mathematical programming techniques on digital computers. The main focus lies on sparse problem instances, although also dense ones can be solved. We enhance several algorithmic components such as reduction techniques and cutting-plane separation algorithms, and combine them in an exact branch-and-cut solver. Furthermore, we provide a parallel implementation. The new solver is shown to significantly outperform existing state-of-the-art software for sparse MaxCut and QUBO instances. Furthermore, we improve the best known bounds for several instances from the 7th DIMACS Challenge and the QPLIB, and solve some of them (for the first time) to optimality.
Abstract Cutting planes from the Boolean Quadric Polytope can be used to reduce the optimality gap of the NP\mathcal {NP}NP NP -hard nonconvex quadratic program with box constraints (BoxQP). It is known that all cuts of the Chvátal–Gomory closure of the Boolean Quadric Polytope are A-odd cycle inequalities. We obtain a compact extended relaxation of allA-odd cycle inequalities, which allows to optimize over the Chvátal–Gomory closure without repeated calls to separation algorithms and has less inequalities than the formulation provided by Boros et al. (SIAM J Discrete Math 5(2):163–177, 1992) for sparse matrices. In a computational study, we confirm the strength of this relaxation and show that we can provide very strong bounds for the BoxQP, even with a plain linear program. The resulting bounds are significantly stronger than these from Bonami et al. (Math Program Comput 10(3):333–382, 2018), which arise from separating A-odd cycle inequalities heuristically.
My work focuses on cutting planes technology in Mixed Integer Programming. I explore novel classes of valid linear inequalities to strengthen linear relaxations of both Linear and Nonlinear Mixed Integer problems. My dissertation consists of three chapters that investigate theoretically and computationally the families of cuts considered. The first chapter is based on joint work with Prof. Pietro Belotti and Prof. Francois Margot. We study linear relaxations of Quadratically Constrained Quadratic Programs. The proposed relaxations are models with both semi-definite constraints (PSD) and linear constraints given by the Reformulation Linearization Technique (RLT). It is known from the literature (Anstreicher, 2007) that PSD and RLT used together yield better bounds than each technique used separately. We adopt a linear outer-approximation of the PSD cone, and we use exclusively linear programming tools to enforce the PSD condition via a cutting plane approach in the lifted space containing the Yij = xixj variables. We study new classes of valid linear inequalities and we test their effectiveness empirically. These include sparse PSD cuts and cuts derived from principal minors. Computational results based on instances from the GLOBALLib and Boxed Constrained Quadratic Programs show that this approach yields better bounds than using solely the standard PSD cuts on top of the RLT inequalities. The C++ code developed for this study has been included in Coin-OR as part of the Couenne project (exact solver for MINLPs). In the second chapter I present a work closely related to the recent developments in the area of \cuts from multiple rows of the simplex tableau" (Andersen et al., 2007). This chapter is based on joint work with Prof. Egon Balas. We generate intersection cuts from lattice-free convex sets as lift-and-project cuts from multiple-term disjunctions. We use the concept of \Disjunctive Hull" defined for a Mixed Integer Program at a fractional vertex v of its linear relaxation P as the convex hull of points in P satisfying all basic disjunctions that cut o v but no integer point. We examine the relationship between the Disjunctive Hull and the Integer Hull and we give procedures to generate inequalities for the Integer Hull derived from the Cut Generating Linear Program associated to the Disjunctive Hull. Strengthening techniques based on coefficient modularization and monoidal strengthening are also discussed. In this chapter we also analyze the case of 0-1 programming which has not been covered in the literature. Our framework applies to this setting with minor changes and produces valid families of cuts for the 0-1 case but invalid for general integer programs. These cuts include the triangle, quadrilateral, split cuts for the MIP case, and cuts from cones and truncated cones for the 0-1 setting. Moreover we run an experiment in which we separate two families of non facet defining cuts: we consider cuts from fixed shape triangles of type 1 and conic disjunctions having the apex at one vertex of the unit cube and two extreme rays, each containing an additional vertex of the cube. In practice, in order to match the strength of the non facet defining inequalities, a large number of facet defining inequalities is typically needed. Triangles of type 1 produce a consistent improvement over the Gomory cuts, as shown in experiments on the MIPLIB 3 0-1 instances. In the third chapter I present a theoretical result on strengthening valid inequalities for Mixed Integer Linear Programs. This chapter is also based on joint work with Prof. Egon Balas. There are two distinct strengthening methods for disjunctive cuts; they differ in the way they modularize the coefficients associated to integer constrained variables. We introduce a new variant of one of these methods, the monoidal cut strengthening procedure (Balas and Jeroslow, 1980), based on the paradox that sometimes weakening a disjunction helps the strengthening procedure and results in sharper cuts. We first derive a general result that applies to cuts from disjunctions with any number of terms. It defines the coefficients of the cut in a way that takes advantage of the option of adding new terms to the disjunction. We then specialize this result to the case of split cuts, in particular Gomory Mixed Integer cuts, and to intersection cuts from multiple rows of a simplex tableau. In both instances we give conditions for the new cuts to have stronger coefficients than the cuts obtained by either of the two currently known strengthening procedures.
While the Maximum Cut Problem and Unconstrained Binary Quadratic Optimization are of high interest in the scientific community and gain increasing importance, state-of-the-art solvers for these problems are publicly available, yet not for sparse instances of larger scale. We present the novel solver McSparse to fill this gap. It is installed as an internet service similar to the well-known services Biq Mac and BiqCrunch. We explain details of the algorithmic innovations based on integer linear programming and polyhedral combinatorics leading to the branch-and-cut algorithm implemented in McSparse. Substantial improvements with respect to former such approaches are demonstrated and the sustained performance is compared to those of other state-of-the-art methods using a broad set of benchmark instances.
We propose new cutting planes for strengthening the linear relaxations that appear in the solution of nonconvex quadratic problems with linear constraints. By a famous result of Motzkin and Straus, these problems are connected to the clique number of a graph. Our cuts are derived in the context of a spatial branch-and-bound algorithm where linearization variables are introduced to represent products. Their validity is based on the result of Motzkin and Straus, in that it depends on the clique number of certain graphs. For convenience, we derive our cuts for the special case of the so-called standard quadratic programs, where the only (linear) constraint imposes that variables must belong to a simplex. We specifically consider cuts that correspond to carefully constructed complete bipartite graphs. We study the relation between these cuts and the classical ones obtained at the first level of the reformulation-linearization technique. By studying this relation, we derive a new type of valid inequalities that generalizes both types of cuts, and thus leading to a large family of cutting planes to strengthen the linear relaxation. We present extensive computational results using the different cutting planes we propose within the spatial branch and bound implemented by the commercial solver CPLEX. We show that our cuts allow us to obtain a significantly better bound than reformulation-linearization cuts and reduce computing times for global optimality. Finally, we formally establish the scaling necessary for using the proposed cuts to solve nonconvex quadratic problems with linear constraints, e.g., quadratic knapsack problems.