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Discrete Applied Mathematics

Bounds and fast approximation algorithms for binary quadratic optimization problems with application to MAX 2SAT

Hans van Maaren, Joost P. Warners

Published Dec 1, 2000• 3 citations
Publisher
DOI
10.1016/s0166-218x(00)00263-8

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The Goemans--Williamson randomized algorithm guarantees a high-quality approximation to the MAX-CUT problem, but the cost associated with such an approximation can be excessively high for large-scale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better practical performance, we propose an alternative, rank-two relaxation and develop a specialized version of the Goemans--Williamson technique. The proposed approach leads to continuous optimization heuristics applicable to MAX-CUT as well as other binary quadratic programs, for example the MAX-BISECTION problem. A computer code based on the rank-two relaxation heuristics is compared with two state-of-the-art semidefinite programming codes that implement the Goemans--Williamson randomized algorithm, as well as with a purely heuristic code for effectively solving a particular MAX-CUT problem arising in physics. Computational results show that the proposed approach is fast and scalable and, more importantly, attains a higher approximation quality in practice than that of the Goemans--Williamson randomized algorithm. An extension to MAX-BISECTION is also discussed, as is an important difference between the proposed approach and the Goemans--Williamson algorithm; namely, that the new approach does not guarantee an upper bound on the MAX-CUT optimal value.
Combinatorial optimization problems are fundamental to many industrial and scientific applications, but are often NP-hard, requiring heuristic and probabilistic approaches to find good solutions fast. This paper presents the first hardware implementation of the neuromorphic-inspired NeuroSA algorithm, for accelerating Quadratic Unconstrained Binary Optimization (QUBO) problems. We show that it is possible to run a standard QUBO benchmark problem with 800 variables on a low-cost FPGA with a clock speed of approximately 144 MHz, which converges to the same solution value found by the numerical solver.
This dissertation investigates the Quadratic Unconstrained Binary Optimization (QUBO) problem, i.e. the problem of minimizing a quadratic function in binary variables. QUBO is studied at two complementary levels. First, there is an algorithmic aspect that tells how to preprocess the problem, how to find heuristics, how to get improved bounds and how to solve the problem with all the above ingredients.Second, there is a practical aspect that uses QUBO to solve various applications from the engineering and social sciences fields including: via minimization, 2D/3D Ising models, 1D Ising chain models, image binarization, hierarchical clustering, greedy graph coloring/partitioning, MAX-2-SAT, MIN-VC, MAX-CLIQUE, MAX-CUT, graph stability and minimum k-partition.Several families of fast heuristics for QUBO are analyzed, which include a novel probabilistic based class of methods.It is shown that there is a unique maximal set of persistencies for the linearization model for QUBO.This set is determined in polynomial time by a maximum flow followed by the computation of the strong components of a network that has 2n+2 nodes, where n is the number of variables. The identification of the above persistencies leads to a unique decomposition of the function, such that each component can be optimized separately. To find further persistencies, two additional techniques are proposed: one is based on the second order derivatives of Hammer et al. [HH81]; the other technique is a probing procedure on the two possible values of the variables.These preprocessing tools work remarkably well for certain classes of problems.We improved the Iterated Roof-Dual bound (IRD) of [BH89] by proposing two combinatorial methods: one was named the squeezed IRD; and the second was called the project-and-lift IRD method.The cubic-dual bound can be found by means of linear programming by adding a set of triangle inequalities to the standard linearization, whose number is cubic in the number of variables. We show that this set can be reduced depending on the coefficients of the terms of the function. This leads to the possibility of computing the cubic-duals of larger QUBOs.