Topics in quadratic binary optimization problems

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(Thesis) Ph.D.
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In this dissertation, we consider the quadratic combinatorial optimization problem (QCOP) and its variations: the generalized vertex cover problem (GVC), the quadratic unconstrained binary optimization problem (QUBO), and the quadratic set covering problem (QSCP). We study these problems as discussed below. For QCOP, we analyze equivalent representations of the pair (Q, c), where Q is a quadratic cost matrix and c is a linear cost vector. We present various procedures to obtain equivalent representations, and study the effect of equivalent representations on the computation time to obtain an optimal solution, on the quality of the lower bound values obtained by various lower bounding schemes, and on the quality of the heuristic solution. The first variation of QCOP is GVC, and we show that GVC is equivalent to QUBO and also equivalent to some other variations of GVC. Some solvable cases are identified and approximation algorithms are suggested for special cases. We also study GVC on bipartite graphs. QUBO is the second variation of QCOP. For QUBO, we analyze several heuristic algorithms using domination analysis. We show that for QUBO, if any algorithm that guarantees a solution no worse than the average, has a domination ratio of at least 1/40. We extend this result to the maximum and minimum cut problems; maximum and minimum uncut problems; and GVC. We also study the QUBO when Q is: 1) (2d + 1)-diagonal matrix, 2) (2d + 1)-reverse-diagonal matrix, and 3) (2d+1)-cross diagonal matrix, and show that these cases are solvable in polynomial time when d is fixed or is of O(log n). The last variation of QCOP is QSCP. For QSCP, we identify various inaccuracies in the paper by R. R. Saxena and S. R. Arora, A linearization technique for solving the Quadratic Set Covering Problem, Optimization, 39 (1997) 33-42. In particular, we observe that their algorithm does not guarantee optimality, contrary to what is claimed. We also present the mixed integer linear programming formulations (MILP) for QSCP. We compare the lower bounds obtained by the linear programming relaxations of MILPs, and study the effect of equivalent representations of QSCP on these MILPs.
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Thesis advisor: Punnen, Abraham
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