The bipartite unconstrained 0–1 quadratic programming problem: Polynomially solvable cases

Abstract

a b s t r a c tWe consider the bipartite unconstrained 0-1 quadratic programming problem (BQP01) which is a generalization of the well studied unconstrained 0-1 quadratic programming problem (QP01). BQP01 has numerous applications and the problem is known to be MAX SNP hard. We show that if the rank of an associated m×n cost matrix Q = (q ij ) is fixed, then BQP01 can be solved in polynomial time. When Q is of rank one, we provide an O(n log n) algorithm and this complexity reduces to O(n) with additional assumptions. Further, if q ij = a i + b j for some a i and b j , then BQP01 is shown to be solvable in O(mn log n) time. By restricting m = O(log n), we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if m = O( k √ n) for a fixed k. Finally, if the minimum number of rows and columns to be deleted from Q to make the remaining matrix nonnegative is O(log n), then we show that BQP01 is polynomially solvable but it is NP-hard if this number is O( k √ n) for any fixed k.

The generalized vertex cover problem and some variations

“…From Theorem 2.8, GVC on the bipartite graph G = (V 1 , V 2 , E) is equivalent to a BQP01 with size m × n and the quadratic cost matrix in BQP01 is as defined in equation (20). Since BQP01 of size m × n is NP-hard if m = O( k √ n) and solvable in polynomial time if m = O(log n) [30], the result follows.…”

“…The bipartite unconstrained 0-1 quadratic program problem (BQP01) [10,30,31] is closely related to GVC on bipartite graphs. Let Q = (q ij ) be an m × n matrix, a = (a 1 , a 2 , .…”

Combinatorial optimization with interaction costs: Complexity and solvable cases

“…Being a generalization of many hard combinatorial optimization problems, the general COPIC is NP-hard. Moreover, even for the "simple" case with no constraints on the feasible solutions it results in the bipartite unconstrained quadratic programming problem which is NP-hard [47]. COPIC (2 [m] , 2 [n] , Q, c, d) can easily be embedded into a COPIC for most sets of feasible solutions F 1 and F 2 , which implies again NP-hardness.…”

“…BAP is a generalization of the well studied quadratic assignment problem [18] and the three-dimensional assignment problem [51] and hence COPIC generalizes these problems as well. When F 1 and F 2 contain all subsets of [m] and [n] respectively, COPIC reduces to the bipartite unconstrained quadratic programming problem [23,47,35,39] studied in the literature by various authors and under different names. Also, when F 1 and F 2 are feasible solutions of generalized upper bound constraints on m and n variables, respectively, COPIC reduces to the bipartite quadratic assignment problem and its variations [21,48].…”

“…Using these definitions, for example, the bipartite unconstrained quadratic programming problem [47] is denoted by COPIC(2 [m] , 2 [n] , Q, c, d).…”