The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases

TL;DR

This paper identifies specific conditions under which the bipartite unconstrained 0-1 quadratic programming problem (BQP01) can be solved in polynomial time, despite its general MAX SNP hardness.

Contribution

It introduces new polynomial-time algorithms for BQP01 when the cost matrix has fixed rank or specific structural properties, expanding the understanding of tractable cases.

Findings

Polynomial algorithms for fixed-rank matrices

Efficient solutions for rank-one and sum-of-two matrices

Complexity results for restricted matrix deletion scenarios

Abstract

We 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 $m\times n$ 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 $m=O(\sqrt[k]{n})$ for a fixed $k$. Finally, if the minimum number of rows and columns to be deleted from $Q$ to make the remaining matrix non-negative is $O(\log n)$ then we show that BQP01 polynomially solvable but it is NP-hard if this number is $O(\sqrt[k]{n})$ for any fixed $k$. Keywords: quadratic programming, 0-1 variables, polynomial algorithms, complexity, pseudo-Boolean programming.