The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases
by Abraham P Punnen, Piyashat Sripratak, Daniel Karapetyan
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 __MATH0__ cost matrix __MATH1__ is fixed, then BQP01 can be solved in polynomial time. When __MATH2__ is of rank one, we provide an __MATH3__ algorithm and this complexity reduces to __MATH4__ with additional assumptions. Further, if __MATH5__ for some __MATH6__ and __MATH7__, then BQP01 is shown to be solvable in __MATH8__ time. By restricting __MATH9__ we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if __MATH10__ for a fixed __MATH11__. Finally, if the number of negative entries in __MATH2__ is fixed, BQP01 is shown to be polynomially solvable whereas it is NP-hard if this number is __MATH13__ for a fixed __MATH11__.
Reference:
The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases (Abraham P Punnen, Piyashat Sripratak, Daniel Karapetyan), Discrete Applied Mathematics 193, 1–10, 2015.
Bibtex Entry:
@Article{Punnen2015,
Title                    = {The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases},
Author                   = {Punnen, Abraham P and Sripratak, Piyashat and Karapetyan, Daniel},
Journal                  = {Discrete Applied Mathematics},
Year                     = {2015},
Pages                    = {1--10},
Volume                   = {193},
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 number of negative entries in $Q$ is fixed, BQP01 is shown to be polynomially solvable whereas it is NP-hard if this number is $O(\sqrt[k]{n})$ for a fixed $k$.},
Arxivid                  = {1212.3736},
DOI                      = {10.1016/j.dam.2015.04.004},
Keywords                 = {0-1 variables,BQP,complexity,polynomial algorithms,pseudo-boolean programming,quadratic programming},
Mendeley-tags            = {BQP}
}