Nonnegative partial s-goodness for the equivalence of a 0-1 linear program to weighted linear programming

TL;DR

This paper establishes conditions under which a 0-1 linear program can be equivalently solved via weighted linear programming, using nonnegative partial s-goodness to connect sparse optimization with combinatorial problems.

Contribution

It introduces the concept of nonnegative partial s-goodness for the constraint matrix, providing verifiable conditions for equivalence between 0-1 LP and weighted LP.

Findings

Derived sufficient and necessary conditions for nonnegative partial s-goodness.

Provided a computable upper bound to verify the conditions.

Validated the theory with three illustrative examples.

Abstract

The 0-1 linear programming problem with nonnegative constraint matrix and objective vector e origins from many NP-hard combinatorial optimization problems. In this paper, we consider recovering an optimal solution to the problem from a weighted linear programming.We first formulate the problem equivalently as a sparse optimization problem. Next, we consider the consistency of the optimal solution of the sparse optimization problem and the weighted linear programming problem. In order to achieve this, we establish nonnegative partial s-goodness of the constraint matrix and the weighted vector. Further, we use two quantities to characterize a sufficient condition and necessary condition for the nonnegative partial s-goodness. However, the two quantities are difficult to calculate, therefore, we provide a computable upper bound for one of the two quantities to verify the nonnegative partial s-goodness. Finally, we give three examples to illustrate that our theory is effective and verifiable.