On Tropical Linear and Integer Programs

TL;DR

This paper provides simple proofs of duality theorems in tropical linear programming, demonstrating no duality gap and offering direct solutions and algorithms for tropical linear and integer programs.

Contribution

It introduces straightforward proofs of duality theorems and develops direct solution methods and algorithms for tropical linear and integer programming problems.

Findings

No duality gap exists for tropical primal-dual problems.

A quadratic complexity algorithm is proposed for the dual tropical integer linear program.

Direct solutions are available for certain tropical linear and integer programs.

Abstract

We present simple compact proofs of the strong and weak duality theorems of tropical linear programming. It follows that there is no duality gap for a pair of tropical primal-dual problems. This result together with known properties of subeigenvectors enables us to directly solve a special tropical linear program with two-sided constraints. We also study the duality gap in tropical integer linear programming. A direct solution is available for the primal problem. An algorithm of quadratic complexity is presented for the dual problem. A direct solution is available provided that all coefficients of the objective function are integer. This solution yields a good estimate of the optimal objective function value in the general case.