Perspectives of Monge properties in optimization

Abstract

An m × n matrix C is called Monge matrix if c ij + c rs ⩽ c is + c rj for all 1 ⩽ i < r ⩽ m, 1 ⩽ j < s ⩽ n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.