Finding Total Unimodularity in Optimization Problems Solved by Linear Programs

Abstract

A popular approach in combinatorial optimization is to model problems as integer linear programs. Ideally, the relaxed linear program would have only integer solutions, which happens for instance when the constraint matrix is totally unimodular. Still, sometimes it is possible to build an integer solution with the same cost from the fractional solution. Examples are two scheduling problems [4,6] and the single disk prefetching/caching problem [3]. We show that problems such as the three previously mentioned can be separated into two subproblems: (1) finding an optimal feasible set of slots, and (2) assigning the jobs or pages to the slots. It is straigthforward to show that the latter can be solved greedily. We are able to solve the former with a totally unimodular linear program, from which we obtain simple combinatorial algorithms with improved worst case running time.

Container Scheduling: Complexity and Algorithms

“…As a special case of , the problem with two job sizes of 1 and m , , referred to as scheduling tall/small multiprocessor tasks, has been studied in the literature. Baptiste and Schieber (2003) proved that it can be solved in polynomial time by providing an algorithm with the complexity of O ( n 4 ), and Durr and Hurand (2009) improved the time complexity to O ( n 3 ) by using a linear programming model and finding a total unimodularity property. Even though we consider the case where containers have a size of either 1 or 2, it can be generalized into a more general case where containers have a size of either 1 or an arbitrary integer μ .…”