Robust two-stage combinatorial optimization problems under convex second-stage cost uncertainty

Abstract

In this paper a class of robust two-stage combinatorial optimization problems is discussed. It is assumed that the uncertain second-stage costs are specified in the form of a convex uncertainty set, in particular polyhedral or ellipsoidal ones. It is shown that the robust two-stage versions of basic network optimization and selection problems are NP-hard, even in a very restrictive cases. Some exact and approximation algorithms for the general problem are constructed. Polynomial and approximation algorithms for the robust two-stage versions of basic problems, such as the selection and shortest path problems, are also provided.

Robust combinatorial optimization problems under budgeted interdiction uncertainty

“…requires us to pack items of total weight strictly greater than V − 1 to avoid having all items interdicted. This means that the partition problem is a Yes-instance if and only if there is a feasible solution x x x ∈ {0, 1} n with objective value less or equal to V. As the partition problem is well-known to be NP-complete (Garey and Johnson 1979), the claim follows. ◻…”