This survey paper prepared for the Handbook of Utility Theory covers the axiomatic foundation of decision making under uncertainty when conditional preferences are allowed to be state dependent, leading to an expected state-dependent utility representation.

In the first part of this paper we prove that the global quadratic optimization problem over a simplex can be solved with a constant relative accuracy. In the second part we consider some natural extensions of the result.

This paper presents a combinatorial polynomial-time algorithm for minimizing submodular set functions. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter.

The cut polyhedron cut(G) of an undirected graph G = (V,E) is the dominant of the convex hull of all of its nonempty edge cutsets. After examining various compact extended formulations for cut(G), we study some of its polyhedral properties. In particular, we characterize all of the facets...

We describe an @( n^4 h min{log U, n^2logn}) capacity scaling algorithm for the minimum cost submodular flow problem. Our algorithm modifies and extends the Edmonds-Karp capacity scaling algorithm for minimum cost flow to solve the minimum cost submodular flow problem. The modification entails...

We study a family of unbounded polyhedra arising in the study of uncapacitated lot-sizing problems with Wagner-Whitin costs. With n the number of periods, we completely characterize the bounded faces of maximal dimension, and derice an O(n^2) alogorithm to express within the polyhedron as a...

We present a convex conic relaxation for a problem of maximising an indefinite quadratic form over a set of convex constraints on the squared variables. We show that for all these problems we get at least 12/37 relative accuracy of the approximation. In the second part of the paper we derive the...