Discrete Convexity and Polynomial Solvability in Minimum 0-Extension Problems: [Extended Abstract]

Abstract

The minimum 0-extension problem 0-Ext[Γ] on a graph Γ is: given a set V including the vertex set VΓ of Γ and a nonnegative cost function c defined on the set of all pairs of V, find a 0-extension d of the path metric dΓ of Γ with Σxy c(xy)d(x, y) minimum, where a 0-extension is a metric d on V such that the restriction of d to VΓ coincides with dΓ and for all x ∊ V there exists a vertex s in Γ with d(x, s) = 0. 0-Ext[Γ] includes a number of basic combinatorial optimization problems, such as minimum (s, t)-cut problem and multiway cut problem. Karzanov proved the polynomial solvability for a certain large class of modular graphs, and raised the question: What are the graphs Γ for which 0-Ext[Γ] can be solved in polynomial time? He also proved that 0-Ext[Γ] is NP-hard if Γ is not modular or not orientable (in a certain sense). In this paper, we prove the converse: if Γ is orientable and modular, then 0-Ext[Γ] can be solved in polynomial time. This completes the classification of the tractable graphs for the 0-extension problem. To prove our main result, we develop a theory of discrete convex functions on orientable modular graphs, analogous to discrete convex analysis by Murota, and utilize a recent result of Thapper and Živný on Valued-CSP.