Discrete convexity and polynomial solvability in minimum 0-extension problems

Abstract

A $$0$$ -extension of graph $$\varGamma $$ is a metric $$d$$ on a set $$V$$ containing the vertex set $$V_{\varGamma }$$ of $$\varGamma $$ such that $$d$$ extends the shortest path metric of $$\varGamma $$ and for all $$x \in V$$ there exists a vertex $$s$$ in $$\varGamma $$ with $$d(x,s) = 0$$ . The minimum $$0$$ -extension problem 0-Ext $$[\varGamma ]$$ on $$\varGamma $$ is: given a set $$V \supseteq V_{\varGamma }$$ and a nonnegative cost function $$c$$ defined on the set of all pairs of $$V$$ , find a $$0$$ -extension $$d$$ of $$\varGamma $$ with $$\sum _{xy}c(xy) d(x,y)$$ minimum. The $$0$$ -extension problem generalizes a number of basic combinatorial optimization problems, such as minimum $$(s,t)$$ -cut problem and multiway cut problem. Karzanov proved the polynomial solvability of 0-Ext $$[\varGamma ]$$ for a certain large class of modular graphs $$\varGamma $$ , and raised the question: What are the graphs $$\varGamma $$ for which 0-Ext $$[\varGamma ]$$ can be solved in polynomial time? He also proved that 0-Ext $$[\varGamma ]$$ is NP-hard if $$\varGamma $$ is not modular or not orientable (in a certain sense). In this paper, we prove the converse: if $$\varGamma $$ is orientable and modular, then 0-Ext $$[\varGamma ]$$ can be solved in polynomial time. This completes the classification of graphs $$\varGamma $$ for which 0-Ext $$[\varGamma ]$$ is tractable. 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.