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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.

Discrete convexity and polynomial solvability in minimum 0-extension problems