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Discrete convexity and polynomial solvability in minimum 0-extension problems

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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.

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Acknowledgments

We thank the referee for helpful comments, and thank Kazuo Murota for careful reading and numerous helpful comments, Kei Kimura for discussion on Valued-CSP, Satoru Iwata for communicating the paper [43] of Kuivinen, Akiyoshi Shioura for the paper [38] of Kolmogorov, and Satoru Fujishige for the paper [27] of Huber-Kolmogorov. This research is partially supported by the Aihara Project, the FIRST program from JSPS, by Global COE Program “The research and training center for new development in mathematics” from MEXT, and by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, 113-8656, Japan

    Hiroshi Hirai

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Hirai, H. Discrete convexity and polynomial solvability in minimum 0-extension problems. Math. Program. 155, 1–55 (2016). https://doi.org/10.1007/s10107-014-0824-7

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