Oracle Tractability of Skew Bisubmodular Functions

In this paper we consider skew bisubmodular functions as recently introduced by the authors and Powell. We construct a convex extension of a skew bisubmodular function which we call Lovasz extension in correspondence to the submodular case. We use this extension to show that skew bisubmodular functions given by an oracle can be minimized in polynomial time.

[1]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[2]  Stanislav Zivny,et al.  The complexity of finite-valued CSPs , 2013, STOC '13.

[3]  Liqun Qi,et al.  Directed submodularity, ditroids and directed submodular flows , 1988, Math. Program..

[4]  Fredrik Kuivinen,et al.  On the complexity of submodular function minimisation on diamonds , 2009, Discret. Optim..

[5]  Anna Huber,et al.  Skew Bisubmodularity and Valued CSPs , 2013, SIAM J. Comput..

[6]  Shin-ichi Tanigawa,et al.  Generalized skew bisubmodularity: A characterization and a min-max theorem , 2014, Discret. Optim..

[7]  André Bouchet,et al.  Greedy algorithm and symmetric matroids , 1987, Math. Program..

[8]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[9]  Martin C. Cooper,et al.  The complexity of soft constraint satisfaction , 2006, Artif. Intell..

[10]  Andrei A. Krokhin,et al.  Maximizing Supermodular Functions on Product Lattices, with Application to Maximum Constraint Satisfaction , 2008, SIAM J. Discret. Math..

[11]  William H. Cunningham,et al.  Delta-Matroids, Jump Systems, and Bisubmodular Polyhedra , 1995, SIAM J. Discret. Math..

[12]  Ramaswamy Chandrasekaran,et al.  On totally dual integral systems , 1990, Discret. Appl. Math..

[13]  László Lovász,et al.  Submodular functions and convexity , 1982, ISMP.

[14]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[15]  Stanislav Zivny,et al.  The Power of Linear Programming for Valued CSPs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[16]  Satoru Iwata,et al.  Submodular function minimization , 2007, Math. Program..

[17]  藤重 悟.  Submodular functions and optimization , 1991 .