L-extendable functions and a proximity scaling algorithm for minimum cost multiflow problem

In this paper, we develop a theory of new classes of discrete convex functions, called L-extendable functions and alternating L-convex functions, defined on the product of trees. We establish basic properties for optimization: a local-to-global optimality criterion, the steepest descend algorithm by successive k -submodular function minimizations, the persistency property, and the proximity theorem. Our theory is motivated by minimum cost free multiflow problem. To this problem, Goldberg and Karzanov gave two combinatorial weakly polynomial time algorithms based on capacity and cost scalings, without explicit running time. As an application of our theory, we present a new simple polynomial proximity scaling algorithm to solve minimum cost free multiflow problem in O ( n log ( n A C ) MF ( k n , k m ) ) time, where n is the number of nodes, m is the number of edges, k is the number of terminals, A is the maximum of edge-costs, C is the total sum of edge-capacities, and MF ( n ' , m ' ) denotes the time complexity to find a maximum flow in a network of n ' nodes and m ' edges. Our algorithm is designed to solve, in the same time complexity, a more general class of multiflow problems, minimum cost node-demand multiflow problem, and is the first combinatorial polynomial time algorithm to this class of problems. We also give an application to network design problem.

[1]  Mihalis Yannakakis,et al.  The Complexity of Multiterminal Cuts , 1994, SIAM J. Comput..

[2]  H. Hirai L-convexity on graph structures , 2016, 1610.02469.

[3]  P. Favati Convexity in nonlinear integer programming , 1990 .

[4]  Dorit S. Hochbaum,et al.  A Cut-Based Algorithm for the Nonlinear Dual of the Minimum Cost Network Flow Problem , 2004, Algorithmica.

[5]  Timothy J. Lowe,et al.  Convex Location Problems on Tree Networks , 1976, Oper. Res..

[6]  Vladimir Kolmogorov,et al.  The Power of Linear Programming for General-Valued CSPs , 2013, SIAM J. Comput..

[7]  Éva Tardos,et al.  Globally optimal pixel labeling algorithms for tree metrics , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[8]  Elliot Anshelevich,et al.  On Survivable Access Network Design: Complexity and Algorithms , 2008, IEEE INFOCOM 2008 - The 27th Conference on Computer Communications.

[9]  Alexander V. Karzanov,et al.  Minimum cost multiflows in undirected networks , 1994, Math. Program..

[10]  Kazuo Murota,et al.  Notes on L-/M-convex functions and the separation theorems , 2000, Math. Program..

[11]  Vladimir Kolmogorov,et al.  Potts Model, Parametric Maxflow and K-Submodular Functions , 2013, 2013 IEEE International Conference on Computer Vision.

[12]  Satoru Iwata,et al.  Conjugate Scaling Algorithm for Fenchel-Type Duality in Discrete Convex Optimization , 2002, SIAM J. Optim..

[13]  Dominique Peeters,et al.  Location on networks , 1992 .

[14]  Dorit S. Hochbaum,et al.  Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations , 2002, Eur. J. Oper. Res..

[15]  Takuro Fukunaga,et al.  Approximating the Generalized Terminal Backup Problem via Half-Integral Multiflow Relaxation , 2014, SIAM J. Discret. Math..

[16]  Hiroshi Hirai,et al.  Half-integrality of node-capacitated multiflows and tree-shaped facility locations on trees , 2010, Math. Program..

[17]  Anna Huber,et al.  Towards Minimizing k-Submodular Functions , 2012, ISCO.

[18]  Satoru Fujishige,et al.  Bisubmodular polyhedra, simplicial divisions, and discrete convexity , 2014, Discret. Optim..

[19]  Andrew V. Goldberg,et al.  Scaling Methods for Finding a Maximum Free Multiflow of Minimum Cost , 1997, Math. Oper. Res..

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

[21]  Vladimir Kolmogorov,et al.  Generalized roof duality and bisubmodular functions , 2010, Discret. Appl. Math..

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

[23]  Elliot Anshelevich,et al.  Terminal Backup, 3D Matching, and Covering Cubic Graphs , 2011, SIAM J. Comput..

[24]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[25]  Yusuke Kobayashi,et al.  The Generalized Terminal Backup Problem , 2014, SIAM J. Discret. Math..

[26]  R. L. Francis,et al.  State of the Art-Location on Networks: A Survey. Part II: Exploiting Tree Network Structure , 1983 .

[27]  S. Thomas McCormick,et al.  Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization , 2008, SODA '08.

[28]  Elliot Anshelevich,et al.  Terminal backup, 3D matching, and covering cubic graphs , 2007, STOC '07.

[29]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[30]  Kazuo Murota,et al.  Exact bounds for steepest descent algorithms of L-convex function minimization , 2014, Oper. Res. Lett..

[31]  Satoru Iwata,et al.  A combinatorial strongly polynomial algorithm for minimizing submodular functions , 2001, JACM.

[32]  Hiroshi Hirai,et al.  Discrete convexity and polynomial solvability in minimum 0-extension problems , 2013, Math. Program..

[33]  F. Y. Wu The Potts model , 1982 .

[34]  D. SIAMJ. BISUBMODULAR FUNCTION MINIMIZATION∗ , 2006 .

[35]  L. Lovász On some connectivity properties of Eulerian graphs , 1976 .

[36]  Kazuo Murota,et al.  Discrete convex analysis , 1998, Math. Program..

[37]  Vladimir Kolmogorov,et al.  New algorithms for convex cost tension problem with application to computer vision , 2009, Discret. Optim..

[38]  H. Hirai Discrete Convexity for Multiflows and 0-extensions , 2013 .

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

[40]  Satoru Iwata,et al.  Submodular Function Minimization under Covering Constraints , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[41]  Alexander Schrijver,et al.  A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time , 2000, J. Comb. Theory B.

[42]  Antoon Kolen,et al.  Tree network and planar rectilinear location theory , 1986 .