Algorithms for submodular flows

The submodular flow problem, introduced by Edmonds and Giles [11], is one of the most important frameworks of efficiently solvable combinatorial optimization problems. It includes the minimum cost flow, the graph orientation, the polymatroid intersection, and the directed cut covering problems as its special cases. See also [23], [24] for further applications of submodular flows. Other frameworks named independent flows [25] and polymatroidal flows [47], [60] are equivalent to submodular flows. These three are collectively called neoflows in [29]. A number of combinatorial algorithms have been proposed as extensions of network flow algorithms. This paper surveys the state of the art in the developments of the submodular flow algorithms and also describes possible directions for further investigations. In Sect. 2, we describe fundamental results concerning submodular functions and submodular flows. Readers should refer to [24], [29], [61] for more details of these results. In Sect. 3, we survey algorithms for submodular flows. We also show some possible future research directions in Sect. 4.

[1]  Éva Tardos,et al.  An O(n2(m + n log n) log n) min-cost flow algorithm , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[2]  Satoru Iwata,et al.  A faster algorithm for minimum cost submodular flows , 1998, SODA '98.

[3]  Ravindra K. Ahuja,et al.  New scaling algorithms for the assignment and minimum mean cycle problems , 1992, Math. Program..

[4]  Lisa Fleischer,et al.  Universally maximum flow with piecewise‐constant capacities , 2001, Networks.

[5]  Satoru Iwata,et al.  Conjugate Scaling Technique for Fenchel - type Duality in Discrete Convex Optimization , 1998 .

[6]  Kazuo Murota.  Submodular Flow Problem with a Nonseparable Cost Function , 1999, Comb..

[7]  A. Hoffman,et al.  Lattice Polyhedra II: Generalization, Constructions and Examples , 1982 .

[8]  Satoru Iwata,et al.  A dual approximation approach to weighted matroid intersection , 1995, Oper. Res. Lett..

[9]  S. Fujishige.  ALGORITHMS FOR SOLVING THE INDEPENDENT-FLOW PROBLEMS , 1978 .

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

[11]  Uwe T. Zimmermann,et al.  A polynomial cycle canceling algorithm for submodular flows , 1999, Math. Program..

[12]  András Frank,et al.  A Weighted Matroid Intersection Algorithm , 1981, J. Algorithms.

[13]  Éva Tardos,et al.  Polynomial dual network simplex algorithms , 2011, Math. Program..

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

[15]  Éva Tardos,et al.  Layered Augmenting Path Algorithms , 1986, Math. Oper. Res..

[16]  J. Edmonds,et al.  A Min-Max Relation for Submodular Functions on Graphs , 1977 .

[17]  Ronald D. Armstrong,et al.  A new strongly polynomial dual network simplex algorithm , 1997, Math. Program..

[18]  Harold N. Gabow.  A framework for cost-scaling algorithms for submodular flow problems , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[19]  Satoru Iwata,et al.  A Strongly Polynomial Cut Canceling Algorithm for the Submodular Flow Problem , 1999, IPCO.

[20]  Satoru Iwata,et al.  A Fast Parametric Submodular Intersection Algorithm for Strong Map Sequences , 1997, Math. Oper. Res..

[21]  A. Frank,et al.  An application of submodular flows , 1989 .

[22]  Kazuo Murota,et al.  Convexity and Steinitz's Exchange Property , 1996, IPCO.

[23]  Satoru Fujishige,et al.  A note on the Frank—Tardos bi-truncation algorithm for crossing-submodular functions , 1992, Math. Program..

[24]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[25]  A. Schrijver,et al.  Proving total dual integrality with cross-free families—A general framework , 1984, Math. Program..

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

[27]  Éva Tardos,et al.  A strongly polynomial minimum cost circulation algorithm , 1985, Comb..

[28]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

[29]  András Frank,et al.  A Primal-Dual Algorithm for Submodular Flows , 1985, Math. Oper. Res..

[30]  Andrew V. Goldberg,et al.  Finding minimum-cost circulations by canceling negative cycles , 1989, JACM.

[31]  Andrew V. Goldberg,et al.  Scaling algorithms for the shortest paths problem , 1995, SODA '93.

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

[33]  Donald M. Topkis,et al.  Minimizing a Submodular Function on a Lattice , 1978, Oper. Res..

[34]  András Frank,et al.  Generalized polymatroids and submodular flows , 1988, Math. Program..

[35]  Andrew V. Goldberg,et al.  A new approach to the maximum flow problem , 1986, STOC '86.

[36]  S. Thomas McCormick,et al.  Two Strongly Polynomial Cut Cancelling Algorithms for Minimum Cost Network Flow , 1993, Discret. Appl. Math..

[37]  S. Fujishige,et al.  A NOTE ON SUBMODULAR FUNCTIONS ON DISTRIBUTIVE LATTICES , 1983 .

[38]  A. Frank.  An Algorithm for Submodular Functions on Graphs , 1982 .

[39]  Satoru Iwata,et al.  A fast cost scaling algorithm for submodular flow , 2000, Inf. Process. Lett..

[40]  Satoru Fujishige,et al.  An out-of-kilter method for submodular flows , 1987, Discret. Appl. Math..

[41]  Satoru Iwata,et al.  A capacity scaling algorithm for convex cost submodular flows , 1996, SODA '96.

[42]  Andrew V. Goldberg,et al.  Beyond the flow decomposition barrier , 1998, JACM.

[43]  Andrew V. Goldberg,et al.  Finding Minimum-Cost Circulations by Successive Approximation , 1990, Math. Oper. Res..

[44]  E. A. Dinic.  Algorithm for solution of a problem of maximal flow in a network with power estimation , 1970 .

[45]  Ravindra K. Ahuja,et al.  A Fast and Simple Algorithm for the Maximum Flow Problem , 2011, Oper. Res..

[46]  Satoru Fujishige,et al.  Structures of polyhedra determined by submodular functions on crossing families , 1984, Math. Program..

[47]  András Frank,et al.  Increasing the rooted-connectivity of a digraph by one , 1999, Math. Program..

[48]  Satoru Iwata,et al.  A faster capacity scaling algorithm for minimum cost submodular flow , 2002, Math. Program..

[49]  S. Thomas McCormick,et al.  Canceling most helpful total cuts for minimum cost network flow , 1993, Networks.

[50]  James B. Orlin,et al.  A polynomial time primal network simplex algorithm for minimum cost flows , 1996, SODA '96.

[51]  S. Thomas McCormick,et al.  Canceling most helpful total submodular cuts for submodular flow , 1993, Conference on Integer Programming and Combinatorial Optimization.

[52]  Satoru Fujishige,et al.  A capacity-rounding algorithm for the minimum-cost circulation problem: A dual framework of the Tardos algorithm , 1986, Math. Program..

[53]  S. Thomas McCormick,et al.  Fast algorithms for parametric scheduling come from extensions to parametric maximum flow , 1996, STOC '96.

[54]  Uwe T. Zimmermann.  Negative circuits for flows and submodular flows , 1992, Discret. Appl. Math..

[55]  András Frank,et al.  An application of simultaneous diophantine approximation in combinatorial optimization , 1987, Comb..

[56]  Satoru Fujishige,et al.  A Strongly Polynomial Algorithm for Minimum Cost Submodular Flow Problems , 1989, Math. Oper. Res..

[57]  James B. Orlin,et al.  A faster strongly polynomial minimum cost flow algorithm , 1993, STOC '88.

[58]  Éva Tardos,et al.  A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs , 1986, Oper. Res..

[59]  Andrew V. Goldberg,et al.  Finding minimum-cost flows by double scaling , 2015, Math. Program..

[60]  U. ZIMMERMANN,et al.  Minimization on submodular flows , 1982, Discret. Appl. Math..

[61]  Alexander Schrijver,et al.  Min-max Relations for Directed Graphs , 1982 .

[62]  Harold N. Gabow.  Centroids, Representations, and Submodular Flows , 1995, J. Algorithms.

[63]  Refael Hassin.  Minimum cost flow with set-constraints , 1982, Networks.

[64]  Remo Guidieri.  Res , 1995, RES: Anthropology and Aesthetics.

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

[66]  Satoru Iwata,et al.  A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions , 2000, STOC '00.

[67]  Satoru Iwata,et al.  Relaxed Most Negative Cycle and Most Positive Cut Canceling Algorithms for Minimum Cost Flow , 2000, Math. Oper. Res..

[68]  E. L. Lawler,et al.  Computing Maximal "Polymatroidal" Network Flows , 1982, Math. Oper. Res..

[69]  Kevin D. Wayne,et al.  A polynomial combinatorial algorithm for generalized minimum cost flow , 1999, STOC '99.

[70]  S. Fujishige,et al.  A PRIMAL ALGORITHM FOR THE SUBMODULAR FLOW PROBLEM WITH MINIMUM-MEAN CYCLE SELECTION , 1988 .

[71]  S. Fujishige,et al.  AN EFFICIENT COST SCALING ALGORITHM FOR THE INDEPENDENT ASSIGNMENT PROBLEM , 1995 .

[72]  M. Iri,et al.  AN ALGORITHM FOR FINDING AN OPTIMAL "INDEPENDENT ASSIGNMENT" , 1976 .

[73]  Alan J. Hoffman,et al.  A generalization of max flow—min cut , 1974, Math. Program..

[74]  William H. Cunningham,et al.  A submodular network simplex method , 1984 .

[75]  András Frank,et al.  How to make a digraph strongly connected , 1981, Comb..

[76]  András Frank,et al.  Finding feasible vectors of Edmonds-Giles polyhedra , 1984, J. Comb. Theory, Ser. B.

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

[78]  Eugene L. Lawler,et al.  Matroid intersection algorithms , 1975, Math. Program..