论文信息 - A Primal-Dual Algorithm for Submodular Flows

A Primal-Dual Algorithm for Submodular Flows

Previously the only polynomial-time solution algorithm to solve the optimal submodular flow problem introduced by Edmonds and Giles was based on the ellipsoid method. Here, modulo an efficient oracle for minimizing certain submodular functions, a polynomial time procedure is presented which uses only combinatorial steps like building auxiliary digraphs, finding augmenting paths. The minimizing oracle is currently available only via the ellipsoid method, in general; however in important special cases, such as network flows, matroid intersections, orientations, and directed cut coverings, the necessary oracle can be provided combinatorially.

András Frank | William H. Cunningham | W. Cunningham | A. Frank

[1] William H. Cunningham,et al. Testing membership in matroid polyhedra , 1984, J. Comb. Theory, Ser. B.

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

[3] D. R. Fulkerson,et al. Flows in Networks. , 1964 .

[4] C. Lucchesi,et al. A Minimax Theorem for Directed Graphs , 1978 .

[5] P. Schönsleben. Ganzzahlige Polymatroid-Intersektions-Algorithmen , 1980 .

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

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

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

[9] F. Dunstan. MATROIDS AND SUBMODULAR FUNCTIONS , 1976 .

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

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

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

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

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

[15] A. Schrijver. Total Dual Integrality from Directed Graphs, Crossing Families, and Sub- and Supermodular Functions , 1984 .