SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Matrix rounding under the Lp-discrepancy measure and its application to digital halftoning
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A Strongly Polynomial Cut Canceling Algorithm for the Submodular Flow Problem
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
Solving the Convex Cost Integer Dual Network Flow Problem
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
Solving the Convex Cost Integer Dual Network Flow Problem
Management Science
The complexity of pure Nash equilibria
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Efficient algorithms for buffer insertion in general circuits based on network flow
ICCAD '05 Proceedings of the 2005 IEEE/ACM International conference on Computer-aided design
Matrix scaling by network flow
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
The notion of a rational convex program, and an algorithm for the Arrow-Debreu Nash bargaining game
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
The notion of a rational convex program, and an algorithm for the arrow-debreu Nash bargaining game
Journal of the ACM (JACM)
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Local optimality conditions for multicommodity flow problems with separable piecewise convex costs
Operations Research Letters
New algorithms for convex cost tension problem with application to computer vision
Discrete Optimization
The one-machine just-in-time scheduling problem with preemption
Discrete Optimization
On project scheduling with irregular starting time costs
Operations Research Letters
Splittable single source-sink routing on CMP grids: a sublinear number of paths suffice
Euro-Par'13 Proceedings of the 19th international conference on Parallel Processing
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We consider the problem of minimizing a separable convex objective function over the linear space given by a system Mx=0 with M a totally unimodular matrix. In particular, this generalizes the usual minimum linear cost circulation and cocirculation problems in a network and the problems of determining the Euclidean distance from a point to the perfect bipartite matching polytope and the feasible flows polyhedron. We first show that the idea of minimum mean cycle canceling originally worked out for linear cost circulations by Goldberg and Tarjan [J. Assoc. Comput. Mach., 36 (1989), pp. 873--886.] and extended to some other problems [T. R. Ervolina and S. T. McCormick, Discrete Appl. Math., 46 (1993), pp. 133--165], [A. Frank and A. V. Karzanov, Technical Report RR 895-M, Laboratoire ARTEMIS IMAG, Université Joseph Fourier, Grenoble, France, 1992], [T. Ibaraki, A. V. Karzanov, and H. Nagamochi, private communication, 1993], [M. Hadjiat, Technical Report, Groupe Intelligence Artificielle, Faculté des Sciences de Luminy, Marseille, France, 1994] can be generalized to give a combinatorial method with geometric convergence for our problem. We also generalize the computationally more efficient cancel-and-tighten method. We then consider objective functions that are piecewise linear, pure and piecewise quadratic, or piecewise mixed linear and quadratic, and we show how both methods can be implemented to find exact solutions in polynomial time (strongly polynomial in the piecewise linear case). These implementations are then further specialized for finding circulations and cocirculations in a network. We finish by showing how to extend our methods to find optimal integer solutions, to linear spaces of larger fractionality, and to the case when the objective functions are given by approximate oracles.