Motivated by various applications to computer vision, we consider the convex cost tension problem, which is the dual of the convex cost flow problem. In this paper, we first propose a primal algorithm for computing an optimal solution of the problem. Our primal algorithm iteratively updates primal variables by solving associated minimum cut problems. We show that the time complexity of the primal algorithm is O(K T(n,m)), where K is the range of primal variables and T(n,m) is the time needed to compute a minimum cut in a graph with n nodes and m edges. We then develop an improved version of the primal algorithm, called the primal-dual algorithm, by making good use of dual variables in addition to primal variables. Although its time complexity is the same as that of the primal algorithm, we can expect a better performance in practice. We finally consider an application to a computer vision problem called the panoramic image stitching.

V. Kolmogorov.
"Primal-dual Algorithm for Convex Markov Random Fields"
Microsoft Technical Report MSR-TR-2005-117, September 2005.

A. Shioura, An unpublished memorandum, August 2003 (see report below).

A. Shioura. "Note on $L^\natural$-convex Function Minimization Algorithms:
Comparison of Murota's and Kolmogorov's Algorithms"
Technical report ZR-06-03, January 2006.