New algorithms for the dual of the convex cost network flow problem with application to computer vision

Motivated by various applications to computer vision, we consider an integer convex optimization problem which is the dual of the convex cost network flow problem. In this paper, we first propose a new primal algorithm for computing an optimal solution of the problem. Our primal algorithm iteratively updates primal variables by solving associated minimum cut problems. The main contribution in this paper is to provide a tight bound for the number of the iterations. We show that the time complexity of the primal algorithm is 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 propose a primal-dual algorithm for the dual of the convex cost network flow problem. The primal-dual algorithm can be seen as a refined version of the primal algorithm by maintaining dual variables (flow) in addition to primal variables. Although its time complexity is the same as that for the primal algorithm, we can expect a better performance practically. We finally consider an application to a computer vision problem called the panoramic stitching problem. We apply several implementations of our primal-dual algorithm to some instances of the panoramic stitching problem and test their practical performance. We also show that our primal algorithm as well as the proofs can be applied to the L\-convex function minimization problem which is a more general problem than the dual of the convex cost network flow problem

On Equivalence of M♮^\natural-concavity of a Set Function and Submodularity of Its Conjugate

A fundamental theorem in discrete convex analysis states that a set function is M$^\natural$-concave if and only if its conjugate function is submodular. This paper gives a new proof to this fact

Note on Minimization of Quasi M♮^\natural-convex Functions

For a class of discrete quasi convex functions called semi-strictly quasi M$^\natural$-convex functions, we investigate fundamental issues relating to minimization, such as optimality condition by local optimality, minimizer cut property, geodesic property, and proximity property. Emphasis is put on comparisons with (usual) M$^\natural$-convex functions. The same optimality condition and a weaker form of the minimizer cut property hold for semi-strictly quasi M$^\natural$-convex functions, while geodesic property and proximity property fail

Note on Steepest Descent Algorithm for Quasi L♮^{\natural}-convex Function Minimization

We define a class of discrete quasi convex functions, called semi-strictly quasi L$^{\natural}$-convex functions, and show that the steepest descent algorithm for L$^{\natural}$-convex function minimization also works for this class of quasi convex functions. The analysis of the exact number of iterations is also extended, revealing the so-called geodesic property of the steepest descent algorithm when applied to semi-strictly quasi L$^{\natural}$-convex functions.Comment: 14 page

Handling Scheduling Problems with Controllable Parameters by Methods of Submodular Optimization

In this paper, we demonstrate how scheduling problems with controllable processing times can be reformulated as maximization linear programming problems over a submodular polyhedron intersected with a box. We explain a decomposition algorithm for solving the latter problem and discuss its implications for the relevant problems of preemptive scheduling on a single machine and parallel machines

Time bounds for iterative auctions : a unified approach by discrete convex analysis

We investigate an auction model where there are many different goods, each good has multiple units and bidders have gross substitutes valuations over the goods. We analyze the number of iterations in iterative auction algo- rithms for the model based on the theory of discrete convex analysis. By making use of L♮-convexity of the Lyapunov function we derive exact bounds on the number of iterations in terms of the ℓ1-distance between the initial price vector and the found equilibrium. Our results extend and unify the price adjustment algorithms for the multi-unit auction model and for the unit-demand auction model, offering computational complexity results for these algorithms, and reinforcing the connection between auction theory and discrete convex analysis

Preemptive models of scheduling with controllable processing times and of scheduling with imprecise computation: A review of solution approaches

This paper provides a review of recent results on scheduling with controllable processing times. The stress is on the methodological aspects that include parametric flow techniques and methods for solving mathematical programming problems with submodular constraints. We show that the use of these methodologies yields fast algorithms for solving problems on single machine or parallel machines, with either one or several objective functions. For a wide range of problems with controllable processing times we report algorithms with the running times which match those known for the corresponding problems with fixed processing times. As a by-product, we present the best possible algorithms for a number of problems on parallel machines that are traditionally studied within the body of research on scheduling with imprecise computation