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

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

Scheduling problems with controllable processing times and a common deadline to minimize maximum compression cost

We consider a range of scheduling problems with controllable processing times, in which the jobs must be completed by a common deadline by compressing appropriately their processing times. The objective is to minimize the maximum compression cost. We present a number of algorithms based on common general principles adapted with a purpose of reducing the resulting running times

Single machine scheduling with controllable processing times by submodular optimization

In scheduling with controllable processing times the actual processing time of each job is to be chosen from the interval between the smallest (compressed or fully crashed) value and the largest (decompressed or uncrashed) value. In the problems under consideration, the jobs are processed on a single machine and the quality of a schedule is measured by two functions: the maximum cost (that depends on job completion times) and the total compression cost. Our main model is bicriteria and is related to determining an optimal trade-off between these two objectives. Additionally, we consider a pair of associated single criterion problems, in which one of the objective functions is bounded while the other one is to be minimized. We reduce the bicriteria problem to a series of parametric linear programs defined over the intersection of a submodular polyhedron with a box. We demonstrate that the feasible region is represented by a so-called base polyhedron and the corresponding problem can be solved by the greedy algorithm that runs two orders of magnitude faster than known previously. For each of the associated single criterion problems, we develop algorithms that deliver the optimum faster than it can be deduced from a solution to the bicriteria problem