Circle packing problems are a class of packing problems in which we are to pack a given set of circles into a container with no overlap. In this paper, we focus on a circle packing problem proposed by López and Beasley. The problem is, given a set of unequal circles to choose and pack a subset of them into a fixed size circular container so as to maximize the total area of the packed circles. López and Beasley formulated this problem as a mixed-integer nonconvex quadratic programming problem and proposed a heuristic method based on its continuous relaxation. By using this method they solved instances with up to 40 circles. Here, we propose heuristic algorithms using mixed-integer DC (Difference-of-Convex) programming. A DC program is an optimization problem whose objective function can be represented by the difference of two convex functions, and a mixed-integer DC program is a DC program where some of the variables are restricted to integer values. Computational results showed that our method obtained solutions at least as good as the López and Beasley algorithm for the problem data used in their experiments, and good solutions for problems with up to the previously unsolvable size of 60 circles within reasonable computation time.

The adwords problem (or the ad-auctions problem) is an online problem which decides allocations of advertisements to search words. The search words arrive one-by-one and a search engine decides what advertisement to display upon arrival of each search word. The owner of the search engine receives revenues from advertisers and the objective of the adwords problem is to maximize the sum of revenues. In this paper, we propose a hybrid algorithm for the adwords problem by combining two algorithms called a greedy algorithm and a PD algorithm. The proposed algorithm has a parameter k ∈ [0,1] and a competitive ratio of the algorithm is 1/(k (e/e-1) + 2 (1-k)) under the small bid assumption. We conduct preliminary numerical experiments, and we observe that the proposed algorithm outperforms other two algorithms with a suitable parameter k.

In discrete convex analysis, various convexity concepts are considered for discrete functions such as separable convexity, L-convexity, M-convexity, integral convexity, and multimodularity. These concepts of discrete convex functions are not mutually independent. For example, M^♮-convexity is a special case of integral convexity, and the combination of L^♮-convexity and M^♮-convexity coincides with separable convexity. This paper aims at a fairly comprehensive analysis of the inclusion and intersection relations for various classes of discrete convex functions. Emphasis is put on the analysis of multimodularity in relation to L^♮-convexity and M^♮-convexity.