Simple games cover voting systems in which a single alternative, such as a bill or anamendment, is pitted against the status quo. A simple game or a yes-no voting system is aset of rules that specifies exactly which collections of “yea” votes yield passage of theissue at hand. Each of these collections of “yea” voters forms a winning coalition. We areinterested in performing a complexity analysis on problems defined on such families ofgames. This analysis as usual depends on the game representation used as input. Weconsider four natural explicit representations: winning, losing, minimal winning, andmaximal losing. We first analyze the complexity of testing whether a game is simple andtesting whether a game is weighted. We show that, for the four types of representations,both problems can be solved in polynomial time. Finally, we provide results on thecomplexity of testing whether a simple game or a weighted game is of a special type. Weanalyze strongness, properness, weightedness, homogeneousness, decisiveness andmajorityness, which are desirable properties to be fulfilled for a simple game. Finally,we consider the possibility of representing a game in a more succinct and natural way andshow that the corresponding recognition problem is hard.

In this paper we introduce some improvements on an approach that we described elsewherefor solving a modification of the well-known extended rapid transit network designproblem. Firstly, we propose an integer programming model for selecting the stations to beconstructed and the links between them, in such a way that a connected rapid transitnetwork is obtained. Secondly, we consider a linear 0-1 programming model for determininga route of minimum length in the rapid transit network between certain pairs of locations,and present a greedy heuristic procedure which attempts to minimize an estimation of thetotal number of transfers that should be made by the users to arrive at theirdestinations. We also report several computational experiments that show that thisprocedure can significantly reduce the estimated total number of transfers required forthe solutions obtained using our previous approach.

Given a simple undirected weighted or unweighted graph, we try to cluster the vertex set into communities and also to quantify the robustness of these clusters. For that task, we propose a new method, called bootstrap clustering which consists in (i) defining a new clustering algorithm for graphs, (ii) building a set of graphs similar to the initial one, (iii) applying the clustering method to each of them, making a profile (set) of partitions, (iv) computing a consensus partition for this profile, which is the final graph partitioning. This allows to evaluate the robustness of a cluster as the average percentage of partitions in the profile joining its element pairs ; this notion can be extended to partitions. Doing so, the initial and consensus partitions can be compared. A simulation protocol, based on random graphs structured in communities is designed to evaluate the efficiency of the Bootstrap Clustering approach.

In this paper, we propose an exact solution method for the windy rural postman problem (WRPP). The motivation to study this problem comes from some real-life applications, such as garbage collecting in a predefined sector with hills, where the traversing or the servicing speed can change following the direction. We present a Dantzig-Wolfe decomposition and a branch-and-price algorithm to solve the WRPP. To the best of our knowledge, Dantzig-Wolfe decomposition has never been used to solve that problem. The numerical results show that optimal solutions are found in a very reasonable amount of time on instances with up to 100 nodes and 180 edges.