Bruno Simeone

Roof duality, complementation and persistency in quadratic 0–1 optimization

The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the ‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones. The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenxi=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the supermodular ones) is exhibited.

The splittance of a graph

The splittance of an arbitrary graph is the minimum number of edges to be added or removed in order to produce a split graph (i.e. a graph whose vertex set can be partitioned into a clique and an independent set). The splittance is seen to depend only on the degree sequence of the graph, and an explicit formula for it is derived. This result allows to give a simple characterization of the degree sequences of split graphs. Worst cases for the splittance are determined for some classes of graphs (the class of all graphs, of all trees and of all planar graphs).

Consensus algorithms for the generation of all maximal bicliques

We describe a new algorithm for generating all maximal bicliques (i.e. complete bipartite, not necessarily induced subgraphs) of a graph. The algorithm is inspired by, and is quite similar to, the consensus method used in propositional logic. We show that some variants of the algorithm are totally polynomial, and even incrementally polynomial. The total complexity of the most efficient variant of the algorithms presented here is polynomial in the input size, and only linear in the output size. Computational experiments demonstrate its high efficiency on randomly generated graphs with up to 2000 vertices and 20,000 edges.

Evaluation and optimization of electoral systems

Preface Part I. Classification and Evaluation of Electoral Systems. 1. The four phases of an electoral process 2. A unified description of electoral systems 3. Performance of an electoral system Part II. Designing Electoral Systems. 4. No electoral system is perfect 5. Basic properties for electoral formulas 6. Integer optimization approach 7. Rewarded and punished parties 8. Mixed electoral systems Part III. Designing Electoral Districts. 9. Traps in electoral district plans 10. Criteria for political districting 11. Indicators for political districting 12. Optimization Models Part IV. The Process of Electoral Reform: A Retrospective Critical View of a Political Scientist. 13. A difficult crossroad 14. The planning and politics of political reform Part V. A Short Guide to the Literature.

Local search algorithms for political districting

Electoral district planning plays an important role in a political election, especially when a majority voting rule is adopted, because it interferes in the translation of votes into seats. The practice of gerrymandering can easily take place if the shape of electoral districts is not controlled. In this paper we consider the following formulation of the political districting problem: given a connected graph (territory) with n nodes (territorial units), partition its set of nodes into k classes such that the subgraph induced by each class (district) is connected and a given vector of functions of the partition is minimized. The nonlinearity of such functions and the connectivity constraints make this network optimization problem a very hard one. Thus, the use of local search heuristics is justified. Experimentation on a sample of medium-large real-life instances has been carried out in order to compare the performance of four local search metaheuristics, i.e., Descent, Tabu Search, Simulated Annealing, and Old Bachelor Acceptance. Our experiments with Italian political districting provided strong evidence in favor of the use of automatic procedures. Actually, except for Descent, all local search algorithms showed a very good performance for this problem. In particular, in our sample of regions, Old Bachelor Acceptance produced the best results in the majority of the cases, especially when the objective function was compactness. Moreover, the district maps generated by this heuristic dominate the institutional district plan with respect to all the districting criteria under consideration. When properly designed, automatic procedures tend to be impartial and yield good districting alternatives. Moreover, they are remarkably fast, and thus they allow for the exploration of a large number of scenarios.

Polynomial-time algorithms for regular set-covering and threshold synthesis

Abstract A set-covering problem is called regular if a cover always remains a cover when any column in it is replaced by an earlier column. From the input of the problem - the coefficient matrix of the set-covering inequalities - it is possible to check in polynomial time whether the problem is regular or can be made regular by permuting the columns. If it is, then all the minimal covers are generated in polynomial time, and one of them is an optimal solution. The algorithm also yields an explicit bound for the number of minimal covers. These results can be used to check in polynomial time whether a given set-covering problem is equivalent to some knapsack problem without additional variables, or equivalently to recognize positive threshold functions in polynomial time. However, the problem of recognizing when an arbitrary Boolean function is threshold is NP-complete. It is also shown that the list of maximal non-covers is essentially the most compact input possible, even if it is known in advance that the problem is regular.

The Maximum Box Problem and its Application to Data Analysis

AbstractGiven two finite sets of points X+ and X− in

On the supermodular knapsack problem

\mathbb{R}^n