Persistency and matroid intersection

In this paper, we show that for any independence system, the problem of finding a persistency partition of the ground set and that of finding a maximum weight independent set are polynomially equivalent. This research has been partially funded by the Greek Ministry of Education under the program Pythagoras.     

On combinatorial optimization problems on matroids with uncertain weights

In this paper the combinatorial optimization problem on weighted matroid is considered. It is assumed that the weights in the problem are ill-known and they are modeled as fuzzy intervals. The optimality of solutions and the optimality of elements are characterized. This characterization is performed in the setting of possibility theory. A method of choosing a solution under uncertainty is also proposed.     

Persistency in the assignment and transportation problems

LetG = (U,V,E) be a bipartite graph with weights of its edgesc _ij . For the assignment and transportation problem given by such a graph we propose efficient procedures for partitioning the edge setE into three classes:E _o is the set of edgesij withx _ij = 0 for each optimum solution (0-persistent edges);E ₁ is the set of edges withx _ij > 0 and constant for each optimum (1-persistent edges) andE _w is the set of edges such that there are two optimum solutions x, x withx _ij x _ij ¹ (weakly persistent edges).     

Random sampling and greedy sparsification for matroid optimization problems

Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems on matroids: finding an optimum matroid basis and packing disjoint matroid bases. Application of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts. An optimum matroid basis is typically found by agreedy algorithm that grows an independent set into an optimum basis one element at a time. This continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm. We simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a givenfixed basis is optimum, showing that the two problems can be solved in roughly the same time. Another application of sampling is to packing matroid bases, also known as matroid partitioning. Sampling reduces the number of bases that must be packed. We combine sampling with a greedy packing strategy that reduces the size of the matroid. Together, these techniques give accelerated packing algorithms. We give particular attention to the problem of packing spanning trees in graphs, which has applications in network reliability analysis. Our results can be seen as generalizing certain results from random graph theory. The techniques have also been effective for other packing problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Some of this work done at Stanford University, supported by National Science Foundation and Hertz Foundation Graduate Fellowships, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation and Xerox Corporation. Also supported by NSF award 962-4239.     

A constrained independent set problem for matroids

In this note, we study a constrained independent set problem for matroids. The problem can be regarded as an ordered version of the matroid parity problem. By a reduction of this problem to matroid intersection, we prove a min-max formula. We show how earlier results of Hefner and Kleinschmidt on the so-called MS-matchings fit in our framework.     

On the intersection of infinite matroids

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by Aharoni and Berger in 2009.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.     

An adaptive membrane algorithm for solving combinatorial optimization problems

Membrane algorithms (MAs), which inherit from P systems, constitute a new parallel and distribute framework for approximate computation. In the paper, a membrane algorithm is proposed with the improvement that the involved parameters can be adaptively chosen. In the algorithm, some membranes can evolve dynamically during the computing process to specify the values of the requested parameters. The new algorithm is tested on a well-known combinatorial optimization problem, the travelling salesman problem. The empirical evidence suggests that the proposed approach is efficient and reliable when dealing with 11 benchmark instances, particularly obtaining the best of the known solutions in eight instances. Compared with the genetic algorithm, simulated annealing algorithm, neural network and a fine-tuned non-adaptive membrane algorithm, our algorithm performs better than them. In practice, to design the airline network that minimize the total routing cost on the CAB data with twenty-five US cities, we can quickly obtain high quality solutions using our algorithm.     

Greedy-type resistance of combinatorial problems

This paper gives a sufficient condition for a combinatorial problem to be greedy-type resistant, i.e. such that, on some instances of the problem, any greedy-type algorithm will output the unique worst possible solution. The condition is used to show that the Equipartition, the k-Clique, the Asymmetric Traveling Salesman, the Hamiltonian Path, the Min–Max Matching, and the Assignment Problems are all greedy-type resistant.     

Packing circuits in matroids

The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: for any nonnegative integral weight function w defined on E(M),

有向拟阵与贪婪算法

有向拟阵是拟阵的一种有向情形.本文证明了有向拟阵可用贪婪算法进行刻划.     

Boolean sublattices connected with minimization problems on matroids

It is noted that the decomposition of a network presented by J.C. Picard and M. Queyranne through an algorithmic argument may be defined from a general lattice-theoretic result for more general problems for which the equalities of maximum-flow minimum-cut type hold.     

Performance guarantees of forward and reverse greedy algorithms for minimizing nonsupermodular nonsubmodular functions on a matroid

This letter studies the problem of minimizing increasing set functions, equivalently, maximizing decreasing set functions, over the base matroid. This setting has received great interest, since it generalizes several applied problems including actuator and sensor placement problems in control, task allocation problems, video summarization, and many others. We study two greedy heuristics, namely, the forward and reverse greedy. We provide two novel performance guarantees for the approximate solutions obtained by them depending on both submodularity ratio and curvature.     

Greedy algorithm and symmetric matroids

Symmetric matroids are set systems which are obtained, in some sense, by a weakening of the structure of a matroid. These set systems are characterized by a greedy algorithm and they are suitable for dealing with autodual properties of matroids. Applications are given to the eulerian tours of 4-regular graphs and the theory ofg-matroids.     

On Tutte polynomial expansion formulas in perspectives of matroids and oriented matroids

We introduce the active partition of the ground set of an oriented matroid perspective (or morphism, or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into Boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the Tutte polynomial in terms of orientation activities. This formula was given in the last unpublished preprint by Michel Las Vergnas; the above equivalence relation and notion of active partition generalize a former construction in oriented matroids by Michel Las Vergnas and the author; and the possibility of such a proof technique in perspectives was announced in the aforementioned preprint. We also briefly highlight how the 5-variable expansion of the Tutte polynomial in terms of subset activities in matroid perspectives comes in a similar way from the known partition of the power set of the ground set into Boolean lattices related to subset activities (and we complete the proof with a property which was missing in the literature). In particular, the paper applies to matroids and oriented matroids on a linearly ordered ground set, and applies to graph and directed graph homomorphisms on a linearly ordered edge-set.     

Efficiently solvable special cases of hard combinatorial optimization problems

We survey some recent advances in the field of polynomially solvable special cases of hard combinatorial optimization problems like the travelling salesman problem, quadratic assignment problems and Steiner tree problems. Such special cases can be found by considering special cost structures, the geometry of the problem, the special topology of the underlying graph structure or by analyzing special algorithms. In particular we stress the importance of recognition algorithms. We comment on open problems in this area and outline some lines for future research in this field. This research has been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.