A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n +m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements.

A module of an undirected graph G = (V, E) is a set X of vertices that have the same set of neighbors in V \ X. The modular decomposition is a unique decomposition of the vertices into nested modules. We give a practical algorithm with an O(n + m(m;n)) time bound and a variant with a linear time bound.

We show that some recent results in slicing of a distributed computation can be applied to developing algorithms to solve problems in combinatorics. A combinatorial problem usually requires enumerating, counting or ascertaining existence of structures that satisfy a given property B. We cast the combinatorial problem as a distributed computation such that there is a bijection between combinatorial structures satisfying B and the global states that satisfy a property equivalent to B. We then apply results in slicing a computation with respect to a predicate to obtain a small representation of only those global states that satisfy B.

This paper gives an O(n²) incremental algorithm for computing the modular decomposition of 2-structure [1, 2]. A 2-structure is a type of edge-colored graph, and its modular decomposition is also known as the prime tree family. Modular decomposition of 2-structures arises in the study of relational systems. The modular decomposition of undirected graphs and digraphs is a special case, and has applications in a number of combinatorial optimization problems. This algorithm generalizes elements of a previous O(n²) algorithm of Muller and Spinrad [3] for the decomposition of undirected graphs. However, Muller and Spinrad's algorithm employs a sophisticated data structure that impedes its generalization to digraphs and 2-structures, and limits its practical use. We replace this data structure with a scheme that labels each edge with at most one node, thereby obtaining an algorithm that is both practical and general to 2-structures.

To deal with computationally hard problems, approximate algorithms are used to provide reasonably good solutions in practical time. Genetic algorithms are an example of the meta-heuristics which were recently introduced and which have been successfully applied to a variety of problems. We propose to use dynamic programming in the process of obtaining new generation solutions in the genetic algorithm, and call it a genetic DP algorithm. To evaluate the eectiveness of this approach, we choose three representative combinatorial optimization problems, the single machine scheduling problem, the optimal linear arrangement problem and the traveling salesman problem, all of which ask to compute optimum permutations of n objects and are known to be NP-hard. Computational results for randomly generated problem instances exhibit encouraging features of genetic DP algorithms. Keywords: Genetic algorithm; Dynamic programming; Heuristics; Combinatorial optimization; Single machine scheduling probl...

Let I be a stable matching instance with N stable matchings. For each man m, order his (not necessarily distinct) N partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as pi(m). Let αi consist of the man-woman pairs where each man m is matched to pi(m). Teo and Sethuraman proved this surprising result: for i = 1 to N, not only is αi a matching, it is also stable. The αi’s are called the generalized median stable matchings of I. Determining if these stable matchings can be computed efficiently is an open problem. In this paper, we present a new characterization of the generalized median stable matchings that provides interesting insights. It implies that the generalized median stable matchings in the middle – α (N+1)/2 when N is odd, α N/2 and α N/2+1 when N is even – are fair not only in a local sense but also in a global sense because they are also medians of the lattice of stable matchings. We then show that there are some families of SM instances for which computing an αi is easy but that the task is NP-hard in general. Finally, we also consider what it means to approximate a median stable matching and present results for this problem. 1

The linear extension diameter of a finite poset P is the maximum distance between a pair of linear extensions of P, where the distance between two linear extensions is the number of pairs of elements of P appearing in different orders in the two linear extensions. We prove a formula for the linear extension diameter of the Boolean Lattice and characterize the diametral pairs of linear extensions. For the more general case of a downset lattice DP of a 2-dimensional poset P, we characterize the diametral pairs of linear extensions of DP and show how to compute the linear extension diameter of DP in time polynomial in |P|. 1

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