A digraph obtained by replacing each edge of a complete m-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete m-partite digraph. We describe results (theorems and algorithms) on directed walks in semicomplete m- partite digraphs including some recent results concerning tournaments. 1

A formula (in conjunctive normal form) is said to be minimal unsatisfiable if it is unsatisfiable and deleting any clause makes it satisfiable. The deficiency of a formula is the difference of the number of clauses and the number of variables. It is known that every minimal unsatisfiable formula has positive deficiency. Until recently, polynomial–time algorithms were known to recognize minimal unsatisfiable formulas with deficiency 1 and 2. We state an algorithm which recognizes minimal unsatisfiable formulas with any fixed deficiency in polynomial time.

We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly non-trivial, even for these ”tournament-like ” digraphs. 1

Trapezoid graphs are a class of cocomparability graphs containing interval graphs and permutation graphs as subclasses. They were introduced by Dagan, Golumbic and Pinter [DGP]. They propose an O(n²) algorithm for chromatic number and a less efficient algorithm for maximum clique on trapezoid graphs. Based on a geometric representation of trapezoid graphs by boxes in the plane we design optimal, i.e., O(n log n), algorithms for chromatic number, weighted independent set, clique cover and maximum weighted clique on such graphs. We also propose generalizations of trapezoid graphs called k-trapezoidal graphs. The ideas behind the clique cover and weighted independent set algorithms for trapezoid graphs carry over to higher dimensions. This leads to O(n log k\Gamma1 n) algorithms for k-trapezoidal graphs. We also propose a new class of graphs called circle trapezoid graphs. This class contains trapezoid graphs, circle graphs and circular-arc graphs as subclasses. We show that cli...

We conduct a computational study of unit capacity flow and bipartite matching algorithms. Our goal is to determine which variant of the push-relabel method is most efficient in practice and to compare push-relabel algorithms with augmenting path algorithms. We have implemented and compared three push-relabel algorithms, three augmenting path algorithms (one of which is new), and one augment-relabel algorithm. The depth-first search augmenting path algorithm was thought to be a good choice for the bipartite matching problem, but our study shows that it is not robust. For the problems we study, our implementations of the fifo and lowest-level selection push-relabel algorithms have the most robust asymptotic rate of growth and work best overall. Augmenting path algorithms, although not as robust, on some problem classes are faster by a moderate constant factor. Our study includes several new problem families and input graphs with as many as 5 \Theta 10 5 vertices.

Abstract. We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size L = f2 (log n) a maximal matching in an n-vertex bipartite graph in time O (n 2 + n2"5/~.) = O (n2"5/log n), how to compute the transitive closure of a digraph with n vertices and m edges in time O(n 2 + nm/,k), how to solve the uncapacitated transportation problem with integer costs in the range [0..C] and integer demands in the range [-U..U] in time O ((n 3 (log log / log n) 1/2 + n 2 log U) log nC), and how to solve the assignment problem with integer costs in the range [0..C] in time O(n 2"5 lognC/(logn/loglogn)l/4). Assuming a suitably compressed input, we also show how to do depth-first and breadth-first search and how to compute strongly connected components and biconnected components in time O(n~. + n2/L), and how to solve the single source shortest-path problem with integer costs in the range [0..C] in time O (n 2 (log C)/log n). For the transitive closure algorithm we also report on the experiences with an implementation. Key Words.

ABSTRACT. The AllDifferent constraint is a crucial component of any constraint toolkit, language or solver, since it is very widely used in a variety of constraint models. The literature contains many different versions of this constraint, which trade strength of inference against computational cost. In this paper, we focus on the highest strength of inference, enforcing a property known as generalised arc consistency (GAC). This work is an analytical survey of optimizations of the main algorithm for GAC for the AllDifferent constraint. We evaluate empirically a number of key techniques from the literature. We also report important implementation details of those techniques, which have often not been described in published papers. We pay particular attention to improving incrementality by exploiting the strongly-connected components discovered during the standard propagation process, since this has not been detailed before. Our empirical work represents by far the most extensive set of experiments on variants of GAC algorithms for AllDifferent. Overall, the best combination of optimizations gives a mean speedup of 168 times over the same implementation without the optimizations. 1.

We characterize weakly Hamiltonian-connected ordinary multipartite tournaments. Our result generalizes such a characterization for tournaments by Thomassen and implies a polynomial algorithm to decide the existence of a Hamiltonian path connecting two given vertices in an ordinary multipartite tournament and find one, if it exists. 1

Partially ordered sets of small width and graphs of small Dilworth number have many interesting properties and have been well studied. Here we show that recognition of such orders and graphs can be done more eÆciently than by using the well-known algorithms based on bipartite matching and matrix multiplication. In particular, we show that deciding deciding if an order has width k can be done in O(kn²) time and whether a graph has Dilworth number k can be done in O(k²n²) time. For very small k we have even better results. We show that orders of width at most 3 can be recognized in O(n) time and of width at most 4 in O(n log n).

The "Constraint Bipartite Vertex Cover" problem (CBVC for short) is: given a bipartite graph G with n vertices and two positive integers k 1 ; k 2 , is there a vertex cover taking at most k 1 vertices from one and at most k 2 vertices from the other vertex set of G? CBVC is NP-complete. It formalizes the spare allocation problem for reconfigurable arrays, an important problem from VLSI manufacturing.