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

We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an n-element set N, compute their subset convolution f ∗g, defined for all S ⊆ N by (f ∗ g)(S) = X f(T)g(S \ T), T ⊆S where addition and multiplication is carried out in an arbitrary ring. Via Möbius transform and inversion, our algorithm evaluates the subset convolution in O(n 2 2 n) additions and multiplications, substantially improving upon the straightforward O(3 n) algorithm. Specifically, if the input functions have an integer range {−M, −M+1,..., M}, their subset convolution over the ordinary sum–product ring can be computed in Õ(2 n log M) time; the notation Õ suppresses polylogarithmic factors. Furthermore, using a standard embedding technique we can compute the subset convolution over the max–sum or min–sum semiring in Õ(2n M) time. To demonstrate the applicability of fast subset convolution, we present the first Õ(2k n 2 + nm) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the Õ(3k n+2 k n 2 +nm) time bound of the classical Dreyfus– Wagner algorithm. We also discuss extensions to recent Õ(2 n)-time algorithms for covering and partitioning problems

One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)) , where e ( P ) = | E(P) |. The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n e(P)) , where n is the number of elements of the poset. Our algorithm is the first constant amortized time algorithm for generating a ``naturally defined'' class of combinatorial objects for which the corresponding counting problem is #P-complete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modified to efficiently count linear extensions, and to compute P(x < y) , for all pairs x,y , in time O( n^2 + e ( P )).

. In this paper, we prove the degenerations of Schubert varieties in a minuscule G=P , as well as the class of Kempf varieties in the flag variety SL(n)=B, to (normal) toric varieties. As a consequence, we obtain that determinantal varieties degenerate to (normal) toric varieties. Introduction In this paper, we carry out the proof of the results announced in [21]. Let G be a semisimple, simply connected algebraic group defined over an algebraically closed field k. Fix a maximal torus T in G, a Borel subgroup B oe T . Let W be the Weyl group of G relative to T . Let Q ' B be a parabolic subgroup of classical type, say Q = T r i=1 P k i , where P k i , 1 i r, is a maximal parabolic subgroup of classical type (see [26] for the definition of a parabolic subgroup of classical type). Let W (Q) be the Weyl group of Q. For w 2 W=W (Q), let X(w)(= BwQ (mod Q) with the canonical reduced structure of a scheme) denote the Schubert variety in G=Q, associated to w. Given m = (m 1 ; : : : ; m ...

Several algorithms [Cas92, MS89, Run92, DDHL94a, DDHL95, GMM95] have been proposed to automatically insert a class into an inheritance hierarchy. But actual hierarchies all include overriden and overloaded properties that these algorithms handle either very partially or not at all. Partially handled means handled provided there is a separate given function f able to compare overloaded properties [DDHL95, GMM95]. In this paper, we describe a new version of our algorithm (named Ares) which handles automatic class insertion more efficiently using such a function f . Although impossible to fully define, this function can be computed for a number of well defined cases of overloading and overriding. We give a classification of such cases and describe the computation process for a well-defined set of nontrivial cases. The algorithm preserves these important properties: - preservation of the maximal factorization of properties - preservation of the underlying structure (Galois lattice) of t...

Balancing networks, originally introduced by Aspnes et al. (Proc. of the 23rd Annual ACM Symposium on Theory of Computing, pp. 348-358, May 1991), represent a new class of distributed, low-contention data structures suitable for solving many fundamental multi-processor coordination problems that can be expressed as balancing problems. In this work, we present a mathematical study of the combinatorial structure of balancing networks, andavariety of its applications. Our study identies important combinatorial transfer parameters of balancing networks. In turn, necessary and sucient combinatorial conditions are established, expressed in terms of transfer parameters, which precisely characterize many important and well studied classes of balancing networks suchascounting networks and smoothing networks.We propose these combinatorial conditions to be \balancing analogs" of the well known Zero-One principle holding for sorting networks.

A lattice theoretic approach is developed to study the properties of functional dependencies in relational databases. The particular attention is paid to the analysis of the semilattice of closed sets, the lattice of all closure operations on a given set and to a new characterization of normal form relation schemes. Relation schemes with restrictions on functional dependencies are also studied. 1. Introduction The relational datamodel was defined by E.F. Codd [14] in 1970, and it is still one of the most powerful database models. In this model a relation is a matrix (table) every row of which corresponds to a record and every column to an attribute. This model has been widely studied. One of the most important branches in the theory of relational databases is that dealing with the design of database schemes. This branch is based on the theory of dependencies and constraints. In this paper we study the functional dependencies. Informally, functional dependency means that some attribu...

Abstract. We describe a new condensation method for computing the submodule lattice of a module for a finite dimensional algebra over a finite field, which exploits the idea of condensation and extends it to the case of primitive idempotents. The method has been implemented in the new C version of the Meat-Axe developed at Aachen, and we give several examples which have been analysed with our method.

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Abstract. In this work, we consider an interesting variant of the wellstudied KP model [18] for selfish routing that reflects some influence from the much older Wardrop model [31]. In the new model, user traffics are still unsplittable, while social cost is now the expectation of the sum, over all links, of a certain polynomial evaluated at the total latency incurred by all users choosing the link; we call it polynomial social cost. The polynomials that we consider have non-negative coefficients. We are interested in evaluating Nash equilibria in this model, and we use the Price of Anarchy as our evaluation measure. We prove the Fully Mixed Nash Equilibrium Conjecture for identical users and two links, and establish an approximate version of the conjecture for arbitrary many links. Moreover, we give upper bounds on the Price of Anarchy. 1