Coalition formation is a key topic in multiagent systems. One may prefer a coalition structure that maximizes the sum of the values of the coalitions, but often the number of coalition structures is too large to allow exhaustive search for the optimal one. Furthermore, finding the optimal coalition structure is NP-complete. But then, can the coalition structure found via a partial search be guaranteed to be within a bound from optimum? We show that none of the previous coalition structure generation algorithms can establish any bound because they search fewer nodes than a threshold that we show necessary for establishing a bound. We present an algorithm that establishes a tight bound within this minimal amount of search, and show that any other algorithm would have to search strictly more. The fraction of nodes needed to be searched approaches zero as the number of agents grows. If additional time remains, our anytime algorithm searches further, and establishes a progressively lower tight bound. Surprisingly, just searching one more node drops the bound in half. As desired, our algorithm lowers the bound rapidly early on, and exhibits diminishing returns to computation. It also significantly outperforms its obvious contenders. Finally, we show how to distribute the desired

We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package.

We introduce analogues of the lattice of non-crossing set partitions for the classical reflection groups of type B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as well-behaved as the original non-crossing set partitions, and the type D analogues almost as well-behaved. In both cases, they are EL-labellable ranked lattices with symmetric chain decompositions (self-dual for type B), whose rank-generating functions, zeta polynomials, rank-selected chain numbers have simple closed forms.

High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.

Abstract. We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy µ on the space of tilings of the plane with dominos. We construct a measure ν on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the ν-measures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for µ and ν, and compute the rate of convergence to mixing (the correlation between distant events). For the measure ν we compute the variance of the height function. Resumé. Soit µ la mesure d’entropie maximale sur l’espace X des pavages du plan par des dominos. On calcule explicitement la mesure des sous-ensembles cylindriques de X. De même, on construit une mesure ν d’entropie maximale sur l’espace X ′ des pavages du plan par losanges, et on calcule explicitement la mesure des sous-ensembles cylindriques. Comme application on calcule, pour µ et ν, les correlations d’évenements distants, ainsi que la ν-variance de la fonction “hauteur ” sur X ′. 1.

An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.

this paper, we consider the problem of enumerating the fully commutative elements of these groups. The main result (Theorem 2.6) is that for six of the seven infinite families (we omit the trivial dihedral family I 2 (m)), the generating function for the number of fully commutative elements can be expressed in terms of three simpler generating functions for certain formal languages over an alphabet with at most four letters. The languages in question vary from family to family, but have a uniform description. The resulting generating function one obtains for each family is algebraic, although in some cases quite complicated. (See (3.7) and (3.11).) In a general Coxeter group, the fully commutative elements index a basis for a natural quotient of the corresponding Iwahori-Hecke algebra [G]. (See also [F1] for the simplylaced case.) For An , this quotient is the Temperley-Lieb algebra. Recently, Fan [F2] has shown that for types A, B, D, E and (in a sketched proof) F , this quotient is generically semisimple, and gives recurrences for the dimensions of the irreducible representations. (For type H, the question of semisimplicity remains open.) This provides another possible approach to computing the number of fully commutative elements in these cases; namely, as the sum of the squares of the dimensions of these representations. Interestingly, Fan also shows that the sum of these dimensions is the number of fully commutative involutions

This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class -- an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.

This paper describes a systematic approach to the enumeration of "noncrossing" geometric configurations built on vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc.

This paper might have been subtitled "An algorithmicist looks at no free lunch." We use simple adversary arguments to redevelop and explore some ofthenofreelunch (NFL) theorems and perhaps extend them a little. A second goal is to clarify the relationship of NFL theorems to algorithm theory. In particular we claim that NFL puts much weaker restrictions on the claims that an evolutionary algorithm can make than does acceptance of the conjectures of traditional complexity theory. And third we take a brief look at whether the notion of natural evolution relates to optimization, and what if any the implications of evolution are for computing. In this part, we mostly try to raise questions concerning the validity of applying the genetic model to the problem of optimization. This is an informal paper -- most of the information presented is not formally proven, and is either "common knowledge" or formally proven elsewhere. Some of the claims are intuitions based on experience with algorithms, and in a more formal setting should be classi ed as conjectures. Thegoalisnotsomuch todevelop theory, asitisto perhaps persuade the reader to adopt a particular viewpoint.