This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial three-dimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published

We show that any k-regular bipartite graph with 2n vertices has at least ( (k\Gamma1) k\Gamma1 k k\Gamma2 ) n perfect matchings (1-factors). Equivalently, this is a lower bound on the permanent of any nonnegative integer n \Theta n matrix with each row and column sum equal to k. For any k, the base (k\Gamma1) k\Gamma1 k k\Gamma2 is largest possible. 1.

The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants -- see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area. 1 Introduction The theory of random graphs as initiated by Erdos and R'enyi [52] and developed along with others, has been mainly concerned with structural properties, in particular the most likely values of various graph invariantss -- see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. We would like in this paper to survey some of the results in this area. We hope to be fairly comprehensive in terms of the areas we tackle and so depth will be sacrificed in favour of breadth. One attractive feature of average case analysis is that it banishes the pessimism o...

We analyze the behavior of augmenting paths in random graphs. Our results show that in almost every graph, any non-maximum 0-1 flow admits a short augmenting path. This enables us to prove that augmenting path algorithms, which are fast in the worst case, also perform exceedingly well on the average. In particular, we show that the O(√(|V|) |E|) algorithms for bipartite and general matchings run in almost linear time with high probability. It is also shown that the expected running time of the matching algorithms is O(|E|) on input graphs chosen uniformly at random from the set of all graphs. We establish that the permanent of almost every bipartite graph can be approximated in polynomial time. We extend our results to the analysis of the running time of Dinic's algorithm for finding factors of graphs.

Abstract. In the random graph G(n, p) with pn bounded, the degrees of the vertices are almost i.i.d Poisson random variables with mean λ: = p(n − 1). Motivated by this fact, we introduce the Poisson cloning model GP C(n, p) for random graphs in which the degrees are i.i.d Poisson random variables with mean λ. Then, we first establish a theorem that shows the new model is equivalent to the classical model G(n, p) in an asymptotic sense. Next, we introduce a useful algorithm, called the cut-off line algorithm, to generate the random graph GP C(n, p). The Poisson cloning model GP C(n, p) equipped the cut-off line algorithm enables us to very precisely analyze the sizes of the largest component and the t-core of G(n, p). This new approach to the problems yields not only elegant proofs but also improved bounds that are essentially best possible. We also consider the Poisson cloning models for random hypergraphs and random k-SAT problems. Then, the t-core problem for random hypergraphs and the pure literal algorithm for random k-SAT problems are analyzed. 1

this paper, c0, cl,... are positive absolute constants whose precise values are not too important to us

We do a probabilistic analysis of the problem of distributing a single piece of information to the vertices of a graph G. Assuming that the input graph G is G n;p , we prove an O(ln n=n) upper bound on the edge density needed so that with high probability the information can be broadcast in dlog 2 ne rounds.

We consider the existence of perfect matchings in random graphs with n vertices (or n+n vertices in the bipartite case) and m random edges, subject to a lower bound on minimum vertex degree. A random bipartite graph without isolated vertices and m > n edges with high probabilty (whp) has a perfect matching i the average vertex degree is 0:5 log n + log log n + cn , cn ! 1 however slow. A random graph with minimum degree at least two whp has a matching that matches all the vertices except "odd-man-out" vertices, one per each isolated cycle of odd length, and one for the remaining vertex set if its cardinality is odd. So, for n even, whp the random graph has a perfect matching i it does not have isolated odd cycles.

For any positive integers r and n, let H(r, n) denote the family of graphs on n vertices with maximum degree r, and let H(r, n, n) denote the family of bipartite graphs H on 2n vertices with n vertices in each vertex class, and with maximum degree r. On one hand, we note that any H(r, n)-universal graph must have Ω(n2−2/r) edges. On the other hand, for any n ≥ n0(r), we explicitly construct H(r, n)-universal graphs G and Λ on n and 2n vertices, and with O(n 2−Ω ( 1 r log r) ) and O(n 2 − 1 r log 1/r n) edges, re-spectively, such that we can efficiently find a copy of any H ∈ H(r, n) in G deterministically. We also achieve sparse universal graphs using random constructions. Finally, we show that the bipartite random graph G = G(n, n, p), with 1 − p = cn 2r log 1/2r n is fault-tolerant; for a large enough constant c, even after deleting any α-fraction of the edges of G, the resulting graph is still H(r, a(α)n, a(α)n)-universal for some a: [0, 1) → (0, 1].

Results are obtained on the likely connectivity properties and sizes of circuits in the column dependence matroid of a random rxn matrix over a finite field, for large rand n. In a sense made precise in the paper, it is shown to be highly probable that when n is less than r such a matroid is the free matroid on n points, while if n exceeds r it is a connected matroid of rank r. Moreover, the connectivity can be strengthened under additional hypotheses on the growth of nand r using the notion of vertical connectivity; and the values of k for which circuits of size k exist can be determined in terms of n and r.