On random cover and matching problems

Abstract: This thesis consists of the following papers.I  J. Larsson, The Minimum Matching in Pseudo-dimension 0 < q < 1, submittedII  V. Falgas-Ravry, J. Larsson, K. Markström, Speed and concentration of the covering time for structured coupon collectors, submittedIII  J. Larsson, K. Markström, Biased random k-SAT problems, manuscriptThese papers can all be seen as variations on the same question: Given a large set V and a family F of subsets of V, each assigned a (random) weight, we assign each subfamily G ⊆ F a cost based on the weights of sets that occur in it. What will the minimal cost of a subfamiliy G that covers V be?In the first paper, we search for a disjoint cover of the ground set V = {u_1,u_2,...u_n,v_1,v_2,...v_n}, using random 2-sets of the form {u_i, v_j}. In other words, we search for matchings in a bipartite graph. Each edge receives a random weight distributed uniformly in [0, 1], and the cost of a perfect matching using edges with weights l_1,l_2,...l_n is Σ_{i=1}^n l_i^{1/q} for some q > 0.The second paper lives in a more general setting. There we search for any cover of the ground set V, for general families F. Each set f ∈ F receives weight w(f) uniformly at random from [0,1]. The cost of a cover f_1,f_2,...f_m is then taken to be max_i w(f_i). This is equivalent (after a rescaling) to drawing sets from F at Poisson times, and the cost of a cover is the first time V is covered. This problem is known under a number of names, perhaps most famously the coupon collector problem. In the classical formulation, single elements of V are drawn, not sets. The classical coupon collector thus corresponds to the family F consisting of singleton sets, and we call the version allowing larger sets structured coupon collector problems. The main concern of this paper is to identify relevant properties of F that affect the covering time (i.e. minimal cost of a cover), and to provide (easily checkable) sufficient conditions for concentration of the covering time.For the third paper we narrow the scopes once more, and study the biased random k-SAT problem. The random k-SAT problem can be seen as a special case of the structured coupon collector, but a special case that has far richer structure than the generic case. The ground set is the hypercube Σ_n = {0, 1}^n, and the coupons are all the k-codimensional subcubes of Σ_n. We study a slight variation on this problem: subcubes are drawn with a constant bias towards 0, so that vertices in Σ_n with fewer 1's and more 0's are easier to cover.