Maximally Disjoint Solutions of the Set Covering Problem

This paper is concerned with finding two solutions of a set covering problem that have a minimum number of variables in common. We show that this problem is NP­ complete, even in the case where we are only interested in completely disjoint solutions. We describe three heuristic methods based on the standard greedy algorithm for set covering problems. Two of these algorithms find the solutions sequentially, while the third finds them simultaneously. A local search method for reducing the overlap of the two given solutions is then described. This method involves the solution of a reduced set covering problem. Finally, extensive computational tests are given demonstrating the nature of these algorithms. These tests are carried out both on randomly generated problems and on problems found in the literature

The monodromy groups of Schwarzian equations on closed Riemann surfaces

Let \theta:\pi_1(R) \to \PSL(2,\C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. Theorem. Necessary and sufficient for \theta to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that \theta(\pi_1(R)) be nonelementary. A branch point is required if and only if the representation \theta does not lift to \SL(2,\C).Comment: 80 pages, published versio

Geometric Aspects of Holographic Bit Threads

We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: spheres and strips in vacuum AdS, and strips in a black brane geometry. Studying more generic bulk metrics, we uncover a sufficient set of conditions on the geometry and matter fields that must hold to be able to use our prescription. (ii) Based on the nesting property of holographic entanglement entropy, we develop a method to construct bit threads that maximize the flux through a given bulk region. As a byproduct, we are able to construct more general thread configurations by combining (i) and (ii) in multiple patches. We apply our methods to study bit threads which simultaneously compute the entanglement entropy and the entanglement of purification of mixed states and comment on their interpretation in terms of entanglement distillation. We also consider the case of disjoint regions for which we can explicitly construct the so-called multi-commodity flows and show that the monogamy property of mutual information can be easily illustrated from our constructions.Comment: 48 pages, multiple figures. v3: matches published versio

Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)

We introduce a theory on marginal values and their core stability for cooperative games with arbitrary coalition structure. The theory is based on the notion of nested sets and the complex of nested sets associated to an arbitrary set system and the M-extension of a game for this set. For a set system being a building set or partition system, the corresponding complex is a polyhedral complex, and the vertices of this complex correspond to maximal strictly nested sets. To each maximal strictly nested set is associated a rooted tree. Given characteristic function, to every maximal strictly nested set a marginal value is associated to a corresponding rooted tree as in [9]. We show that the same marginal value is obtained by using the M-extension for every permutation that is associated to the rooted tree. The GC-solution is defined as the average of the marginal values over all maximal strictly nested sets. The solution can be viewed as the gravity center of the image of the vertices of the polyhedral complex. The GC-solution differs from the Myerson-kind value defined in [2] for union stable structures. The HS-solution is defined as the average of marginal values over the subclass of so-called half-space nested sets. The NT-solution is another solution and is defined as the average of marginal values over the subclass of NT-nested sets. For graphical buildings the collection of NT-nested sets corresponds to the set of spanning normal trees on the underlying graph and the NT-solution coincides with the average tree solution. We also study core stability of the solutions and show that both the HS-solution and NT-solution belong to the core under half-space supermodularity, which is a weaker condition than convexity of the game. For an arbitrary set system we show that there exists a unique minimal building set containing the set system. As solutions we take the solutions for this building covering by extending in a natural way the characteristic function to it by using its Möbius inversion.Core;polytope;building set;nested set complex;Möbius inversion;permutations;normal fan;average tree solution;Myerson value