Maximum flow is approximable by deterministic constant-time algorithm in sparse networks

We show a deterministic constant-time parallel algorithm for finding an almost maximum flow in multisource-multitarget networks with bounded degrees and bounded edge capacities. As a consequence, we show that the value of the maximum flow over the number of nodes is a testable parameter on these networks.Comment: 8 page

Maximum flow is approximable by deterministic constant-time algorithm in sparse networks

We show a deterministic constant-time parallel algorithm for finding an almost maximum flow in multisource-multitarget networks with bounded degrees and bounded edge capacities. As a consequence, we show that the value of the maximum flow over the number of nodes is a testable parameter on these networks

Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points

Let P be a set of n points in the plane in general position, and consider the problem of finding an axis-parallel rectangle with a given perimeter, or area, or diagonal, that encloses the maximum number of points of P. We present an exact algorithm that finds such a rectangle in O(n^{5/2} log n) time, and, for the case of a fixed perimeter or diagonal, we also obtain (i) an improved exact algorithm that runs in O(nk^{3/2} log k) time, and (ii) an approximation algorithm that finds, in O(n+(n/(k epsilon^5))*log^{5/2}(n/k)log((1/epsilon) log(n/k))) time, a rectangle of the given perimeter or diagonal that contains at least (1-epsilon)k points of P, where k is the optimum value. We then show how to turn this algorithm into one that finds, for a given k, an axis-parallel rectangle of smallest perimeter (or area, or diagonal) that contains k points of P. We obtain the first subcubic algorithms for these problems, significantly improving the current state of the art

Parallel and Distributed Algorithms for the Housing Allocation Problem

We give parallel and distributed algorithms for the housing allocation problem. In this problem, there is a set of agents and a set of houses. Each agent has a strict preference list for a subset of houses. We need to find a matching such that some criterion is optimized. One such criterion is Pareto Optimality. A matching is Pareto optimal if no coalition of agents can be strictly better off by exchanging houses among themselves. We also study the housing market problem, a variant of the housing allocation problem, where each agent initially owns a house. In addition to Pareto optimality, we are also interested in finding the core of a housing market. A matching is in the core if there is no coalition of agents that can be better off by breaking away from other agents and switching houses only among themselves. In the first part of this work, we show that computing a Pareto optimal matching of a house allocation is in {\bf CC} and computing the core of a housing market is {\bf CC}-hard. Given a matching, we also show that verifying whether it is in the core can be done in {\bf NC}. We then give an algorithm to show that computing a maximum Pareto optimal matching for the housing allocation problem is in {\bf RNC}^2 and quasi-{\bf NC}^2. In the second part of this work, we present a distributed version of the top trading cycle algorithm for finding the core of a housing market. To that end, we first present two algorithms for finding all the disjoint cycles in a functional graph: a Las Vegas algorithm which terminates in $O(\log l)$ rounds with high probability, where $l$ is the length of the longest cycle, and a deterministic algorithm which terminates in $O(\log^* n \log l)$ rounds, where $n$ is the number of nodes in the graph. Both algorithms work in the synchronous distributed model and use messages of size $O(\log n)$