Lattice Linear Predicate Algorithms for the Constrained Stable Marriage Problem with Ties

We apply Lattice-Linear Predicate Detection Technique to derive parallel and distributed algorithms for various variants of the stable matching problem. These problems are: (a) the constrained stable marriage problem (b) the super stable marriage problem in presence of ties, and (c) the strongly stable marriage in presence of ties. All these problems are solved using the Lattice-Linear Predicate (LLP) algorithm showing its generality. The constrained stable marriage problem is a version of finding the stable marriage in presence of lattice-linear constraints such as ``Peter's regret is less than that of Paul.'' For the constrained stable marriage problem, we present a distributed algorithm that takes $O(n^2)$ messages each of size $O(\log n)$ where $n$ is the number of men in the problem. Our algorithm is completely asynchronous. Our algorithms for the stable marriage problem with ties are also parallel with no synchronization.Comment: arXiv admin note: text overlap with arXiv:1812.1043

Computing With Distributed Information

The age of computing with massive data sets is highlighting new computational challenges. Nowadays, a typical server may not be able to store an entire data set, and thus data is often partitioned and stored on multiple servers in a distributed manner. A natural way of computing with such distributed data is to use distributed algorithms: these are algorithms where the participating parties (i.e., the servers holding portions of the data) collaboratively compute a function over the entire data set by sending (preferably small-size) messages to each other, where the computation performed at each participating party only relies on the data possessed by it and the messages received by it. We study distributed algorithms focused on two key themes: convergence time and data summarization. Convergence time measures how quickly a distributed algorithm settles on a globally stable solution, and data summarization is the approach of creating a compact summary of the input data while retaining key information. The latter often leads to more efficient computation and communication. The main focus of this dissertation is on design and analysis of distributed algorithms for important problems in diverse application domains centering on the themes of convergence time and data summarization. Some of the problems we study include convergence time of double oral auction and interdomain routing, summarizing graphs for large-scale matching problems, and summarizing data for query processing

Matching Theory for Future Wireless Networks: Fundamentals and Applications

The emergence of novel wireless networking paradigms such as small cell and cognitive radio networks has forever transformed the way in which wireless systems are operated. In particular, the need for self-organizing solutions to manage the scarce spectral resources has become a prevalent theme in many emerging wireless systems. In this paper, the first comprehensive tutorial on the use of matching theory, a Nobelprize winning framework, for resource management in wireless networks is developed. To cater for the unique features of emerging wireless networks, a novel, wireless-oriented classification of matching theory is proposed. Then, the key solution concepts and algorithmic implementations of this framework are exposed. Then, the developed concepts are applied in three important wireless networking areas in order to demonstrate the usefulness of this analytical tool. Results show how matching theory can effectively improve the performance of resource allocation in all three applications discussed

Locally Optimal Load Balancing

This work studies distributed algorithms for locally optimal load-balancing: We are given a graph of maximum degree $\Delta$, and each node has up to $L$ units of load. The task is to distribute the load more evenly so that the loads of adjacent nodes differ by at most $1$. If the graph is a path ($\Delta = 2$), it is easy to solve the fractional version of the problem in $O(L)$ communication rounds, independently of the number of nodes. We show that this is tight, and we show that it is possible to solve also the discrete version of the problem in $O(L)$ rounds in paths. For the general case ($\Delta > 2$), we show that fractional load balancing can be solved in $\operatorname{poly}(L,\Delta)$ rounds and discrete load balancing in $f(L,\Delta)$ rounds for some function $f$, independently of the number of nodes.Comment: 19 pages, 11 figure

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 $O(\log n)$

Editorial: special issue on matching under preferences

This special issue of Algorithms is devoted to the study of matching problems involving ordinal preferences from the standpoint of algorithms and complexit