Access to Data and Number of Iterations: Dual Primal Algorithms for Maximum Matching under Resource Constraints

In this paper we consider graph algorithms in models of computation where the space usage (random accessible storage, in addition to the read only input) is sublinear in the number of edges $m$ and the access to input data is constrained. These questions arises in many natural settings, and in particular in the analysis of MapReduce or similar algorithms that model constrained parallelism with sublinear central processing. In SPAA 2011, Lattanzi etal. provided a $O(1)$ approximation of maximum matching using $O(p)$ rounds of iterative filtering via mapreduce and $O(n^{1+1/p})$ space of central processing for a graph with $n$ nodes and $m$ edges. We focus on weighted nonbipartite maximum matching in this paper. For any constant $p>1$, we provide an iterative sampling based algorithm for computing a $(1-epsilon)$-approximation of the weighted nonbipartite maximum matching that uses $O(p/epsilon)$ rounds of sampling, and $O(n^{1+1/p})$ space. The results extends to $b$-Matching with small changes. This paper combines adaptive sketching literature and fast primal-dual algorithms based on relaxed Dantzig-Wolfe decision procedures. Each round of sampling is implemented through linear sketches and executed in a single round of MapReduce. The paper also proves that nonstandard linear relaxations of a problem, in particular penalty based formulations, are helpful in mapreduce and similar settings in reducing the adaptive dependence of the iterations.

Near Linear Time Approximation Schemes for Uncapacitated and Capacitated b--Matching Problems in Nonbipartite Graphs

We present the first near optimal approximation schemes for the maximum weighted (uncapacitated or capacitated) $b$--matching problems for non-bipartite graphs that run in time (near) linear in the number of edges. For any $delta>3/sqrt{n}$ the algorithm produces a $(1-delta)$ approximation in $O(m poly(delta^{-1},log n))$ time. We provide fractional solutions for the standard linear programming formulations for these problems and subsequently also provide (near) linear time approximation schemes for rounding the fractional solutions. Through these problems as a vehicle, we also present several ideas in the context of solving linear programs approximately using fast primal-dual algorithms. First, even though the dual of these problems have exponentially many variables and an efficient exact computation of dual weights is infeasible, we show that we can efficiently compute and use a sparse approximation of the dual weights using a combination of (i) adding perturbation to the constraints of the polytope and (ii) amplification followed by thresholding of the dual weights. Second, we show that approximation algorithms can be used to reduce the width of the formulation, and faster convergence.

Approximation Algorithms for Bayesian Multi-Armed Bandit Problems

In this paper, we consider several finite-horizon Bayesian multi-armed bandit problems with side constraints which are computationally intractable (NP-Hard) and for which no optimal (or near optimal) algorithms are known to exist with sub-exponential running time. All of these problems violate the standard exchange property, which assumes that the reward from the play of an arm is not contingent upon when the arm is played. Not only are index policies suboptimal in these contexts, there has been little analysis of such policies in these problem settings. We show that if we consider near-optimal policies, in the sense of approximation algorithms, then there exists (near) index policies. Conceptually, if we can find policies that satisfy an approximate version of the exchange property, namely, that the reward from the play of an arm depends on when the arm is played to within a constant factor, then we have an avenue towards solving these problems. However such an approximate version of the idling bandit property does not hold on a per-play basis and are shown to hold in a global sense. Clearly, such a property is not necessarily true of arbitrary single arm policies and finding such single arm policies is nontrivial. We show that by restricting the state spaces of arms we can find single arm policies and that these single arm policies can be combined into global (near) index policies where the approximate version of the exchange property is true in expectation. The number of different bandit problems that can be addressed by this technique already demonstrate its wide applicability.

Information Acquisition and Exploitation in Multichannel Wireless Networks

A wireless system with multiple channels is considered, where each channel has several transmission states. A user learns about the instantaneous state of an available channel by transmitting a control packet in it. Since probing all channels consumes significant energy and time, a user needs to determine what and how much information it needs to acquire about the instantaneous states of the available channels so that it can maximize its transmission rate. This motivates the study of the trade-off between the cost of information acquisition and its value towards improving the transmission rate. A simple model is presented for studying this information acquisition and exploitation trade-off when the channels are multi-state, with different distributions and information acquisition costs. The objective is to maximize a utility function which depends on both the cost and value of information. Solution techniques are presented for computing near-optimal policies with succinct representation in polynomial time. These policies provably achieve at least a fixed constant factor of the optimal utility on any problem instance, and in addition, have natural characterizations. The techniques are based on exploiting the structure of the optimal policy, and use of Lagrangean relaxations which simplify the space of approximately optimal solutions.