Gregory Schwartzman

Fast Distributed Algorithms for Testing Graph Properties

We provide a thorough study of distributed property testing – producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called dense graph testing model we emulate sequential tests for nearly all graph properties having 1-sided tests, while in the general and sparse models we obtain faster tests for triangle-freeness, cycle-freeness and bipartiteness, respectively. In addition, we show a logarithmic lower bound for testing bipartiteness and cycle-freeness, which holds even in the LOCAL model.

A (2 + ϵ )-approximation for maximum weight matching in the semi-streaming model

We present a simple deterministic single-pass (2 + ϵ)-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of (3.5 + ϵ). Our algorithm uses O(n log2 n) space for constant values of ϵ. It relies on a variation of the local-ratio theorem, which may be of independent interest in the semi-streaming model.

A Distributed (2+ε)-Approximation for Vertex Cover in O(logδ/ε log log δ) Rounds

We present a simple deterministic distributed (2+ε) approximation algorithm for minimum weight vertex cover, which completes in O(logδ/εlog logδ) rounds, where δ is the maximum degree in the graph, for any ε > 0 which is at most O(1). For a constant ε, this implies a constant approximation in Ologδ/log log δ) rounds, which contradicts the lower bound of [KMW10].

Distributed Approximation of Maximum Independent Set and Maximum Matching

We present a simple distributed Δ-approximation algorithm for maximum weight independent set (MaxIS) in the CONGEST model which completes in O(MIS ⋅ log W) rounds, where Δ is the maximum degree, MIS is the number of rounds needed to compute a maximal independent set (MIS) on G, and W is the maximum weight of a node. Plugging in the best known algorithm for MIS gives a randomized solution in O(log n log W) rounds, where n is the number of nodes. We also present a deterministic O(Δ +log* n)-round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a 2-approximation for maximum weight matching without incurring any additional round penalty in the CONGEST model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of local aggregation algorithms for which we describe a mechanism that allows the simulation to run in the CONGEST model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to (2+ε) allows us to devise a distributed algorithm requiring O((log Δ)/(log logΔ)) rounds for any constant ε>0. For the unweighted case, we can even obtain a (1+ε)-approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on Δ.

Fast distributed algorithms for testing graph properties

We initiate a thorough study of distributed property testing—producing algorithms for the approximation problems of property testing in the CONGEST model. In particular, for the so-called dense graph testing model we emulate sequential tests for nearly all graph properties having 1-sided tests, while in the general model we obtain faster tests for triangle-freeness and cycle-freeness, and in the sparse model we obtain a faster test for bipartiteness. In addition, we show a logarithmic lower bound for testing bipartiteness and cycle-freeness, which holds even in the stronger LOCAL model. In most cases, aided by parallelism, the distributed algorithms have a much shorter running time than their counterparts from the sequential querying model of traditional property testing. More importantly, the distributed algorithms we develop for testing graph properties are in many cases much faster than what is known for exactly deciding whether the property holds. The simplest property testing algorithms allow a relatively smooth transition to the distributed model. For the more complex tasks we develop new machinery that may be of independent interest.

A Distributed (2 + ε)-Approximation for Vertex Cover in O(log Δ / ε log log Δ) Rounds

We present a simple deterministic distributed (2 + ε)-approximation algorithm for minimum-weight vertex cover, which completes in O(log Δ/εlog log Δ) rounds, where Δ is the maximum degree in the graph, for any ε > 0 that is at most O(1). For a constant ε, this implies a constant approximation in O(log Δ/log log Δ) rounds, which contradicts the lower bound of [KMW10].