68 research outputs found
Near-Quadratic Lower Bounds for Two-Pass Graph Streaming Algorithms
We prove that any two-pass graph streaming algorithm for the -
reachability problem in -vertex directed graphs requires near-quadratic
space of bits. As a corollary, we also obtain near-quadratic space
lower bounds for several other fundamental problems including maximum bipartite
matching and (approximate) shortest path in undirected graphs.
Our results collectively imply that a wide range of graph problems admit
essentially no non-trivial streaming algorithm even when two passes over the
input is allowed. Prior to our work, such impossibility results were only known
for single-pass streaming algorithms, and the best two-pass lower bounds only
ruled out space algorithms, leaving open a large gap between
(trivial) upper bounds and lower bounds
Set Covering with Our Eyes Wide Shut
In the stochastic set cover problem (Grandoni et al., FOCS '08), we are given
a collection of sets over a universe of size
, and a distribution over elements of . The algorithm draws
elements one-by-one from and must buy a set to cover each element on
arrival; the goal is to minimize the total cost of sets bought during this
process. A universal algorithm a priori maps each element
to a set such that if is formed by drawing
times from distribution , then the algorithm commits to outputting .
Grandoni et al. gave an -competitive universal algorithm for this
stochastic set cover problem.
We improve unilaterally upon this result by giving a simple, polynomial time
-competitive universal algorithm for the more general prophet
version, in which is formed by drawing from different distributions
. Furthermore, we show that we do not need full foreknowledge
of the distributions: in fact, a single sample from each distribution suffices.
We show similar results for the 2-stage prophet setting and for the
online-with-a-sample setting.
We obtain our results via a generic reduction from the single-sample prophet
setting to the random-order setting; this reduction holds for a broad class of
minimization problems that includes all covering problems. We take advantage of
this framework by giving random-order algorithms for non-metric facility
location and set multicover; using our framework, these automatically translate
to universal prophet algorithms
Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs
As massive graphs become more prevalent, there is a rapidly growing need for
scalable algorithms that solve classical graph problems, such as maximum
matching and minimum vertex cover, on large datasets. For massive inputs,
several different computational models have been introduced, including the
streaming model, the distributed communication model, and the massively
parallel computation (MPC) model that is a common abstraction of
MapReduce-style computation. In each model, algorithms are analyzed in terms of
resources such as space used or rounds of communication needed, in addition to
the more traditional approximation ratio.
In this paper, we give a single unified approach that yields better
approximation algorithms for matching and vertex cover in all these models. The
highlights include:
* The first one pass, significantly-better-than-2-approximation for matching
in random arrival streams that uses subquadratic space, namely a
-approximation streaming algorithm that uses space
for constant .
* The first 2-round, better-than-2-approximation for matching in the MPC
model that uses subquadratic space per machine, namely a
-approximation algorithm with memory per
machine for constant .
By building on our unified approach, we further develop parallel algorithms
in the MPC model that give a -approximation to matching and an
-approximation to vertex cover in only MPC rounds and
memory per machine. These results settle multiple open
questions posed in the recent paper of Czumaj~et.al. [STOC 2018]
A Simple -Approximation Semi-Streaming Algorithm for Maximum (Weighted) Matching
We present a simple semi-streaming algorithm for -approximation
of bipartite matching in passes. This matches the
performance of state-of-the-art "-efficient" algorithms, while being
considerably simpler.
The algorithm relies on a "white-box" application of the multiplicative
weight update method with a self-contained primal-dual analysis that can be of
independent interest. To show case this, we use the same ideas, alongside
standard tools from matching theory, to present an equally simple
semi-streaming algorithm for -approximation of weighted matchings
in general (not necessarily bipartite) graphs, again in
passes
- …