38 research outputs found
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]
Multiplicative Bidding in Online Advertising
In this paper, we initiate the study of the multiplicative bidding language
adopted by major Internet search companies. In multiplicative bidding, the
effective bid on a particular search auction is the product of a base bid and
bid adjustments that are dependent on features of the search (for example, the
geographic location of the user, or the platform on which the search is
conducted). We consider the task faced by the advertiser when setting these bid
adjustments, and establish a foundational optimization problem that captures
the core difficulty of bidding under this language. We give matching
algorithmic and approximation hardness results for this problem; these results
are against an information-theoretic bound, and thus have implications on the
power of the multiplicative bidding language itself. Inspired by empirical
studies of search engine price data, we then codify the relevant restrictions
of the problem, and give further algorithmic and hardness results. Our main
technical contribution is an -approximation for the case of
multiplicative prices and monotone values. We also provide empirical
validations of our problem restrictions, and test our algorithms on real data
against natural benchmarks. Our experiments show that they perform favorably
compared with the baseline.Comment: 25 pages; accepted to EC'1
Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP
Abstract — We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. PATH-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem. We present (2 − ɛ)-approximation algorithms for all three problems, connected by a unified technique for improving prizecollecting algorithms that allows us to circumvent the integrality gap barrier. 1
Metric Clustering and MST with Strong and Weak Distance Oracles
We study optimization problems in a metric space where we
can compute distances in two ways: via a ''strong'' oracle that returns exact
distances , and a ''weak'' oracle that returns distances
which may be arbitrarily corrupted with some probability. This
model captures the increasingly common trade-off between employing both an
expensive similarity model (e.g. a large-scale embedding model), and a less
accurate but cheaper model. Hence, the goal is to make as few queries to the
strong oracle as possible. We consider both so-called ''point queries'', where
the strong oracle is queried on a set of points and
returns for all , and ''edge queries'' where it is queried
for individual distances .
Our main contributions are optimal algorithms and lower bounds for clustering
and Minimum Spanning Tree (MST) in this model. For -centers, -median, and
-means, we give constant factor approximation algorithms with only
strong oracle point queries, and prove that queries
are required for any bounded approximation. For edge queries, our upper and
lower bounds are both . Surprisingly, for the MST problem
we give a approximation algorithm using no strong oracle
queries at all, and a matching lower bound. We
empirically evaluate our algorithms, and show that their quality is comparable
to that of the baseline algorithms that are given all true distances, but while
querying the strong oracle on only a small fraction () of points