3 research outputs found
A Lower Bound for the Distributed Lov\'asz Local Lemma
We show that any randomised Monte Carlo distributed algorithm for the
Lov\'asz local lemma requires communication rounds,
assuming that it finds a correct assignment with high probability. Our result
holds even in the special case of , where is the maximum degree
of the dependency graph. By prior work, there are distributed algorithms for
the Lov\'asz local lemma with a running time of rounds in
bounded-degree graphs, and the best lower bound before our work was
rounds [Chung et al. 2014].Comment: 17 pages, 3 figure
On Approximating (Sparse) Covering Integer Programs
We consider approximation algorithms for covering integer programs of the
form min over subject to and ; where , , and all have
nonnegative entries. We refer to this problem as , and the
special case without the multiplicity constraints as
. These problems generalize the well-studied Set
Cover problem. We make two algorithmic contributions.
First, we show that a simple algorithm based on randomized rounding with
alteration improves or matches the best known approximation algorithms for
and in a wide range of
parameter settings, and these bounds are essentially optimal. As a byproduct of
the simplicity of the alteration algorithm and analysis, we can derandomize the
algorithm without any loss in the approximation guarantee or efficiency.
Previous work by Chen, Harris and Srinivasan [12] which obtained near-tight
bounds is based on a resampling-based randomized algorithm whose analysis is
complex.
Non-trivial approximation algorithms for are based on
solving the natural LP relaxation strengthened with knapsack cover (KC)
inequalities [5,24,12]. Our second contribution is a fast (essentially
near-linear time) approximation scheme for solving the strengthened LP with a
factor of speed up over the previous best running time [5].
Together, our contributions lead to near-optimal (deterministic)
approximation bounds with near-linear running times for
and .Comment: SODA 201
An Extension of the Lovász Local Lemma, and its Applications to Integer Programming
The Lov'asz Local Lemma (LLL) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. We consider three classes of NP-hard integer programs: minimax, packing, and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan & Thompson to derive good approximation algorithms for such problems. We use our extended LLL to prove that randomized rounding produces, with non-zero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are sparse (e.g., VLSI routing using short paths, problems on hypergraphs with small dimension/degree). We also generalize the method of pessimistic estimators due to Raghavan, to constructivize our packing and covering results.