22 research outputs found
LIPIcs
The Lovász Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. Besides conditions for existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?", etc. These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs
Commutative Algorithms Approximate the LLL-distribution
Following the groundbreaking Moser-Tardos algorithm for the Lovasz Local
Lemma (LLL), a series of works have exploited a key ingredient of the original
analysis, the witness tree lemma, in order to: derive deterministic, parallel
and distributed algorithms for the LLL, to estimate the entropy of the output
distribution, to partially avoid bad events, to deal with super-polynomially
many bad events, and even to devise new algorithmic frameworks. Meanwhile, a
parallel line of work, has established tools for analyzing stochastic local
search algorithms motivated by the LLL that do not fall within the Moser-Tardos
framework. Unfortunately, the aforementioned results do not transfer to these
more general settings. Mainly, this is because the witness tree lemma,
provably, no longer holds. Here we prove that for commutative algorithms, a
class recently introduced by Kolmogorov and which captures the vast majority of
LLL applications, the witness tree lemma does hold. Armed with this fact, we
extend the main result of Haeupler, Saha, and Srinivasan to commutative
algorithms, establishing that the output of such algorithms well-approximates
the LLL-distribution, i.e., the distribution obtained by conditioning on all
bad events being avoided, and give several new applications. For example, we
show that the recent algorithm of Molloy for list coloring number of sparse,
triangle-free graphs can output exponential many list colorings of the input
graph
Using deep learning to construct stochastic local search SAT solvers with performance bounds
The Boolean Satisfiability problem (SAT) is the most prototypical NP-complete
problem and of great practical relevance. One important class of solvers for
this problem are stochastic local search (SLS) algorithms that iteratively and
randomly update a candidate assignment. Recent breakthrough results in
theoretical computer science have established sufficient conditions under which
SLS solvers are guaranteed to efficiently solve a SAT instance, provided they
have access to suitable "oracles" that provide samples from an
instance-specific distribution, exploiting an instance's local structure.
Motivated by these results and the well established ability of neural networks
to learn common structure in large datasets, in this work, we train oracles
using Graph Neural Networks and evaluate them on two SLS solvers on random SAT
instances of varying difficulty. We find that access to GNN-based oracles
significantly boosts the performance of both solvers, allowing them, on
average, to solve 17% more difficult instances (as measured by the ratio
between clauses and variables), and to do so in 35% fewer steps, with
improvements in the median number of steps of up to a factor of 8. As such,
this work bridges formal results from theoretical computer science and
practically motivated research on deep learning for constraint satisfaction
problems and establishes the promise of purpose-trained SAT solvers with
performance guarantees.Comment: 15 pages, 9 figures, code available at
https://github.com/porscheofficial/sls_sat_solving_with_deep_learnin
A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations
Computing Delaunay triangulations in involves evaluating the
so-called in\_sphere predicate that determines if a point lies inside, on
or outside the sphere circumscribing points . This
predicate reduces to evaluating the sign of a multivariate polynomial of degree
in the coordinates of the points . Despite
much progress on exact geometric computing, the fact that the degree of the
polynomial increases with makes the evaluation of the sign of such a
polynomial problematic except in very low dimensions. In this paper, we propose
a new approach that is based on the witness complex, a weak form of the
Delaunay complex introduced by Carlsson and de Silva. The witness complex
is defined from two sets and in some metric space
: a finite set of points on which the complex is built, and a set of
witnesses that serves as an approximation of . A fundamental result of de
Silva states that if .
In this paper, we give conditions on that ensure that the witness complex
and the Delaunay triangulation coincide when is a finite set, and we
introduce a new perturbation scheme to compute a perturbed set close to
such that . Our perturbation
algorithm is a geometric application of the Moser-Tardos constructive proof of
the Lov\'asz local lemma. The only numerical operations we use are (squared)
distance comparisons (i.e., predicates of degree 2). The time-complexity of the
algorithm is sublinear in . Interestingly, although the algorithm does not
compute any measure of simplex quality, a lower bound on the thickness of the
output simplices can be guaranteed.Comment: 24 page
Uniform Sampling through the Lovász Local Lemma
We propose a new algorithmic framework, called `"partial rejection sampling'', to draw samples exactly from a product distribution, conditioned on none of a number of bad events occurring. Our framework builds new connections between the variable framework of the Lovász Local Lemma and some classical sampling algorithms such as the "cycle-popping"' algorithm for rooted spanning trees. Among other applications, we discover new algorithms to sample satisfying assignments of k-CNF formulas with bounded variable occurrences
Algorithms for Fundamental Problems in Computer Networks.
Traditional studies of algorithms consider the sequential setting, where the whole input data is fed into a single device that computes the solution. Today, the network, such as the Internet, contains of a vast amount of information. The overhead of aggregating all the information into a single device is too expensive, so a distributed approach to solve the problem is often preferable. In this thesis, we aim to develop efficient algorithms for the following fundamental graph problems that arise in networks, in both sequential and distributed settings.
Graph coloring is a basic symmetry breaking problem in distributed computing. Each node is to be assigned a color such that adjacent nodes are assigned different colors. Both the efficiency and the quality of coloring are important measures of an algorithm. One of our main contributions is providing tools for obtaining colorings of good quality whose existence are non-trivial. We also consider other optimization problems in the distributed setting. For example, we investigate efficient methods for identifying the connectivity as well as the bottleneck edges in a distributed network. Our approximation algorithm is almost-tight in the sense that the running time matches the known lower bound up to a poly-logarithmic factor. For another example, we model how the task allocation can be done in ant colonies, when the ants may have different capabilities in doing different tasks.
The matching problems are one of the classic combinatorial optimization problems. We study the weighted matching problems in the sequential setting. We give a new scaling algorithm for finding the maximum weight perfect matching in general graphs, which improves the long-standing Gabow-Tarjan's algorithm (1991) and matches the running time of the best weighted bipartite perfect matching algorithm (Gabow and Tarjan, 1989). Furthermore, for the maximum weight matching problem in bipartite graphs, we give a faster scaling algorithm whose running time is faster than Gabow and Tarjan's weighted bipartite {it perfect} matching algorithm.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113540/1/hsinhao_1.pd