Distributed Symmetry Breaking in Hypergraphs

Morsy

Pandurangan

et al. 2016

Random Struct. Alg.

Self Cite

Section: Resultsmentioning

confidence: 99%

Efficient computation of sparse structures

Morsy

Pandurangan

et al. 2016

Random Struct. Alg.

Self Cite

“…al. [12] described a number of distributed algorithms for hypergraph MIS. These are also based on the REDUCE procedure, but there is a complication: in Kelsen's original algorithm, the probability p used in REDUCE is a function of the maximum degree of the current residual hypergraph.…”

Section: Producing the Full Mismentioning

confidence: 99%

“…This is a global statistic, and so it cannot be computed easily in a distributed (local) algorithm. As a result, the algorithm of [12] is fairly complex and has some additional technical limitations.…”

Section: Producing the Full Mismentioning

confidence: 99%

Derandomized Concentration Bounds for Polynomials, and Hypergraph Maximal Independent Set

2019

ACM Trans. Algorithms

A parallel algorithm for maximal independent set (MIS) in hypergraphs has been a longstanding algorithmic challenge, dating back nearly 30 years to a survey of Karp & Ramachandran (1990). The best randomized parallel algorithm for hypergraphs of fixed rank r was developed by Beame & Luby (1990) and Kelsen (1992), running in time roughly (log n) r! .We improve the randomized algorithm of Kelsen, reducing the runtime to roughly (log n) 2 r and simplifying the analysis through the use of more-modern concentration inequalities. We also give a method for derandomizing concentration bounds for low-degree polynomials, which are the key technical tool used to analyze that algorithm. This leads to a deterministic PRAM algorithm also running in (log n) 2 r+3 time and poly(m, n) processors. This is the first deterministic algorithm with sub-polynomial runtime for hypergraphs of rank r > 3.Our analysis can also apply when r is slowly growing; using this in conjunction with a strategy of Bercea et al. (2015) gives a deterministic MIS algorithm running in time exp (O( log(mn) log log(mn) )). This is an extended version of a paper appearing in the ACM-SIAM Symposium on Discrete Algorithms (SODA) 2018. 0 In [10], Kelsen discussed one straightforward approach to derandomize the randomized algorithm by drawing the random bits from a probability space with k-wise independence. When k is constant, this space has polynomial size, and so can be searched efficiently. However, this leads to relatively weak concentration bounds, and so the resulting algorithm runs in n ǫ time and poly(n) processors for any constant r and ǫ > 0. When k ≥ Ω( log n log log n ), the algorithm runs in polylog(n) time, but requires super-polynomial processor count.The probabilistic methods underpinning polynomial concentration bounds are similar to those for sums of independent random variables. There have been numerous powerful techniques developed for the latter, much more powerful than k-wise-independence for constant k. Unfortunately, there are severe technical roadblocks to applying these to non-linear polynomials. To the best of our knowledge, polynomial concentration bounds have not led to any efficient deterministic algorithms. Our contributionsIn Section 2, we give a slightly modified form of Kelsen's randomized algorithm, and show an improved bound on its running time.Theorem 1.1. There is a randomized parallel algorithm, running in (log n) 2 r+3 +O(1) expected time and O(n + m log n) processors, to produce an MIS of a rank-r hypergraph.There are two main ingredients to this improvement. First, we use a concentration inequality due to Schudy & Sviridenko [16], which is much tighter than the bounds originally developed by Kelsen. Second, we use an alternate degree statistic to measure the algorithm's progress. This measure is defined in terms of a single scalar value, which is used globally to bound the degrees throughout the graph. In addition to better running time, this substantially simplifies the analysis of [2, 3, 10], which used multiple, int...

“…(The definition of the distributed model of computing for hypergraphs is not as formalized as for graphs; these issues are elucidated more in [11].) These use the algorithm of Kelsen as a core subroutine.…”

Section: Distributed Algorithmsmentioning

confidence: 99%

Derandomized concentration bounds for polynomials, and hypergraph maximal independent set

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

2018

A parallel algorithm for maximal independent set (MIS) in hypergraphs has been a long-standing algorithmic challenge, dating back nearly 30 years to a survey of Karp & Ramachandran (1990). Despite its apparent simplicity, there have been no general sub-polynomialtime algorithms or hardness reductions. The best randomized parallel algorithm for hypergraphs of fixed rank r was developed by Beame & Luby (1990) and Kelsen (1992), running in time roughly (log n) r! . The key probabilistic tool underlying this algorithm is a concentration bound for low-degree polynomials applied to independent input variables; this is a natural generalization of concentration bounds for sums of independent random variables, which are ubiquitous in combinatorics and computer science.These concentration bounds for polynomials do not lend themselves to standard derandomization techniques. Thus, the algorithm of Kelsen cannot be derandomized in any known way. There are no deterministic parallel algorithms for hypergraph MIS for any fixed rank r > 3.We improve the randomized algorithm of Kelsen to obtain a running time of (log n) 2 r . We also give a method for derandomizing concentration bounds for polynomials, thus obtaining a deterministic algorithm running in (log n) 2 r+3 time and (mn) O(1) processors. Our analysis can also apply when r is slowly growing; using this in conjunction with a strategy of Bercea et al. (2015) gives a deterministic MIS algorithm running in time exp(O( log m log log m + log log n).