Regular graphs of large girth and arbitrary degree

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message:Fast ∆-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al.[11], we prove that the randomized complexity of ∆-coloring a tree with maximum degree ∆ is Θ(log ∆ log n), for any ∆ ≥ 55, whereas its deterministic complexity is Θ(log ∆ n) for any ∆ ≥ 3. 1 This also establishes a large separation between the deterministic complexity of ∆-coloring and (∆ + 1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(log ∆ n) rounds can be transformed to run in O(log * n − log * ∆ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(log ∆ n) time deterministically. (This gives an alternate proof that deterministically ∆-coloring a tree with small ∆ takes Ω(log ∆ n) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √ log n. This shows that a deterministic Ω(log ∆ n) lower bound for any problem (∆-coloring a tree, for example) implies a randomized Ω(log ∆ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2 O( √ log log n) terms in the complexities of the best MIS and (∆ + 1)-coloring algorithms without also improving the 2 O( √ log n) -round Panconesi-Srinivasan algorithms.

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“…For any ∆ ≥ 3 there exist a bipartite ∆-regular graphs with girth Ω(log ∆ n); see [13,9]. Such graphs are trivially ∆-edge colorable.…”

Section: Lemma 2 ([11]mentioning

confidence: 99%

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Chang

Kopelowitz

Pettie

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

184

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“…Next, we show two lower bounds on the approximation ratio of PS-d by the class (d + 1)-CH. The following is from [9].…”

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confidence: 99%

Simple Mechanisms for Agents with Complements

Feldman

Friedler

Morgenstern

et al. 2016

Proceedings of the 2016 ACM Conference on Economics and Computation

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We study the efficiency of simple auctions in the presence of complements. Devanur et al. [11] introduced the single-bid auction, and showed that it has a price of anarchy (PoA) of O(log m) for complement-free (i.e., subadditive) valuations. Prior to our work, no non-trivial upper bound on the PoA of single bid auctions was known for valuations exhibiting complements. We introduce a hierarchy over valuations, where levels of the hierarchy correspond to the degree of complementarity, and the PoA of the single bid auction degrades gracefully with the level of the hierarchy. This hierarchy is a refinement of the Maximum over Positive Hypergraphs (MPH) hierarchy [16], where the degree of complementarity d is captured by the maximum number of neighbors of a node in the positive hypergraph representation. We show that the price of anarchy of the single bid auction for valuations of level d of the hierarchy is O(d 2 log(m/d)), where m is the number of items. We also establish an improved upper bound of O(d log m) for a subclass where every hyperedge in the positive hypergraph representation is of size at most 2 (but the degree is still d). Finally, we show that randomizing between the single bid auction and the grand bundle auction has a price of anarchy of at most O( √ m) for general valuations. All of our results are derived via the smoothness framework, thus extend to coarse-correlated equilibria and to Bayes Nash equilibria.

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“…To show the results, we assume that G is a -regular n-vertex graph with girth at least ( log log n) and = (log n). It is worth mentioning that there exist several explicit families of -regular n-vertex graphs with arbitrary degree ⩾ 3 and girth Ω(log n) (e.g., see [9]). Let us first consider high-degree graphs (i.e., = (log n)) on n nodes.…”

Section: Our Resultsmentioning

confidence: 99%

“…1. The construction of the scheme is explicit (a graph G with the desired degree and girth properties can be constructed as in [9]). 2.…”

Section: Comparison With Related Workmentioning

confidence: 99%

Balanced allocation on graphs: A random walk approach

Pourmiri

2019

Random Struct Algorithms

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We propose algorithms for allocating n sequential balls into n bins that are interconnected as a -regular n-vertex graph G, where ⩾ 3 can be any integer. In general, the algorithms proceeds in n succeeding rounds. Let > 0 be an integer, which is given as an input to the algorithms. In each round, ball 1 ⩽ t ⩽ n picks a node of G uniformly at random and performs a nonbacktracking random walk of length from the chosen node and simultaneously collects the load information of a subset of the visited nodes. It then allocates itself to one of them with the minimum load (ties are broken uniformly at random). For graphs with sufficiently large girths, we obtain upper and lower bounds for the maximum number of balls at any bin after allocating all n balls in terms of , with high probability.KEYWORDS balls-into-bins models, balanced allocation, nonbacktracking random walks 1 * A preliminary version of this paper appeared in the proceedings of the 22nd International Computing and Combinatorics Conference (COCOON'16).Random Struct Alg. 2019;55:980-1009.wileyonlinelibrary.com/journal/rsa

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Regular graphs of large girth and arbitrary degree

Cited by 26 publications

References 13 publications

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Simple Mechanisms for Agents with Complements

Balanced allocation on graphs: A random walk approach

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