Locality of Not-so-Weak Coloring

Balliu, Alkida; Hirvonen, Juho; Lenzen, Christoph; Olivetti, Dennis; Suomela, Jukka

doi:10.1007/978-3-030-24922-9_3

“…This is in stark contrast with MaxCut, as we have seen already. It was recently proved by Balliu, Hirvonen, Lenzen, Olivetti, and Suomela [2] that any deterministic algorithm finding a maximal cut in d-regular graphs (d 3) takes Ω(log n) rounds and any randomized algorithm takes Ω(log log n) rounds in the LOCAL model. It would be interesting to find algorithms matching these round complexities.…”

Section: Resultsmentioning

confidence: 99%

“…The following claim immediately holds as well: Knowing these claims, we now prove that CUT 2 /OPT is greater than some f d (α, β) with α + β > 0. To show this, we need to prove refined versions of inequalities (2) and (3).…”

Section: éTienne Bamas and Louis Esperetmentioning

confidence: 99%

“…Recall the proof of inequality (2) (CUT 0 OPT − d−1 2 · |M |): Start with some optimum cut and remove the edges of the cut leaving M ∩ V − and the edges entering M ∩ V + . By definition we remove at most d−1 2 · |M | edges, which implies (2). Let E 0 be the set of edges with one end in V − and the other in M ∩ V + , or with one end in M ∩V − and the other in V + , or between two vertices of M ∩V + , or between two vertices of M ∩V + .…”

Section: éTienne Bamas and Louis Esperetmentioning

confidence: 99%

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Local approximation of the Maximum Cut in regular graphs

Bamas

¹

2020

Theoretical Computer Science

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This paper is devoted to the distributed complexity of finding an approximation of the maximum cut (MaxCut) in graphs. A classical algorithm consists in letting each vertex choose its side of the cut uniformly at random. This does not require any communication and achieves an approximation ratio of at least 1 2 in expectation. When the graph is d-regular and triangle-free, a slightly better approximation ratio can be achieved with a randomized algorithm running in a single round. Here, we investigate the round complexity of deterministic distributed algorithms for MaxCut in regular graphs. We first prove that if G is d-regular, with d even and fixed, no deterministic algorithm running in a constant number of rounds can achieve a constant approximation ratio. We then give a simple one-round deterministic algorithm achieving an approximation ratio of 1 d for d-regular graphs when d is odd. We show that this is best possible in several ways, and in particular no deterministic algorithm with approximation ratio 1 d + (with > 0) can run in a constant number of rounds. We also prove results of a similar flavour for the MaxDiCut problem in regular oriented graphs, where we want to maximize the number of arcs oriented from the left part to the right part of the cut.

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“…Existing results cover many LVLs that are potentially interesting from the perspective of game theoretical applications (see e.g. [7,9,16,17,19,21,38]). Second, it is a strong distributed model in the sense that algorithms in the LOCAL model are only limited by information propagation, and not e.g.…”

Section: Distributed Computing and Graphical Gamesmentioning

confidence: 99%

“…in distributed computing, the corresponding LVL is known as locally optimal cut. It is known to be a hard problem [9]: on 3-regular graphs it requires Ω(log 𝑛) deterministic time and Ω(log log 𝑛) randomized time.…”

Section: Minority Gamementioning

confidence: 99%

Classifying Convergence Complexity of Nash Equilibria in Graphical Games Using Distributed Computing Theory

Hirvonen,

Schmid,

Chatterjee

et al. 2021

Preprint

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0

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Graphical games are a useful framework for modeling the interactions of (selfish) agents who are connected via an underlying topology and whose behaviors influence each other. They have wide applications ranging from computer science to economics and biology.Yet, even though a player's payoff only depends on the actions of their direct neighbors in graphical games, computing the Nash equilibria and making statements about the convergence time of "natural" local dynamics in particular can be highly challenging. In this work, we present a novel approach for classifying complexity of Nash equilibria in graphical games by establishing a connection to local graph algorithms, a subfield of distributed computing. In particular, we make the observation that the equilibria of graphical games are equivalent to locally verifiable labelings (LVL) in graphs; vertex labelings which are verifiable with a constant-round local algorithm. This connection allows us to derive novel lower bounds on the convergence time to equilibrium of best-response dynamics in graphical games. Since we establish that distributed convergence can sometimes be provably slow, we also introduce and give bounds on an intuitive notion of "time-constrained" inefficiency of best responses. We exemplify how our results can be used in the implementation of mechanisms that ensure convergence of best responses to a Nash equilibrium. Our results thus also give insight into the convergence of strategy-proof algorithms for graphical games, which is still not well understood.

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Local Approximation of the Maximum Cut in Regular Graphs

Bamas¹

2019

Graph-Theoretic Concepts in Computer Science

0

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This paper is devoted to the distributed complexity of finding an approximation of the maximum cut (MaxCut) in graphs. A classical algorithm consists in letting each vertex choose its side of the cut uniformly at random. This does not require any communication and achieves an approximation ratio of at least 1 2 in expectation. When the graph is d-regular and triangle-free, a slightly better approximation ratio can be achieved with a randomized algorithm running in a single round. Here, we investigate the round complexity of deterministic distributed algorithms for MaxCut in regular graphs. We first prove that if G is d-regular, with d even and fixed, no deterministic algorithm running in a constant number of rounds can achieve a constant approximation ratio. We then give a simple one-round deterministic algorithm achieving an approximation ratio of 1 d for d-regular graphs when d is odd. We show that this is best possible in several ways, and in particular no deterministic algorithm with approximation ratio 1 d + (with > 0) can run in a constant number of rounds. We also prove results of a similar flavour for the MaxDiCut problem in regular oriented graphs, where we want to maximize the number of arcs oriented from the left part to the right part of the cut.

show abstract

Locality of Not-so-Weak Coloring

Cited by 11 publications

References 23 publications

Local approximation of the Maximum Cut in regular graphs

Local approximation of the Maximum Cut in regular graphs

Classifying Convergence Complexity of Nash Equilibria in Graphical Games Using Distributed Computing Theory

Local Approximation of the Maximum Cut in Regular Graphs

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