Nearly complete graphs decomposable into large induced matchings and their applications

Alon, Noga; Moitra, Ankur; Sudakov, Benny

doi:10.1145/2213977.2214074

Cited by 40 publications

(29 citation statements)

References 22 publications

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“…Multitasking capability, as we have considered it here, bears a close relationship to issues that arise in the design of electronic communication systems, that seek to optimize the efficiency of transmission through distributed, parallel communications while avoiding the risks of cross talk introduced by shared communication channels (Alon, Moitra, & Sudakov, 2012;Birk, Linial, & Meshulam, 1993;Chlamtac & Kutten, 1985). Communication channels require balancing channel capacity, i.e.…”

Section: Shared Communication Channelsmentioning

confidence: 99%

On the Rational Boundedness of Cognitive Control: Shared Versus Separated Representations

Musslick

Saxe²,

Hoskin

et al. 2020

Preprint

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One of the most fundamental and striking limitations of human cognition appears to be a constraint in the number of control-dependent processes that can be executed at one time. This constraint motivates one of the most influential tenets of cognitive psychology: that cognitive control relies on a central, limited capacity processing mechanism that imposes a seriality constraint on processing. Here we provide a formally explicit challenge to this view. We argue that the causality is reversed: the constraints on control-dependent behavior reflect a rational bound that control mechanisms impose on processing, to prevent processing interference that arises if two or more tasks engage the same resource to be executed. We use both mathematical and numerical analyses of shared representations in neural network architectures to articulate the theory, and demonstrate its ability to explain a wide range of phenomena associated with control-dependent behavior. Furthermore, we argue that the need for control, arising from the shared use of the same resources by different tasks, reflects the optimization of a fundamental tradeoff intrinsic to network architectures: the increase in learning efficacy associated with the use of shared representations, versus the efficiency of parallel processing (i.e., multitasking) associated with task-dedicated representations. The theory helps frame a formally rigorous, normative approach to the tradeoff between control-dependent processing versus automaticity, and relates to a number of other fundamental principles and phenomena concerning cognitive function, and computation more generally.

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Section: Shared Communication Channelsmentioning

confidence: 99%

On the Rational Boundedness of Cognitive Control: Shared Versus Separated Representations

Musslick

Saxe²,

Hoskin

et al. 2020

Preprint

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“…It would be a major breakthrough to substantially improve either of these bounds on RS(n). Some of the notable constructions of Ruzsa-Szemerédi -like graphs include [4,39,28,11]. As this is an old problem with a very rich history, it would be impractical to fully survey its importance in mathematics here; instead, we refer the reader to the survey by Conlon and Fox [18].…”

Section: Upper Bound Lower Boundmentioning

confidence: 99%

Linear Size Distance Preservers

Bodwin

2017

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

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The famous shortest path tree lemma states that, for any node s in a graph G = (V, E), there is a subgraph on O(n) edges that preserves all distances between node pairs in the set {s} × V . A very basic question in distance sketching research, with applications to other problems in the field, is to categorize when else graphs admit sparse subgraphs that preserve distances between a set P of p node pairs, where P has some different structure than {s} × V or possibly no guaranteed structure at all. Trivial lower bounds of a path or a clique show that such a subgraph will need Ω(n + p) edges in the worst case. The question is then to determine when these trivial lower bounds are sharp; that is, when do graphs have linear size distance preservers on O(n + p) edges?In this paper, we make the first new progress on this fundamental question in over ten years. We show:

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“…Several related questions have been studied, as well as applications and generalizations of the Ruzsa-Szemerédi problem: see for example [4] and [2]. A natural generalization that we believe has not been considered is the following generalized Turán problem.…”

Section: Introductionmentioning

confidence: 99%

Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques

Gowers¹,

Janzer²

2020

Combinator. Probab. Comp.

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Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing $n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$ copies of K s such that any K r is contained in at most one K s . We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with $n^{r-1-o(1)}$ copies of K r such that no K r is rainbow.

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Nearly complete graphs decomposable into large induced matchings and their applications

Abstract: We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on N vertices with

Cited by 40 publications

References 22 publications

On the Rational Boundedness of Cognitive Control: Shared Versus Separated Representations

On the Rational Boundedness of Cognitive Control: Shared Versus Separated Representations

Linear Size Distance Preservers

Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques

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