Complete Minors in Graphs Without Sparse Cuts

Krivelevich, Michael; Nenadov, Rajko

doi:10.1093/imrn/rnz086

Cited by 4 publications

(7 citation statements)

References 33 publications

(51 reference statements)

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“…This generalizes various classical results of a similar avor (e.g. [39,44,18,43]). See the next subsection for a discussion comparing these (and other) existing embedding results to ours.…”

Section: T H E O R E M 1 1supporting

confidence: 89%

“…The lower bounds for Card(𝐺, ì 𝑡) for 𝑡 > 1 can then be obtained by a reduction to the 𝑡 = For the proof of our embedding theorem (Theorem 1.2), we extend an argument due to Krivelevich and Nenadov [44] (see also [43]) for ordinary minors (rather than topological minors). In order to obtain a minor embedding of 𝐻 in 𝐺, the idea there is to embed the vertices one by one from 𝐻 in 𝐺 while maintaining an "unused" subgraph 𝐺 of 𝐺 which is a slightly worse expander than 𝐺 is.…”

Section: Overview Of Proof Techniquesmentioning

confidence: 99%

“…and Nenadov [44] simpli ed and strengthened this by improving the bound on the size of 𝐻 to 𝑂(𝑛/log 𝑛). These results di er from ours in two key ways: (i) we want topological minors, and (ii) we want to be able to control the parities of the path lengths in the embedding.…”

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confidence: 99%

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Perfect Matching in Random Graphs is as Hard as Tseitin

Austrin¹,

Risse²

2022

TheoretiCS

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We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n / \log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lov\'asz-Schrijver proof system requires $n^\delta$ rounds to refute these formulas for some $\delta > 0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.

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“…This generalizes various classical results of a similar avor (e.g. [39,44,18,43]). See the next subsection for a discussion comparing these (and other) existing embedding results to ours.…”

Section: T H E O R E M 1 1supporting

confidence: 89%

Section: Overview Of Proof Techniquesmentioning

confidence: 99%

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Perfect Matching in Random Graphs is as Hard as Tseitin

Austrin¹,

Risse²

2022

TheoretiCS

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show abstract

“…, and this bound holds for c = 2 [KN21], see [CN19] for more details. This bound implies that for any fixed H, the maximum degree of an n-vertex φ-expander…”

Section: Our Frameworkmentioning

confidence: 86%

Narrowing the LOCAL$\unicode{x2013}$CONGEST Gaps in Sparse Networks via Expander Decompositions

Chang¹,

Su²

2022

Preprint

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Many combinatorial optimization problems, including maximum weighted matching and maximum independent set, can be approximated within (1 ± ǫ) factors in poly(log n, 1/ǫ) rounds in the LOCAL model via network decompositions [Ghaffari, Kuhn, and Maus, STOC 2018]. These approaches, however, require sending messages of unlimited size, so they do not extend to the more realistic CONGEST model, which restricts the message size to be O(log n) bits. For example, despite the long line of research devoted to the distributed matching problem, it still remains a major open problem whether an (1 − ǫ)-approximate maximum weighted matching can be computed in poly(log n, 1/ǫ) rounds in the CONGEST model.In this paper, we develop a generic framework for obtaining poly(log n, 1/ǫ)-round (1 ± ǫ)approximation algorithms for many combinatorial optimization problems, including maximum weighted matching, maximum independent set, and correlation clustering, in graphs excluding a fixed minor in the CONGEST model. This class of graphs covers many sparse network classes that have been studied in the literature, including planar graphs, bounded-genus graphs, and bounded-treewidth graphs.Furthermore, we show that our framework can be applied to give an efficient distributed property testing algorithm for an arbitrary minor-closed graph property that is closed under taking disjoint union, significantly generalizing the previous distributed property testing algorithm for planarity in [Levi, Medina, and Ron, PODC 2018 & Distributed Computing 2021].Our framework uses distributed expander decomposition algorithms [Chang and Saranurak, FOCS 2020] to decompose the graph into clusters of high conductance. We show that any graph excluding a fixed minor admits small edge separators. Using this result, we show the existence of a high-degree vertex in each cluster in an expander decomposition, which allows the entire graph topology of the cluster to be routed to a vertex. Similar to the use of network decompositions in the LOCAL model, the vertex will be able to perform any local computation on the subgraph induced by the cluster and broadcast the result over the cluster.

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“…Let us mention that [37] presents a result about large complete minors in expanding graph with growing degrees. Since the result's statement is somewhat involved and assumes stronger edge expansion of small sets, we decided not to state it here, referring the interested reader to [37] for details.…”

Section: Complete Minors In Expanding Graphsmentioning

confidence: 99%

Expanders - how to find them, and what to find in them

Krivelevich¹

2018

Preprint

Self Cite

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A graph G = (V, E) is called an expander if every vertex subset U of size up to |V |/2 has an external neighborhood whose size is comparable to |U |. Expanders have been a subject of intensive research for more than three decades and have become one of the central notions of modern graph theory.We first discuss the above definition of an expander and its alternatives. Then we present examples of families of expanding graphs and state basic properties of expanders. Next, we introduce a way to argue that a given graph contains a large expanding subgraph. Finally we research properties of expanding graphs, such as existence of small separators, of cycles (including cycle lengths), and embedding of large minors.

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Complete Minors in Graphs Without Sparse Cuts

Cited by 4 publications

References 33 publications

Perfect Matching in Random Graphs is as Hard as Tseitin

Perfect Matching in Random Graphs is as Hard as Tseitin

Narrowing the LOCAL$\unicode{x2013}$CONGEST Gaps in Sparse Networks via Expander Decompositions

Expanders - how to find them, and what to find in them

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