A geometric Achlioptas process

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some predetermined objective in an online randomized environment. They have algorithmic implications in various areas of computer science, as well as connections to biological processes involving decision making. In this paper, we consider a recently proposed semi-random graph process, described as follows: we start with an empty graph on n vertices, and in each round, the decision-maker, called Builder, receives a uniformly random vertex v, and must immediately (in an online manner) choose another vertex u, adding the edge {u, v} to the graph. Builder's end goal is to make the constructed graph satisfy some predetermined monotone graph property.We consider the property P H of containing a spanning graph H as a subgraph. It was asked by N. Alon whether for any bounded-degree H, Builder can construct a graph satisfying P H with high probability in O(n) rounds. We answer this question positively in a strong sense, showing that any graph with maximum degree ∆ can be constructed with high probability in (3∆/2 + o(∆))n rounds, where the o(∆) term tends to zero as ∆ → ∞. This is tight (even for the offline case) up to a multiplicative factor of 3. Furthermore, for the special case where H is a spanning forest of maximum degree ∆, we show that H can be constructed with high probability in O(n log ∆) rounds. This is tight up to a multiplicative constant, even for the offline setting. Finally, we show a separation between adaptive and non-adaptive strategies, proving a lower bound of Ω(n √ log n) on the number of rounds necessary to eliminate all isolated vertices w.h.p. using a non-adaptive strategy. This bound is tight, and in fact O(n √ log n) rounds are sufficient to construct a K r -factor w.h.p. using a non-adaptive strategy.

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“…Achlioptas-like processes involving two choices were also investigated in other contexts, see e.g. [25] for a geometric perspective.…”

Section: Semi-random Processesmentioning

confidence: 99%

Very fast construction of bounded-degree spanning graphs via the semi-random graph process

Ben‐Eliezer¹,

Gishboliner²,

Hefetz³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Similar problems have also been studied for other properties or objectives; for example, the problem of avoiding a fixed subgraph [15,20], or the problem of speeding up the appearance of a Hamilton cycle [16]. Achlioptas-like processes involving two choices were also investigated in other contexts, see for example [19] for a geometric perspective.…”

Section: Related Work: Semi-random Processesmentioning

confidence: 99%

Very fast construction of bounded‐degree spanning graphs via the semi‐random graph process

Ben‐Eliezer

Gishboliner

Hefetz

et al. 2020

Random Struct Algorithms

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In this paper, we study the following recently proposed semi-random graph process: starting with an empty graph on n vertices, the process proceeds in rounds, where in each round we are given a uniformly random vertex v, and must immediately (in an online manner) add to our graph an edge incident with v. The end goal is to make the constructed graph satisfy some predetermined monotone graph property. Alon asked whether every given bounded-degree spanning graph can be constructed with high probability in O(n) rounds. We answer this question positively in a strong sense, showing that any n-vertex graph with maximum degree Δ can be constructed with high probability in (3Δ∕2 + o(Δ))n rounds. This is tight up to a multiplicative factor of 3 + o Δ (1). We also obtain tight bounds for the number of rounds necessary to embed bounded-degree spanning trees, and consider a nonadaptive variant of this setting. KEYWORDS embedding spanning graphs, semi-random graph process 1 INTRODUCTION Recently, the following semi-random graph process was proposed by Peleg Michaeli, and analyzed by Ben-Eliezer, Hefetz, Kronenberg, Parczyk, Shikhelman, and Stojaković [3]. A single adaptive player, called Builder, starts with an empty graph G on a set V of n vertices. The process then proceeds in rounds, where in each round Builder is offered a uniformly random vertex v, and chooses an edge of the form {v, u} to add to the graph G. Builder's objective is typically to construct a graph that satisfies some predetermined monotone graph property; for example, to make G an expander with certain

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