Fast construction on a restricted budget

TL;DR

This paper introduces a controlled random graph process model where a builder strategically purchases edges within constraints to achieve specific graph properties, extending classical hitting time results with optimal strategies.

Contribution

It presents new strategies for constructing various graph properties under budget and time constraints, extending classical results and proving their optimality.

Findings

Builder can achieve k-vertex-connectivity with fewer edges than k.

Optimal strategies for creating Hamilton cycles and perfect matchings within given budgets.

Conditions for constructing specific subgraphs like trees and cycles are established and shown to be optimal.

Abstract

We introduce a model of a controlled random graph process. In this model, the edges of the complete graph $K_n$ are ordered randomly and then revealed, one by one, to a player called Builder. He must decide, immediately and irrevocably, whether to purchase each observed edge. The observation time is bounded by parameter $t$, and the total budget of purchased edges is bounded by parameter $b$. Builder's goal is to devise a strategy that, with high probability, allows him to construct a graph of purchased edges possessing a target graph property $\mathcal{P}$, all within the limitations of observation time and total budget. We show the following: (a) Builder has a strategy to achieve $k$-vertex-connectivity at the hitting time for this property by purchasing at most $c_kn$ edges for an explicit $c_k<k$; and a strategy to achieve minimum degree $k$ (slightly) after the threshold for minimum degree $k$ by purchasing at most $(1+\varepsilon)kn/2$ edges (which is optimal). (b) Builder has a strategy to create a Hamilton cycle at the hitting time for Hamiltonicity by purchasing at most $Cn$ edges for an absolute constant $C>1$; this is optimal in the sense that $C$ cannot be arbitrarily close to $1$. This substantially extends the classical hitting time result for Hamiltonicity due to Ajtai--Koml\'os--Szemer\'edi and Bollob\'as. (c) Builder has a strategy to create a perfect matching by time $(1+\varepsilon)n\log{n}/2$ while purchasing at most $(1+\varepsilon)n/2$ edges (which is optimal). (d) Builder has a strategy to create a copy of a given $k$-vertex tree if $t\ge b\gg\max\{(n/t)^{k-2},1\}$, and this is optimal; (e) For $\ell=2k+1$ or $\ell=2k+2$, Builder has a strategy to create a copy of a cycle of length $\ell$ if $b\gg\max \{n^{k+2}/t^{k+1},n/\sqrt{t}\}$, and this is optimal.