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We consider a variation on Maker-Breaker games on graphs or digraphs where the edges have random costs. We assume that Maker wishes to choose the edges of a spanning tree, but wishes to minimise his cost. Meanwhile Breaker wants to make Maker's cost as large as possible.more » « lessFree, publicly-accessible full text available April 11, 2026
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The greedy and nearest-neighbor TSP heuristics can both have $$\log n$$ approximation factors from optimal in worst case, even just for $$n$$ points in Euclidean space. In this note, we show that this approximation factor is only realized when the optimal tour is unusually short. In particular, for points from any fixed $$d$$-Ahlfor's regular metric space (which includes any $$d$$-manifold like the $$d$$-cube $[0,1]^d$ in the case $$d$$ is an integer but also fractals of dimension $$d$$ when $$d$$ is real-valued), our results imply that the greedy and nearest-neighbor heuristics have additive errors from optimal on the order of the optimal tour length through random points in the same space, for $d>1$.more » « less
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Let $$N=\binom{n}{2}$$ and $$s\geq 2$$. Let $$e_{i,j},\,i=1,2,\ldots,N,\,j=1,2,\ldots,s$$ be $$s$$ independent permutations of the edges $$E(K_n)$$ of the complete graph $$K_n$$. A MultiTree is a set $$I\subseteq [N]$$ such that the edge sets $$E_{I,j}$$ induce spanning trees for $$j=1,2,\ldots,s$$. In this paper we study the following question: what is the smallest $m=m(n)$ such that w.h.p. $[m]$ contains a MultiTree. We prove a hitting time result for $s=2$ and an $$O(n\log n)$$ bound for $$s\geq 3$$.more » « less
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Abstract Let be chosen independently and uniformly at random from the unit ‐dimensional cube . Let be given and let . The random geometric graph has vertex set and an edge whenever . We show that if each edge of is colored independently from one of colors and has the smallest value such that has minimum degree at least two, then contains a rainbow Hamilton cycle asymptotically almost surely.more » « less
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We study the minimum spanning arborescence problem on the complete digraph [Formula: see text], where an edge e has a weight W e and a cost C e , each of which is an independent uniform random variable U s , where [Formula: see text] and U is uniform [Formula: see text]. There is also a constraint that the spanning arborescence T must satisfy [Formula: see text]. We establish, for a range of values for [Formula: see text], the asymptotic value of the optimum weight via the consideration of a dual problem.more » « less
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