These algorithms combine exterior algebra with random matrix estimators, using the tensor-train moment bound of Rakhshan and Rabusseau~[AISTATS 2020].
Abstract
We give randomized approximation algorithms for counting k-paths and k-forests in a host graph. Here k denotes the number of pattern vertices, n and m denote the numbers of host vertices and edges or arcs, {\epsilon} is the relative error, and {\delta} is the failure probability. Our main results are: 1. Paths: We approximate the number of directed paths on $k$ vertices in $2^k k^{O(1)}(n+m)\varepsilon^{-2}\log(2/\delta)$ arithmetic operations. 2. Trees and forests: For every fixed $\eta>0$, we approximate the number of non-induced copies of a given forest on $k$ vertices in $(2+\eta)^k n^{O_\eta(1)}\varepsilon^{-2}\log(2/\delta)$ arithmetic operations. Our path algorithm resolves a conjecture of Koutis and Williams~[CACM 2016] and answers an open question of Lokshtanov, Saurabh, and Zehavi~[SODA 2021] by giving a $2^k poly(n,\varepsilon^{-1})$-time approximation scheme. Our algorithms combine exterior algebra with random matrix estimators, using the tensor-train moment bound of Rakhshan and Rabusseau~[AISTATS 2020]. For forests, we use a small-component separator to evaluate the estimator efficiently.
We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$...
Improper edges were introduced by Shor to refine Cayley's formula for rooted labeled trees. Zeng established a connection between Shor's refinement and the Ramanujan polynomials. Let $\mathscr{T}_n$ denote the set of rooted labeled trees on $[n]=\{1,\ldots,n\}$. We prove that the number of improper edges in a uniformly...
We prove a sparse version of the four-colour theorem of Brandt and Thomass\'{e}, answering a question of Allen, B\"ottcher, Kohayakawa and Roberts. For every fixed $0<\gamma\le1/10$ and every $p=p(n)\in(0,1]$, asymptotically almost surely every spanning triangle-free $H\subseteq G(n,p)$ with $\delta(H)\ge(1/3+\gamma)pn...
We prove the following two upper bounds for ordered Ramsey numbers: (1) Every ordered forest $F$ on $n$ vertices satisfies $R_{<}(F,F)=O(n^{1+\lceil\log\chi_{<}(F)\rceil})$. This in particular answers a question of Geneson, Holmes, Liu, Neidinger, Pehova and Wass. (2) There is a function $f$ such that, for every fixed...
Let $G$ be a simple graph on $n$ vertices, and let $F(G)$ denote the number of its spanning forests. Bencs and Csikv\'ari [Upper bound for the number of spanning forests of regular graphs, European J. Combin. 110 (2023) 103677] proved that every $r$-regular graph $G$ with $r\geq 2$ satisfies $F(G) \leq r^{n}$. They fur...
Ting-Zeng Wu, S. Lu, Xiang-Shuai Dong· 0 citations
We prove that the number of labelled embeddings of any $n$-vertex tree $T$ into an $n$-vertex graph $G$ of maximum degree $d$ satisfies $$ \mathrm{inj}(T,G) \leq (d/e)^n \exp(o_d(1) n). $$ The bound is sharp up to determining $o_d(1)$, even for paths, and the dependence of the error $\exp(o_d(1)n)$ on $d$ is necessary....
Csongor Beke, V. Boskovic, Nina Kamčev et al.· 0 citations
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