A randomized algorithm returns a $(1\pm\varepsilon)-approximation with failure probability at most $\delta$ in $O^*(2^k\varepsilon^{-2}\log(1/\delta)$ time.
Abstract
We resolve the conjecture of Koutis and Williams that $k$-paths can be approximately counted in $O^*(2^k)$ time. For a simple directed graph with $n$ vertices and $m$ arcs, our randomized algorithm returns a $(1\pm\varepsilon)$-approximation with failure probability at most $\delta$ in $O^*(2^k\varepsilon^{-2}\log(1/\delta))$ time.
We show that set cover on a universe of size $n$ and with sets of size at most $k$ can be solved in time $2^{(1-1/k+O(1/k^{3/2}))n}$. This improves on a $2^{(1-0.929/k)n}$-time algorithm of Bj\"orklund (STACS 2010) for all sufficiently large $k$.
We determine, up to a factor of $2^{o(k)}$, the number of $k$-sets $A \subset \{1, \ldots, n\}$ such that $|A + A| \leq m$, where $k = \Theta(\log n)$ and $m \leq k^{1 + \alpha}$, for small $\alpha>0$, answering a question of Green and Morris.
Marcelo Campos, Gabriel Dahia, João Pedro Marciano· 0 citations
The $k$-core of a graph is its (unique) largest subgraph with minimum degree at least $k$. For any $k \geq 3$, deciding whether a given vertex belongs to the $k$-core is a P-complete problem, meaning that it is inherently sequential and highly unlikely to admit efficient parallel algorithms, even on graphs of maximum d...
Let $f(k)$ denote the smallest integer such that every oriented graph $D$ with chromatic number at least $f(k)$ contains every oriented tree on $k$ vertices. Burr (1980) showed that $f(k)\le (k-1)^2$ and conjectured that $f(k)=2k-2$. Bessy, Gon\c{c}alves and Reinald (2025) proved that $f(k)=O(k^{3/2})$. In this paper,...
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 appr...
Fahad Panolan, Saket Saurabh, M. Zehavi et al.· 0 citations
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