Packing even directed circuits quarter-integrally.
CoRR(2023)
摘要
We prove the existence of a computable function
$f\colon\mathbb{N}\to\mathbb{N}$ such that for every integer $k$ and every
digraph $D$ either contains a collection $\mathcal{C}$ of directed cycles of
even length such that no vertex of $D$ belongs to more than four cycles in
$\mathcal{C}$, or there exists a set $S\subseteq V(D)$ of size at most $f(k)$
such that $D-S$ has no directed cycle of even length. Moreover, we provide an
algorithm that finds one of the two outcomes of this statement in time
$g(k)n^{\mathcal{O}(1)}$ for some computable function $g\colon
\mathbb{N}\to\mathbb{N}$.
Our result unites two deep fields of research from the algorithmic theory for
digraphs: The study of the Erd\H{o}s-P\'osa property of digraphs and the study
of the Even Dicycle Problem. The latter is the decision problem which asks if a
given digraph contains an even dicycle and can be traced back to a question of
P\'olya from 1913. It remained open until a polynomial time algorithm was
finally found by Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and,
independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC
1997). The Even Dicycle Problem is equivalent to the recognition problem of
Pfaffian bipartite graphs and has applications even beyond discrete mathematics
and theoretical computer science. On the other hand, Younger's Conjecture
(1973), states that dicycles have the Erd\H{o}s-P\'osa property. The conjecture
was proven more than two decades later by Reed, Robertson, Seymour, and Thomas
(Combinatorica 1996) and opened the path for structural digraph theory as well
as the algorithmic study of the directed feedback vertex set problem. Our
approach builds upon the techniques used to resolve both problems and combines
them into a powerful structural theorem that yields further algorithmic
applications for other prominent problems.
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