ACM Transactions on Algorithms (TALG)
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Journal of Combinatorial Theory Series B
On the approximability of some degree-constrained subgraph problems
Discrete Applied Mathematics
Theoretical Computer Science
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Let $\ell$ be the number of edges in a longest cycle containing a given vertex $v$ in an undirected graph. We show how to find a cycle through $v$ of length $\exp(\Omega(\sqrt {\log \ell/\log\log \ell}))$ in polynomial time. This implies the same bound for the longest cycle, longest $vw$-path, and longest path. The previous best bound for longest path is length $\Omega( (\log \ell )^2/\, \log\log \ell)$ due to Bjo¨rklund and Husfeldt. Our approach, which builds on Bjo¨rklund and Husfeldt’s, uses cycles to enlarge cycles. This self-reducibility allows the approximation method to be iterated.