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- W2745680580 abstract "The kTree problem is a special case of Subgraph Isomorphism where the pattern graph is a tree, that is, the input is an $n$-node graph $G$ and a $k$-node tree $T$, and the goal is to determine whether $G$ has a subgraph isomorphic to $T$. We provide evidence that this problem cannot be computed significantly faster than $2^{k} textsf{poly}(n)$, which matches the fastest algorithm known for this problem by Koutis and Williams [ICALP 2009 and TALG 2016]. Specifically, we show that if kTree can be solved in time $(2-varepsilon)^k textsf{poly}(n)$ for some constant $varepsilon>0$, then Set Cover with $n'$ elements and $m'$ sets can be solved in time $(2-delta)^{n'} textsf{poly}(m')$ for a constant $delta(varepsilon) > 0$, which would refute the Set Cover Conjecture by Cygan et al. [CCC 2012 and TALG 2016]. Our techniques yield a new algorithm for the p-Partial Cover problem, a parameterized version of Set Cover that requires covering at least $p$ elements (rather than all elements). Its running time is $(2+varepsilon)^p (m')^{O(1/varepsilon)}$ for any fixed $varepsilon>0$, which improves the previous $2.597^p textsf{poly}(m')$-time algorithm by Zehavi [ESA 2015]. Our running time is nearly optimal, as a $(2-varepsilon')^p textsf{poly}(m')$-time algorithm would refute the Set Cover Conjecture." @default.
- W2745680580 created "2017-08-31" @default.
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- W2745680580 date "2017-08-25" @default.
- W2745680580 modified "2023-09-27" @default.
- W2745680580 title "Conditional Lower Bound for Subgraph Isomorphism with a Tree Pattern" @default.
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