Abstract
In algorithmic graph theory, a classic open question is to determine the complexity of the Maximum Independent Set problem on P_t-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for tle 5 (Lokshtanov et al., in: Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms, SODA 2014, Portland, OR, USA, January 5–7, 2014, pp 570–581, 2014), and an algorithm for t=6 announced recently (Grzesik et al. in Polynomial-time algorithm for maximum weight independent set on {P}_6-free graphs. CoRR, arXiv:1707.05491, 2017). Here we study the existence of subexponential-time algorithms for the problem: we show that for any tge 1, there is an algorithm for Maximum Independent Set on P_t-free graphs whose running time is subexponential in the number of vertices. Even for the weighted version MWIS, the problem is solvable in 2^{mathcal {O}(sqrt{tn log n})} time on P_t-free graphs. For approximation of MIS in broom-free graphs, a similar time bound is proved. Scattered Set is the generalization of Maximum Independent Set where the vertices of the solution are required to be at distance at least d from each other. We give a complete characterization of those graphs H for which d-Scattered Set on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus the number of edges):If every component of H is a path, then d-Scattered Set on H-free graphs with n vertices and m edges can be solved in time 2^{mathcal {O}(|V(H)|sqrt{n+m}log (n+m))}, even if d is part of the input.Otherwise, assuming the Exponential-Time Hypothesis (ETH), there is no 2^{o(n+m)}-time algorithm for d-Scattered Set for any fixed dge 3 on H-free graphs with n-vertices and m-edges.
Highlights
There are some problems in discrete optimization that can be considered fundamental
We study the existence of subexponential-time algorithms for the problem: we show that for any t ≥ 1, there is an algorithm for Maximum Independent Set on Pt -free graphs whose running time is subexponential in the the problem is solvable innum2Obe(√r tonflovgenr)titciemse
It is NP-hard (its decision version “Is α(G) ≥ k?” being NP-complete), but APX-hard as well, and, not even approximable within O(n1−ε) in polynomial time for any ε > 0 unless P = NP, as proved by Zuckerman [30]. Those classes of graphs on which Maximum Independent Set (MIS) becomes tractable are of definite interest. One direction of this area is to study the complexity of MIS on H -free graphs, that is, on graphs not containing any induced subgraph isomorphic to a given graph H
Summary
There are some problems in discrete optimization that can be considered fundamental. The Maximum Independent Set problem (MIS, for short) is one of them. We can consider with d being part of the input, or assume that d ≥ 2 is a fixed constant, in which case we call the problem d- Scattered Set. Clearly, MIS is exactly the same as 2- Scattered Set. Despite its similarity to MIS, the branching algorithm of Theorem 1 cannot be generalized: we give evidence that there is no subexponential-time algorithm for 3- Scattered Set on P5-free graphs. The algorithmic side of Theorem 4 is based on the combinatorial observation that the treewidth of Pt -free graphs is sublinear in the number of edges, which means that standard algorithms on bounded-treewidth graphs can be invoked to solve the problem in time subexponential in the number of edges.
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