Subexponential-Time Algorithms for Maximum Independent Set in P_t-Free and Broom-Free Graphs

In algorithmic graph theory, a classic open question is to determine the complexity of the Maximum Independent Set problem on Pt -free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t≤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 P6 -free graphs. CoRR, arXiv:1707.05491, 2017). Here 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 number of vertices. Even for the weighted version MWIS, the problem is solvable in 2O(tnlogn√) time on Pt -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 2O(|V(H)|n+m√log(n+m)) , even if d is part of the input.

Otherwise, assuming the Exponential-Time Hypothesis (ETH), there is no 2o(n+m) -time algorithm for d-Scattered Set for any fixed d≥3 on H-free graphs with n-vertices and m-edges.