In the Maximum Weight Independent Set problem, the input is a graph G, every vertex has a nonnegative integer weight, and the task is to find a set S of pairwise non-adjacent vertices, maximizing the total weight of the vertices in S. We give an n~(O(log~2 n)) time algorithm for this problem on graphs excluding the path P_6 on 6 vertices as an induced subgraph. Currently, there is no constant k known for which Maximum Weight Independent Set on P_k-free graphs becomes NP-complete, and our result implies that if such a k exists, then k > 6 unless all problems in NP can be decided in (quasi)polynomial time. Using the combinatorial tools that we develop for the above algorithm, we also give a polynomial-time algorithm for Maximum Weight Efficient Dominating Set on P_6-free graphs. In this problem, the input is a graph G, every vertex has an integer weight, and the objective is to find a set S of maximum weight such that every vertex in G has exactly one vertex in S in its closed neighborhood, or to determine that no such set exists. Prior to our work, the class of P6-free graphs was the only class of graphs defined by a single forbidden induced subgraph on which the computational complexity of Maximum Weight Efficient Dominating Set was unknown.
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