On r-Simple k-Path and Related Problems Parameterized by k/r
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Abasi et al. (2014) and Gabizon et al. (2015) studied the following problems. In the r-Simple k-Path problem, given a digraph G on n vertices and integers r,k, decide whether G has an r-simple k-path, which is a walk where every vertex occurs at most r times and the total number of vertex occurrences is k. In the (r,k)-Monomial Detection problem, given an arithmetic circuit that encodes some polynomial P on n variables and integers k,r, decide whether P has a monomial of degree k where the degree of each variable is at most r. In the p-Set (r,q)-Packing problem, given a universe V, positive integers p,q,r, and a collection H of sets of size p whose elements belong to V, decide whether there exists a subcollection H' of H of size q where each element occurs in at most r sets of H'. Abasi et al. and Gabizon et al. proved that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by (k/r) r, where k=pq for p-Set (r,q)-Packing and asked whether the r factor in the exponent can be avoided. We consider their question from a wider perspective: are the above problems FPT when parameterized by k/r only? We resolve the wider question by (a) obtaining a 2^O((k/r)^2(k/r)) (n+ k)^O(1)-time algorithm for r-Simple k-Path on digraphs and a 2^O(k/r) (n+ k)^O(1)-time algorithm for r-Simple k-Path on undirected graphs (i.e., for undirected graphs we answer the original question in affirmative), (b) showing that p-Set (r,q)-Packing is FPT, and (c) proving that (r,k)-Monomial Detection is para-NP-hard. For p-Set (r,q)-Packing, we obtain a polynomial kernel for any fixed p, which resolves a question posed by Gabizon et al. regarding the existence of polynomial kernels for problems with relaxed disjointness constraints.
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