Using Parametric Transformations Toward Polynomial Kernels for Packing Problems Allowing Overlaps

Fernau, Henning; López-Ortíz, Alejandro; Romero, Jazmín

doi:10.1145/2786015

Cited by 4 publications

(4 citation statements)

References 24 publications

(22 reference statements)

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“…Note that the size of the kernel depends on k. Prior to the work by Gabizon et al [24], packing problems with relaxed disjointness conditions have already been considered from the viewpoint of parameterized complexity (see, e.g., [33,19,38,39]). Roughly speaking, these papers do not exhibit behaviors where relaxed disjointness conditions substantially (or at all) simplify the problem at hand, but rather provide parameterized algorithms and kernels with respect to k. Here, the work most relevant to us is that by Fernau et al [19], who studied the p-Set (r, q)-Packing problem. In particular, for any r ≥ 1, Fernau et al proved that several very restricted versions of p-Set (r, q)-Packing with p = 3 are already NP-hard.…”

Section: Introductionmentioning

confidence: 99%

On r-Simple k-Path and Related Problems Parameterized by k/r

Gutin

Wahlström

Zehavi

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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Abasi et al. (2014) introduced the following two problems. In the r-Simple k-Path problem, given a digraph G on n vertices and positive 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 succinctly encodes some polynomial P on n variables and positive integers k, r, decide whether P has a monomial of total degree k where the degree of each variable is at most r. Abasi et al. obtained randomized algorithms of running time 4 (k/r) log r · n O(1) for both problems. Gabizon et al. (2015) designed deterministic 2 O((k/r) log r) · n O(1) -time algorithms for both problems (however, for the (r, k)-Monomial Detection problem the input circuit is restricted to be non-canceling). Gabizon et al. also studied the following problem. 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 . Gabizon et al. obtained a deterministic 2 O((pq/r) log r) · n O(1) -time algorithm for p-Set (r, q)-Packing.The above results prove that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by the product of two parameters, that is, k/r and log r, where k = pq for p-Set (r, q)-Packing. Abasi et al. and Gabizon et al. asked whether the log r factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for (r, k)-Monomial Detection by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2 o((k/r) log r) · (n + log k) O(1) -time algorithm for (r, k)-Monomial Detection, i.e. (r, k)-Monomial Detection is highly unlikely to be single-exponentially FPT when parameterized by k/r alone. The question remains open for r-Simple k-Path and p-Set (r, q)-Packing.We consider the question from a wider perspective: are the above problems FPT when parameterized by k/r only, i.e. whether there exists a computable function f such that the problems admit a f (k/r)(n + log k) O(1) -time algorithm? Since r can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon et al. do not even show that any of these three problems is in XP parameterized by k/r alone. We resolve the wider question by (a) obtaining a 2 O((k/r) 2 log(k/r)) · (n + log k) O(1) -time algorithm for r-Simple k-Path on digraphs and a 2 O(k/r) · (n + log 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 (in contrast, we prove that p-Multiset (r, q)-Packing is W[1]-hard), and (c) proving that (r, k)-Monomial Detection is para-NP-hard even if only two distinct variables are in polynomial P and the circuit is non-canceling. For the special case of (r, k)-Monomial De...

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Section: Introductionmentioning

confidence: 99%

On r-Simple k-Path and Related Problems Parameterized by k/r

Gutin

Wahlström

Zehavi

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In practice communities may overlap by sharing one or more of their members [10,4,22]. In [15,8], the H-Packing with t-Overlap was proposed as an abstraction for the community discovering problem. The goal is to find k subgraphs in a given graph G (the network) where each subgraph (a community) should be isomorphic to a graph H ∈ H where H is a family of graphs (the community models).…”

Section: Introductionmentioning

confidence: 99%

“…Related Work. H. Fernau et al, [8] provide an O(r r k r−t−1 ) kernel for the H-Packing and r-Set Packing with t-Overlap problems. In addition, an O(r rk k (r−t−1) k+2 n r ) algorithm for these problems can be found in [15].…”

Section: Introductionmentioning

confidence: 99%

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Arbitrary Overlap Constraints in Graph Packing Problems

López-Ortíz

Romero

2018

Int. J. Found. Comput. Sci.

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In earlier versions of the community discovering problem, the overlap between communities was restricted by a simple count upperbound [17,5,11,8]. In this paper, we introduce the Π-Packing with α()-Overlap problem to allow for more complex constraints in the overlap region than those previously studied. Let V r be all possible subsets of vertices of V (G) each of size at most r, and α : V r × V r → {0, 1} be a function. The Π-Packing with α()-Overlap problem seeks at least k induced subgraphs in a graph G subject to: (i) each subgraph has at most r vertices and obeys a property Π, and (ii) for any pair Hi, Hj, with i = j, α(Hi, Hj) = 0 (i.e., Hi, Hj do not conflict). We also consider a variant that arises in clustering applications: each subgraph of a solution must contain a set of vertices from a given collection of sets C, and no pair of subgraphs may share vertices from the sets of C. In addition, we propose similar formulations for packing hypergraphs. We give an O(r rk k (r+1)k n cr ) algorithm for our problems where k is the parameter and c and r are constants, provided that: i) Π is computable in polynomial time in n and ii) the function α() satisfies specific conditions. Specifically, α() is hereditary, applicable only to overlapping subgraphs, and computable in polynomial time in n. Motivated by practical applications we give several examples of α() functions which meet those conditions.

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