On r-Simple k-Path

Abasi, Hasan; Bshouty, Nader H.; Gabizon, Ariel; Haramaty, Elad

doi:10.1007/978-3-662-44465-8_1

Cited by 8 publications

(105 citation statements)

References 21 publications

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“…By Lemma 4.11, |D r,k | = 2 O(k/r) . Thus, it is clear that our algorithm runs in time 2 O(k/r) · f (k/r) · (n + log k) O (1) . In what follows, we prove that our algorithm is correct.…”

Section: Bounding Rmentioning

confidence: 95%

“…We remark that DPs over so-called hidden tree decompositions are a well-known tool to design subexponential-time algorithms for parameterized problems in Computational Geometry (see, e.g., [28,6]). We use a DP Step Each entry in N is computed in time 2 O(k/r) · (r + n + log k) O (1) . Since there are only 2 O(k/r) · (r + n + log k) O(1) entries in N, the total running time is 2 O(k/r) · (r + n + log k) O(1) .…”

Section: Two-level Dynamic Programming (Dp)mentioning

confidence: 99%

“…Proof. By Lemma 4.10, it suffices to show that the special case of Undirected Colorful r-Simple k-Path where r > √ k is solvable in time 2 O(k/r) · f (k/r) · (n + log k) O (1) . Let A be an algorithm that solves Special Undirected Colorful r-Simple k-Path in time f (k/r) · (n + log k) O (1) .…”

Section: Bounding Rmentioning

confidence: 99%

“…By Lemma 4.10, it suffices to show that the special case of Undirected Colorful r-Simple k-Path where r > √ k is solvable in time 2 O(k/r) · f (k/r) · (n + log k) O (1) . Let A be an algorithm that solves Special Undirected Colorful r-Simple k-Path in time f (k/r) · (n + log k) O (1) . Then, given an instance (G, k, r) of Undirected r-Simple k-Path, we first use the algorithm in Corollary 4.2 to either correctly conclude that (G, k, r) is a Yesinstance or find a vertex cover U of G of size at most 3k/r.…”

Section: Bounding Rmentioning

confidence: 99%

“…The most technical parts of the paper deal with the Directed r-Simple k-Path and Undirected r-Simple k-Path problems. We prove that Directed r-Simple k-Path can be solved in time 2 O((k/r) 2 log(k/r)) · (n + log k) O (1) and polynomial space using a chain of reductions from Directed r-Simple k-Path that includes three auxiliary problems. The first of these problems is the Directed r-Simple Long (s, t)-Path problem, where we are given a strongly connected digraph G, positive integers k, r, and vertices s, t ∈ V (G).…”

Section: Introductionmentioning

confidence: 99%

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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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“…By Lemma 4.11, |D r,k | = 2 O(k/r) . Thus, it is clear that our algorithm runs in time 2 O(k/r) · f (k/r) · (n + log k) O (1) . In what follows, we prove that our algorithm is correct.…”

Section: Bounding Rmentioning

confidence: 95%

Section: Two-level Dynamic Programming (Dp)mentioning

confidence: 99%

Section: Bounding Rmentioning

confidence: 99%

Section: Bounding Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Almost Optimal Cover-Free Families

Bshouty

Gabizon

2017

Lecture Notes in Computer Science

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Roughly speaking, an (n, (r, s))-Cover Free Family (CFF) is a small set of n-bit strings such that: "in any d := r + s indices we see all patterns of weight r". CFFs have been of interest for a long time both in discrete mathematics as part of block design theory, and in theoretical computer science where they have found a variety of applications, for example, in parametrized algorithms where they were introduced in the recent breakthrough work of Fomin, Lokshtanov and Saurabh [16] under the name 'lopsided universal sets'. In this paper we give the first explicit construction of cover-free families of optimal size up to lower order multiplicative terms, for any r and s. In fact, our construction time is almost linear in the size of the family. Before our work, such a result existed only for r = d o(1) . and r = ω(d/(log log d log log log d)). As a sample application, we improve the running times of parameterized algorithms from the recent work of Gabizon, Lokshtanov and Pilipczuk [18].

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Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints

Gabizon

Lokshtanov

Pilipczuk

2015

Algorithms - ESA 2015

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In parameterized complexity, it is a natural idea to consider different generalizations of classic problems. Usually, such generalization are obtained by introducing a "relaxation" variable, where the original problem corresponds to setting this variable to a constant value. For instance, the problem of packing sets of size at most p into a given universe generalizes the Maximum Matching problem, which is recovered by taking p = 2. Most often, the complexity of the problem increases with the relaxation variable, but very recently Abasi et al. have given a surprising example of a problemr-Simple k-Path -that can be solved by a randomized algorithm with running time O * (2 O(k log r r ) ). That is, the complexity of the problem decreases with r.In this paper we pursue further the direction sketched by Abasi et al. Our main contribution is a derandomization tool that provides a deterministic counterpart of the main technical result of Abasi et al.: the O * (2 O(k log r r ) ) algorithm for (r, k)-Monomial Detection, which is the problem of finding a monomial of total degree k and individual degrees at most r in a polynomial given as an arithmetic circuit. Our technique works for a large class of circuits, and in particular it can be used to derandomize the result of Abasi et al. for r-Simple k-Path. On our way to this result we introduce the notion of representative sets for multisets, which may be of independent interest.Finally, we give two more examples of problems that were already studied in the literature, where the same relaxation phenomenon happens. The first one is a natural relaxation of the Set Packing problem, where we allow the packed sets to overlap at each element at most r times. The second one is Degree Bounded Spanning Tree, where we seek for a spanning tree of the graph with a small maximum degree.

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On r-Simple k-Path

Cited by 8 publications

References 21 publications

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

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

Almost Optimal Cover-Free Families

Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints

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