Spotting Trees with Few Leaves

Björklund, Andreas; Kamat, Vikram; Kowalik, Łukasz; Zehavi, Meirav

doi:10.1137/15m1048975

Cited by 8 publications

(4 citation statements)

References 45 publications

(77 reference statements)

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“…given different parameters 1 ≤ c, appear separately in Appendix A. 4. By choosing c = 1.497, we get the bound O * (5.139 k ).…”

Section: Upper Bounds For Omentioning

confidence: 96%

“…The first reduction (of [10]) allows to focus on finding a small out-tree rather than an out-branching, while the second allows to focus on finding an even smaller outtree, but we need to find it along with a set of paths on 2 nodes (Appendix A.1). The second reduction might be of independent interest (indeed, the paper [4] applies our reduction). We then use a representative sets-based procedure in a manner that does not directly solve the problem, but returns a family of partial solutions that are trees.…”

Section: Our Mixing Strategiesmentioning

confidence: 99%

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Mixing Color Coding-Related Techniques

Zehavi

2015

Algorithms - ESA 2015

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Abstract. Narrow sieves, representative sets and divide-and-color are three breakthrough color coding-related techniques, which led to the design of extremely fast parameterized algorithms. We present a novel family of strategies for applying mixtures of them. This includes: (a) a mix of representative sets and narrow sieves; (b) a faster computation of representative sets under certain separateness conditions, mixed with divide-and-color and a new technique, "balanced cutting"; (c) two mixtures of representative sets, iterative compression and a new technique, "unbalanced cutting". We demonstrate our strategies by obtaining, among other results, significantly faster algorithms for k-Internal Out-Branching and Weighted 3-Set k-Packing, and a framework for speeding-up the previous best deterministic algorithms for k-Path, kTree, r-Dimensional k-Matching, Graph Motif and Partial Cover. IntroductionA problem is fixed-parameter tractable (FPT) with respect to a parameter k if it can be solved in time O * (f (k)) for some function f , where O * hides factors polynomial in the input size. The color coding technique, introduced by Alon et al. [1], led to the discovery of the first single exponential time FPT algorithms for many subcases of Subgraph Isomorphism. In the past decade, three breakthrough techniques improved upon it, and led to the development of extremely fast FPT algorithms for many fundamental problems. This includes the combinatorial divide-and-color technique [8], the algebraic multilinear detection technique [26,27,43] (which was later improved to the more powerful narrow sieves technique [2,3]), and the combinatorial representative sets technique [22].Divide-and-color was the first technique that resulted in (both randomized and deterministic) FPT algorithms for weighted problems that are faster than those relying on color coding. Later, representative sets led to the design of deterministic FPT algorithms for weighted problems that are faster than the randomized ones based on divide-and-color. The fastest FPT algorithms, however, rely on narrow sieves. Unfortunately, narrow sieves is only known to be relevant to the design of randomized algorithms for unweighted problems. arXiv:1410.5062v3 [cs.DS] 20 Apr 2015We present novel strategies for applying these techniques, combining the following elements (see Section 3).• Mixing narrow sieves and representative sets, previously considered to be two independent color coding-related techniques.• Under certain "separateness conditions", speeding-up the best known computation of representative sets.• Mixing divide-and-color-based preprocessing with the computation in the previous item, speeding-up any standard representative sets-based algorithm.• Cutting the universe into small pieces in two special manners, one used in the mix in the previous item, and the other mixed with a non-standard representative sets-based algorithm to improve its running time (by decreasing the size of the partial solutions it computes).To demonstrate our strategies, we consider th...

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“…given different parameters 1 ≤ c, appear separately in Appendix A. 4. By choosing c = 1.497, we get the bound O * (5.139 k ).…”

Section: Upper Bounds For Omentioning

confidence: 96%

Section: Our Mixing Strategiesmentioning

confidence: 99%

Mixing Color Coding-Related Techniques

Zehavi

2015

Algorithms - ESA 2015

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“…Finally, let us remark that k-Path (on both directed and undirected graph) and p-Set q-Packing are both among the most extensively studied problems in Parameterized Complexity. In particular, after a long sequence of works during the past three decades, the current best known parameterized algorithms for k-Path have running times 1.657 k n O(1) (randomized, undirected only) [10,9] (extended in [11]), 2 k n O(1) (randomized) [43] and 2.597 k n O(1) (deterministic) [44,20,40]. In addition, k-Path is known not to admit any polynomial kernel unless NP ⊆ coNP/poly [12].…”

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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“…Our warmup result involves a generalization of the directed Hamiltonian path problem, namely the k-Internal Out-Branching problem, where the goal is to detect whether a given directed graph contains a spanning out-branching that has at least k internal vertices. This is a wellstudied problem on its own, with several successive improvements the latest of which is an O * (3.617 k ) algorithm by Zehavi [25] and an O * (3.455 k ) algorithm by Björklund et al [9] for the undirected version of the problem.…”

Section: Introductionmentioning

confidence: 99%

Directed Hamiltonicity and Out-Branchings via Generalized Laplacians

Björklund,

Kaski,

Koutis

2016

Preprint

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We are motivated by a tantalizing open question in exact algorithms: can we detect whether an n-vertex directed graph G has a Hamiltonian cycle in time significantly less than 2 n ?We present new randomized algorithms that improve upon several previous works: a. We show that for any constant 0 < λ < 1 and prime p we can count the Hamiltonian cycles modulo p ⌊(1−λ) n 3p ⌋ in expected time less than c n for a constant c < 2 that depends only on p and λ. Such an algorithm was previously known only for the case of counting modulo two [Björklund and Husfeldt, FOCS 2013].b. We show that we can detect a Hamiltonian cycle in O * (3 n−α(G) ) time and polynomial space, where α(G) is the size of the maximum independent set in G. In particular, this yields an O * (3 n/2 ) time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013].Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix-Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix-Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an O * (2 k )-time randomized algorithm for detecting out-branchings with at least k internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Björklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed k-Leaf problem, based on a non-standard monomial detection problem.

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Spotting Trees with Few Leaves

Cited by 8 publications

References 45 publications

Mixing Color Coding-Related Techniques

Mixing Color Coding-Related Techniques

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

Directed Hamiltonicity and Out-Branchings via Generalized Laplacians

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