Edge Estimation with Independent Set Oracles

Beame, Paul; Har-Peled, Sariel; Ramamoorthy, Sivaramakrishnan Natarajan; Rashtchian, Cyrus; Sinha, Makrand

doi:10.4230/lipics.itcs.2018.38

Cited by 5 publications

(4 citation statements)

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“…In subsequent independent work, Beame et al [3] obtain a result similar to Theorem 1, with an overall running time of ε −4 • Õ(1) but with no further bound on the number of independence queries used. Thus their result outperforms Theorem 1 when independence queries are fast, and underperforms when they are slow.…”

Section: Our Main Results Is As Followsmentioning

confidence: 86%

“…Step 5a is executed at depth i, then |X ′ | ≤ 3 4 |X| holds and ∂(X ′ ) is a γapproximation of ∂(X).…”

Section: Theorem 1 (Restated)mentioning

confidence: 99%

“…Form an instance (G ′ , w ′ ) of APSP as follows. Let V (G ′ ) = (V × [3]), and let E(G ′ ) = i∈{1,2} {u,v}∈E {((u, i), (v, i + 1)), ((v, i), (u, i + 1))}.…”

Section: Theorem 7 (Restated)mentioning

confidence: 99%

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Fine-grained reductions from approximate counting to decision

Dell

Lapinskas

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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In this paper, we introduce a general framework for fine-grained reductions of approximate counting problems to their decision versions. (Thus we use an oracle that decides whether any witness exists to multiplicatively approximate the number of witnesses with minimal overhead.) This mirrors a foundational result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the polynomial-time setting, and a similar result of Müller (IWPEC 2006) in the FPT setting. Using our framework, we obtain such reductions for some of the most important problems in fine-grained complexity: the Orthogonal Vectors problem, 3SUM, and the Negative-Weight Triangle problem (which is closely related to All-Pairs Shortest Path). While all these problems have simple algorithms over which it is conjectured that no polynomial improvement is possible, our reductions would remain interesting even if these conjectures were proved; they have only polylogarithmic overhead, and can therefore be applied to subpolynomial improvements such as the n 3 / exp(Θ( √ log n))-time algorithm for the Negative-Weight Triangle problem due to Williams (STOC 2014). Our framework is also general enough to apply to versions of the problems for which more efficient algorithms are known. For example, the Orthogonal Vectors problem over GF(m) d for constant m can be solved in time n · poly(d) by a result of Williams and Yu (SODA 2014); our result implies that we can approximately count the number of orthogonal pairs with essentially the same running time.We also provide a fine-grained reduction from approximate #SAT to SAT. Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for some 1 < c < 2 and all k there is an O(c n )-time algorithm for k-SAT. Then we prove * We thank Rahul Santhanam and Ryan Williams for some valuable discussions. that for all k, there is an O((c + o(1)) n )-time algorithm for approximate #k-SAT. In particular, our result implies that the Exponential Time Hypothesis (ETH) is equivalent to the seemingly-weaker statement that there is no algorithm to approximate #3-SAT to within a factor of 1 + ε in time 2 o(n) /ε 2 (taking ε > 0 as part of the input).

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Section: Our Main Results Is As Followsmentioning

confidence: 86%

“…Step 5a is executed at depth i, then |X ′ | ≤ 3 4 |X| holds and ∂(X ′ ) is a γapproximation of ∂(X).…”

Section: Theorem 1 (Restated)mentioning

confidence: 99%

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Fine-grained reductions from approximate counting to decision

Dell

Lapinskas

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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“…Research involving queries to identify particular graph structures or elements, or queries to verify certain properties, can be found in various flavors. A well-studied problem class is property testing [31], and there are many more, see e.g., [4,10,17,43,47,50]. Without describing such problems in detail, we emphasize a fundamental difference to our work.…”

Section: Related Workmentioning

confidence: 99%

Orienting (hyper)graphs under explorable stochastic uncertainty

Bampis¹,

Dürr²,

Erlebach³

et al. 2021

Preprint

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Given a hypergraph with uncertain node weights following known probability distributions, we study the problem of querying as few nodes as possible until the identity of a node with minimum weight can be determined for each hyperedge. Querying a node has a cost and reveals the precise weight of the node, drawn from the given probability distribution. Using competitive analysis, we compare the expected query cost of an algorithm with the expected cost of an optimal query set for the given instance. For the general case, we give a polynomial-time f (α)-competitive algorithm, where f (α) ∈ [1.618 + ϵ, 2] depends on the approximation ratio α for an underlying vertex cover problem. We also show that no algorithm using a similar approach can be better than 1.5-competitive. Furthermore, we give polynomial-time 4/3-competitive algorithms for bipartite graphs with arbitrary query costs and for hypergraphs with a single hyperedge and uniform query costs, with matching lower bounds. ACM Subject ClassificationTheory of Computation → Design and analysis of algorithms; Mathematics of computing → Discrete mathematics

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Approximately counting and sampling small witnesses using a colourful decision oracle

Dell¹,

Lapinskas²,

Meeks³

2020

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

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In this paper, we prove "black box" results for turning algorithms which decide whether or not a witness exists into algorithms to approximately count the number of witnesses, or to sample from the set of witnesses approximately uniformly, with essentially the same running time. We do so by extending the framework of Dell and Lapinskas (STOC 2018), which covers decision problems that can be expressed as edge detection in bipartite graphs given limited oracle access; our framework covers problems which can be expressed as edge detection in arbitrary k-hypergraphs given limited oracle access. (Simulating this oracle generally corresponds to invoking a decision algorithm.) This includes many key problems in both the finegrained setting (such as k-SUM, k-OV and weighted k-Clique) and the parameterised setting (such as induced subgraphs of size k or weight-k solutions to CSPs). From an algorithmic standpoint, our results will make the development of new approximate counting algorithms substantially easier; indeed, it already yields a new state-of-the-art algorithm for approximately counting graph motifs, improving on Jerrum and Meeks (JCSS 2015) unless the input graph is very dense and the desired motif very small. Our k-hypergraph reduction framework generalises and strengthens results in the graph oracle literature due to Beame et al. (ITCS 2018) and Bhattacharya et al. (CoRR abs/1808.00691).

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Edge Estimation with Independent Set Oracles

Cited by 5 publications

References 0 publications

Fine-grained reductions from approximate counting to decision

Fine-grained reductions from approximate counting to decision

Orienting (hyper)graphs under explorable stochastic uncertainty

Approximately counting and sampling small witnesses using a colourful decision oracle

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