Into the Square: On the Complexity of Some Quadratic-time Solvable Problems

Borassi, Michele; Crescenzi, Pierluigi; Habib, Michel

doi:10.1016/j.entcs.2016.03.005

Cited by 65 publications

(99 citation statements)

References 39 publications

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“…For sets of small cardinality, we hash the universe U to a smaller universe, where complementing the sets does not take too much time and space. From this reduction, the claim follows: Claim 4.2 is itself an interesting result: in [11], conditional lower bounds for many problems stem from the above three problems, forming a tree of reductions. By our equivalence, the root of the tree can be replaced by the quadratic-time hardness conjecture on any of the three problems, simplifying the reduction tree.…”

Section: Complementing Sparse Relationsmentioning

confidence: 95%

“…, n k and n u , and let the input size m (corresponding to the number of edges in the input graph) be the sum of all sets' sizes in all set families. Borassi et al [11] showed that when k = 2, these Basic Problems require time m 2−o(1) under SETH, and that if the size of universe U is poly-logarithmic in the input size, then the three problems are equivalent under subquadratic-time reductions. The main idea of the reductions between these problems is to complement all sets in S 1 , or S 2 , or both.…”

Section: Complementing Sparse Relationsmentioning

confidence: 99%

“…Thus far, fine-grained complexity has remained focused on specific problems, rather than organizing problems into classes as in traditional complexity. As the field has grown, many fundamental relationships between problems have been discovered, making the graph of known results a somewhat tangled web of reductions ( [36,5,9,11,12,1,13,2,27]). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Gao¹,

Impagliazzo²,

Kolokolova³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Properties definable in first-order logic are algorithmically interesting for both theoretical and pragmatic reasons. Many of the most studied algorithmic problems, such as Hitting Set and Orthogonal Vectors, are first-order, and the first-order properties naturally arise as relational database queries. A relatively straightforward algorithm for evaluating a property with k + 1 quantifiers takes time O(m k ) and, assuming the Strong Exponential Time Hypothesis (SETH), some such properties require O(m k− ) time for any > 0. (Here, m represents the size of the input structure, i.e. the number of tuples in all relations.)We give algorithms for every first-order property that improves this upper bound to m k /2 Θ( √ log n) , i.e., an improvement by a factor more than any poly-log, but less than the polynomial required to refute SETH. Moreover, we show that further improvement is equivalent to improving algorithms for sparse instances of the well-studied Orthogonal Vectors problem. Surprisingly, both results are obtained by showing completeness of the Sparse Orthogonal Vectors problem for the class of first-order properties under fine-grained reductions. To obtain improved algorithms, we apply the fast Orthogonal Vectors algorithm of [3,16].While fine-grained reductions (reductions that closely preserve the conjectured complexities of problems) have been used to relate the hardness of disparate specific problems both within P and beyond, this is the first such completeness result for a standard complexity class.

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Section: Complementing Sparse Relationsmentioning

confidence: 95%

Section: Complementing Sparse Relationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Gao¹,

Impagliazzo²,

Kolokolova³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

show abstract

“…A natural approach to prove Radius limitations in sparse graphs is to base them on the OV conjecture. However, such a lower bound has remained elusive [2,16]. This is due to the following type mismatch.…”

mentioning

confidence: 99%

Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs

Abboud¹,

Williams²,

Wang³

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

136

409

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The radius and diameter are fundamental graph parameters, with several natural definitions for directed graphs. Each definition is well-motivated in a variety of applications. All versions of diameter and radius can be solved via solving all-pairs shortest paths (APSP), followed by a fast postprocessing step. However, solving APSP on n-node graphs requires Ω(n 2 ) time even in sparse graphs. We study the question: when can diameter and radius in sparse graphs be solved in truly subquadratic time, and when is such an algorithm unlikely? Motivated by our conditional lower bounds on computing these measures exactly in truly subquadratic time, we search for approximation and fixed parameter subquadratic algorithms, and alternatively, for reasons why they do not exist.We find that:• Most versions of Diameter and Radius can be solved in truly subquadratic time with optimal approximation guarantees, under plausible assumptions. For example, there is a 2-approximation algorithm for directed Radius with one-way distances that runs inÕ(m √ n) time, while a (2 − δ)-approximation algorithm in O(n 2−ε ) time is considered unlikely.• On graphs with treewidth k, we can solve all versions in 2 O(k log k) n 1+o(1) time. We show that these algorithms are near optimal since even a (3/2 − δ)-approximation algorithm that runs in time 2 o(k) n 2−ε would refute plausible assumptions.Two conceptual contributions of this work that we hope will incite future work are: the introduction of a Fixed Parameter Tractability in P framework, and the statement of a differently-quantified variant of the Orthogonal Vectors Conjecture, which we call the Hitting Set Conjecture. * The full version of the paper can be found at: http://arxiv.org/abs/1506. 01799. A.A and V

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“…Our work is an example of hardness results in P, that is a growing area of research (e.g., see [12]). Roughly, finer-grained notions of reduction between polynomial-time solvable problems are used in order to prove that if A can be reduced to B and B can be solved inÕ(n q−ε )-time 2 then A can also be solved inÕ(n q−ε )-time [43].…”

Section: Introductionmentioning

confidence: 99%

Revisiting Decomposition by Clique Separators

Coudert¹,

Ducoffe²

2018

SIAM J. Discrete Math.

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We study the complexity of decomposing a graph by means of clique separators. This common algorithmic tool, first introduced by Tarjan, allows to cut a graph into smaller pieces, and so, it can be applied to preprocess the graph in the computation of optimization problems. However, the best-known algorithms for computing a decomposition have respective O(nm)-time and O(n (3+α)/2 ) = o(n 2.69 )-time complexity, with α < 2.3729 being the exponent for matrix multiplication. Such running times are prohibitive for large graphs. Here we prove that for every graph G, a decomposition can be computed in O(T (G) + min{n α , ω 2 n})-time with T (G) and ω being respectively the time needed to compute a minimal triangulation of G and the clique-number of G. In particular, it implies that every graph can be decomposed by clique separators in O(n α log n)-time. Based on prior work from Kratsch et al., we prove in addition that decomposing a graph by clique-separators is as least as hard as triangle detection. Therefore, the existence of any o(n α )-time algorithm for this problem would be a significant breakthrough in the field of algorithmic. Finally, our main result implies that planar graphs, bounded-treewidth graphs and bounded-degree graphs can be decomposed by clique separators in linear or quasi-linear time.

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Into the Square: On the Complexity of Some Quadratic-time Solvable Problems

Cited by 65 publications

References 39 publications

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs

Revisiting Decomposition by Clique Separators

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