Towards tight approximation bounds for graph diameter and eccentricities

Bačkurs, Artūrs; Roditty, Liam; Segal, Gilad; Williams, Virginia Vassilevska; Wein, Nicole

doi:10.1145/3188745.3188950

Cited by 33 publications

(53 citation statements)

References 55 publications

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“…We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes -where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radius and center in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| − 1. number of edges on a shortest path. Beyond its many practical applications, this fundamental problem in Graph Theory has attracted a lot of attention in the fine-grained complexity study of polynomial-time solvable problems [1,4,8,15,18,22,25,32,52]. More precisely, for every n-vertex m-edge unweighted graph the textbook algorithm for computing its diameter runs in time O(nm).…”

mentioning

confidence: 99%

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Ducoffe¹,

Habib²,

Viennot³

2020

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

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Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic-time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes -where the distance VC-dimension of a graph G is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radius and center in G. In particular for any fixed H, the class of H-minor free graphs has distance VC-dimension at most |V (H)| − 1. number of edges on a shortest path. Beyond its many practical applications, this fundamental problem in Graph Theory has attracted a lot of attention in the fine-grained complexity study of polynomial-time solvable problems [1,4,8,15,18,22,25,32,52]. More precisely, for every n-vertex m-edge unweighted graph the textbook algorithm for computing its diameter runs in time O(nm). In a seminal paper [52] this roughly quadratic running-time was matched by a quadratic lower-bound, assuming the Strong Exponential-Time Hypothesis (SETH). We stress that for graphs with millions of nodes and edges, quadratic time is already too prohibitive.The conditional lower-bound of [52] also holds for sparse graphs i.e., with only m = O(n) edges [1]. However it does not hold for many well-structured graph classes [1,11,13,20,14,21,23,25,31,33,35,49]. Our work proposes some new advances on the characterization of graph families for which we can compute the diameter in truly subquadratic time. Related workBefore we detail our contributions, we wish to mention a few recent (and not so recent) results that are most related to our approach.

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mentioning

confidence: 99%

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Ducoffe¹,

Habib²,

Viennot³

2020

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

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“…In recent work, Backurs et al [33] show that unless the 3-OV Hypothesis (and hence SETH) is false, any 3/2-approximation algorithm to the Diameter in sparse graphs needs n 1.5−o(1) time, thus resolving the question. They also obtain a variety of other tight conditional lower bounds based on k-OV for different k for graph Eccentricities, and variants of Diameter.…”

Section: Hardness Results From Sethmentioning

confidence: 99%

“…When it comes to exact computation in sparse weighted graphs, S-T Diameter is (n 2 , n 2 )-equivalent to Diameter (see [33]). When it comes to approximation, the problems differ a bit.…”

Section: Hardness Results From Sethmentioning

confidence: 99%

On Some Fine-Grained Questions in Algorithms and Complexity

Williams

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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In recent years, a new "fine-grained" theory of computational hardness has been developed, based on "fine-grained reductions" that focus on exact running times for problems. Mimicking NP-hardness, the approach is to (1) select a key problem X that for some function t, is conjectured to not be solvable by any O(t(n) 1−ε ) time algorithm for ε > 0, and (2) reduce X in a fine-grained way to many important problems, thus giving tight conditional time lower bounds for them. This approach has led to the discovery of many meaningful relationships between problems, and to equivalence classes.The main key problems used to base hardness on have been: the 3-SUM problem, the CNF-SAT problem (based on the Strong Exponential Time Hypothesis (SETH)) and the All Pairs Shortest Paths Problem. Research on SETH-based lower bounds has flourished in particular in recent years showing that the classical algorithms are optimal for problems such as Approximate Diameter, Edit Distance, Frechet Distance and Longest Common Subsequence.This paper surveys the current progress in this area, and highlights some exciting new developments. IntroductionArguably the main goal of the theory of algorithms is to study the worst case time complexity of fundamental computational problems. When considering a problem P , we fix a computational model, such as a Random Access Machine (RAM) or a Turing machine (TM). Then we strive to develop an efficient algorithm that solves P and to prove that for a (hopefully slow growing) function t(n), the algorithm solves P on instances of size n in O(t(n)) time in that computational model. The gold standard for the running time t(n) is linear time, O(n); to solve most problems, one needs to at least read the input, and so linear time is necessary. The theory of algorithms has developed a wide variety of techniques. These have yielded near-linear time algorithms for many diverse problems. For instance, it is known since the 1960s and 70s (e.g. [143,144,145,99]) that Depth-First Search (DFS) and Breadth-First Search (BFS) run in linear time in graphs, and that using these techniques one can obtain linear time algorithms (on a RAM) for many interesting graph problems: Single-Source Shortest paths, Topological Sort of a Directed Acyclic Graph, Strongly Connected Components, Testing Graph Planarity etc. More recent work has shown that even more complex problems such as Approximate Max Flow, Maximum Bipartite Matching, Linear Systems on Structured Matrices, and many others, admit close to linear time algorithms, by combining combinatorial and linear algebraic techniques (see e.g. [140,64,141,116,117,67,68,65,66,113]).Nevertheless, for most problems of interest, the fastest known algorithms run much slower than linear time. This is perhaps not too surprising. Time hierarchy theorems show that for most computational models, for any computable function t(n) ≥ n, there exist problems that are solvable in O(t(n)) time but are NOT solvable in O(t(n) 1−ε ) time for ε > 0 (this was first proven for TMs [95], see [124] for more).Ti...

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“…First, having an f (k)n 2.3 time algorithm with respect to a parameter k for which Diameter is GP-hard would yield a faster Diameter algorithm. Moreover, from the known SETH-based hardness results [3,11,34] we get the following. Observation 1.…”

Section: Lemma 1 ([4]mentioning

confidence: 99%

“…The domination number of G is four since {t 1 , t 2 , t 3 , t 4 } is a dominating set. The acyclic chromatic number of G is at most four as V 1 ∪ V 2 ∪ B, S 1 ∪ S 2 ∪ {t 1 , t 2 }, {t 3 Hence, such an algorithm for Diameter would refute the SETH.…”

Section: Parameters Related To Both Diameter and H-indexmentioning

confidence: 99%

Parameterized Complexity of Diameter

Bentert

Nichterlein

2019

Lecture Notes in Computer Science

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Diameter-the task of computing the length of a longest shortest path-is a fundamental graph problem. Assuming the Strong Exponential Time Hypothesis, there is no O(n 1.99 )time algorithm even in sparse graphs [Roditty and Williams, 2013]. To circumvent this lower bound we aim for algorithms with running time f (k)(n + m) where k is a parameter and f is a function as small as possible. We investigate which parameters allow for such running times. To this end, we systematically explore a hierarchy of structural graph parameters.1 The h-index of a graph G is the largest number ℓ such that G contains at least ℓ vertices of degree at least ℓ. 1 See e. g. Borassi et al. [6] for a recent example which also yields good performance bounds using average-case analysis [7]. Concerning worst-case analysis, the theoretically fastest algorithms are based on matrix multiplication and run in O(n 2.373 ) time [35] and any O((n + m) 2−ε )-time algorithm refutes the SETH [34].The following results on approximating Diameter are known: It is folklore that a simple breadth-first search gives a linear-time 2-approximation. Aingworth et al. [2] improved the approximation factor to 3/2 at the expense of the higher running time of O(n 2 log n + m √ n log n). The lower bound of Roditty and Williams [34] also implies that approximating Diameter within a factor of 3/2 − δ in O(n 2−ε ) time refutes the SETH. Moreover, a 3/2 − δ-approximation in O(m 2−ε ) time or a 5/3 − δ-approximation in O(m 3/2−ε ) time also refute the SETH [3,11]. On planar graphs, there is an approximation scheme with near linear running time [38]; the fastest exact algorithm for Diameter on planar graphs runs in O(n 1.667 ) time [19].Concerning FPT in P, Diameter can be solved in 2 O(k) n 1+o(1) time where k is the treewidth of the graph [10]; however, an 2 o(k) n 2−ε -time algorithm refutes the SETH [1]. In fact, the construction actually proves the same running time lower bound with k being the vertex cover number. The reduction for the lower bound of Roditty and Williams [34] also implicitly implies that the SETH is refuted by any f (k)(n+m) 2−ε -time algorithm for Diameter for any computable function f when k is either the the (vertex deletion) distance to chordal graphs or the combined parameter h-index and domination number. Evald and Dahlgaard [16] adapted the reduction by Roditty and Williams and proved that any f (k)(n + m) 2−ε -time algorithm for Diameter parameterized by the maximum degree k for any computable function f refutes the SETH.

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Towards tight approximation bounds for graph diameter and eccentricities

Cited by 33 publications

References 55 publications

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

On Some Fine-Grained Questions in Algorithms and Complexity

Parameterized Complexity of Diameter

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