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

Abstract

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

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

“…Strong Exponential Time Hypothesis (SETH) 4 : For every > 0, there exists a k ≥ 2 so that k-CNF-SAT cannot be solved in time O(2 n(1− ) ).…”

“…It implies LDOVC; subquadratic approximation algorithms for Diameter-2 and Radius-2 would respectively refute the LDOVC and the Hitting Set Conjecture. [4] …”

“…For each i and each j, we create sets In this section we present an exact complexity reduction from any M C(∀∃ k−1 ∀) problem to a M C(∃ k ∀) problem, establishing the hardness of k-OV for these problems. This reduction gives an extension of the reduction from Hitting Set to Orthogonal Vectors in [4] to sparse structures.…”

Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n5/3) Time

“…This holds even for undirected, unweighted graphs with treewidth O(log n). For graphs of bounded treewidth, the diameter can be computed in near-linear time [5] (see also [27,40] for algorithms with time bounds that depend on the value of the diameter itself). Near-linear time algorithms were developed for many other restricted graph families, see e.g.…”

Fully polynomial-time parameterized computations for graphs and matrices of low treewidth

“…As mentioned earlier, standard dynamic programming on tree decompositions seems difficult to apply here, due to inherently exponential number of states. Interestingly, the recent work of Abboud et al [1] indicates that for some polynomial-time solvable problems this seems to be a real obstacle. In particular, Abboud et al [1] proved that the Diameter and Radius problems can be solved in time 2…”