We prove that , assuming the Strong Exponential Time Hypothesis , for any \delta>0, a 5/3-\delta approximation of the diameter of an undirected unweighted graph with nvertices needs n ^{ 3/2-o(1) time. This result improves on lower bounds of Backurs, Roditty, Segal, Vassilevska-Williams, and Wein ) for the Diameter of a sparse directed unweighted graph can be ruled out by a deterministic SETH-based reduction for every \delta\in(0,1) and essentially every \varepsilon\in(0,1), assuming NSETH. We make the same characterization for randomized SETH-based reductions, assuming NUNSETH. This settles the SETH-hardness of approximating the diameter of sparse directed unweighted graphs for deterministic reductions, up to NSETH, and for randomized reductions, up to NUNSETH. We prove additional hardness and non-reducibility results for undirected graphs} .
We prove several tight results on the fine-grained complexity of approximating the diameter of a graph. First, we prove that, for any \varepsilon>0 , assuming the Strong Exponential Time Hypothesis (SETH), there are no near-linear time 2-\varepsilon-approximation algorithms for the Diameter of a sparse directed graph, even in unweighted graphs. This result shows that a simple near-linear time 2-approximation algorithm for Diameter is optimal under SETH, answering a question from a survey of Rubinstein and Vassilevska-Williams (SIGACT '19) for the case of directed graphs. In the same survey, Rubinstein and Vassilevska-Williams also asked if it is possible to show that there are no 2-\varepsilon approximation algorithms for Diameter in a directed graph in O(n ^{ 1.499 ) time. We show that, assuming a hypothesis called NSETH, one cannot use a deterministic SETH-based reduction to rule out the existence of such algorithms. Furthermore, assuming another hypothesis called Non-Uniform NSETH (NUNSETH), one cannot use a even a randomized SETH-based reduction to rule out the existence of such algorithms. Extending the techniques in these two results, we characterize whether a 2-\varepsilon approximation algorithm running in time O(n^{1+\delta) for the Diameter of a sparse directed unweighted graph can be ruled out by a deterministic SETH-based reduction for every \delta\in(0,1) and essentially every \varepsilon\in(0,1), assuming NSETH. We make the same characterization for randomized SETH-based reductions, assuming NUNSETH. This settles the SETH-hardness of approximating the diameter of sparse directed unweighted graphs for deterministic reductions, up to NSETH, and for randomized reductions, up to NUNSETH. We prove additional hardness and non-reducibility results for undirected graphs} .
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several tight results on the fine-grained complexity of approximating the diameter of a graph. First, we prove that, for any \varepsilon>0
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, for any \delta>0, a 5/3-\delta approximation of the diameter of an undirected unweighted graph with nvertices needs n
(SETH), there are no near-linear time 2-\varepsilon-approximation algorithms for the Diameter of a sparse directed graph, even in unweighted graphs. This result shows that a simple near-linear time 2-approximation algorithm for Diameter is optimal under SETH, answering a question from a survey of Rubinstein and Vassilevska-Williams (SIGACT '19) for the case of directed graphs. In the same survey, Rubinstein and Vassilevska-Williams also asked if it is possible to show that there are no 2-\varepsilon approximation algorithms for Diameter in a directed graph in O(n
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3/2-o(1)
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time. This result improves on lower bounds of Backurs, Roditty, Segal, Vassilevska-Williams, and Wein
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) time. We show that, assuming a hypothesis called NSETH, one cannot use a deterministic SETH-based reduction to rule out the existence of such algorithms. Furthermore, assuming another hypothesis called Non-Uniform NSETH (NUNSETH), one cannot use a even a randomized SETH-based reduction to rule out the existence of such algorithms. Extending the techniques in these two results, we characterize whether a 2-\varepsilon approximation algorithm running in time O(n^{1+\delta