by Amir Abboud

A central goal of complexity theory is to understand and prove lower bounds for the time complexity of fundamental problems. One of the most important computational problems is Edit Distance, the problem of computing the minimum number of edit operations (insertions, deletions, and substitutions) required to transform one sequence into another. A classical dynamic programming algorithm that is taught in basic algorithms courses solves Edit Distance on sequences of length n in O(n^{2}) time. Generalizations of Edit Distance are of fundamental importance in computational biology and genomics, where n is typically a few billions, and this quadratic runtime is prohibitive. A faster, e.g. linear-time, algorithm would have far reaching consequences: the paper introducing BLAST, a heuristic algorithm for a generalization of Edit Distance, that runs in linear time but is not guaranteed to return an optimal solution, has been cited more than fifty thousand times. Despite decades of attempts, no upper bound below the O(n^{2}/log^{2}n) bound of Masek and Paterson is known for Edit Distance.

This is a situation where lower bounds are highly desirable, but unfortunately, the current state of the art in complexity is far from providing a lower bound that is close to quadratic for any natural problem in NP, let alone Edit Distance. Therefore, researchers have turned their attention to conditional lower bounds, and a recent celebrated result by Backurs and Indyk shows that Edit Distance cannot be solved in truly subquadratic O(n^{2−ε}) time, for some ε>0, unless the Strong Exponential Time Hypothesis (SETH) is false (this is among the few STOC papers that made it to the Boston Globe news website). SETH is a pessimistic version of P ≠ NP, which essentially states that there is no ε>0 such that SAT on CNF formulas on n variables and O(n) clauses can be solved in O((2−ε)^{n}) time.

Other interesting recent results show that under SETH, the current algorithms for many central problems in diverse areas of computer science are optimal, up to n^{o(1)} factors. These areas include pattern matching, graph algorithms, computational geometry, data mining, and economics, and the list is growing by the day. These conditional lower bounds are a part of a more general line of work in which one bases the hardness of important problems in P on well-known conjectures about the exact complexity of other famous problems. Other conjectures concern the exact complexity of 3-SUM (given n integers, do three of them sum to zero?) and All-Pairs-Shortest-Path (compute all pairwise distances in a weighted graph). This line of work was the focus of the third workshop in the Fall 2015 Simons Institute Program on Fine-Grained Complexity and Algorithm Design, entitled Computational Complexity of Low-Polynomial Time Problems. It seems that we are rapidly approaching an exciting state of affairs in which a tight lower bound for many problems of practical significance can be derived from a small set of such conjectures.

One of the major goals of this research is to strengthen the foundations on which these results are based. We do not have strong evidence supporting these conjectures, and perhaps the main reason to believe them is the fact that no one knows how to refute them. For example, SETH was introduced by Impagliazzo and Paturi as a plausible explanation for the lack of (2−ε)^{n} algorithms for CNF-SAT, despite its being one of the most extensively studied problems in computer science. Can we find more evidence for these conjectures? Can we show surprising consequences of refuting them? Alternatively, can we replace them with more believable ones?

In the rest of this Vignette I will discuss a recent paper with Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Ryan Williams, in which we show that many of the SETH lower bounds can be based on an assumption that is much more believable than SETH.

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