Tight(er) Bounds for Similarity Measures, Smoothed Approximation and Broadcasting

The first part approaches the question why a number of fundamental polynomial-time problems - specifically, Dynamic Time Warping, Longest Common Subsequence (LCS), and the Levenshtein distance - resists decades-long attempts to obtain polynomial improvements over their simple dynamic programming solutions. We prove that any (strongly) subquadratic algorithm for these and related sequence similarity measures would refute the Strong Exponential Time Hypothesis (SETH). Focusing particularly on LCS, we determine a tight running time bound (up to lower order factors and conditional on SETH) when the running time is expressed in terms of all input parameters that have been previously exploited in the extensive literature.

In the second part, we investigate the approximation performance of the popular 2-Opt heuristic for the Traveling Salesperson Problem using the smoothed analysis paradigm. For the Fréchet distance, we design an improved approximation algorithm for the natural input class of c-packed curves, matching a conditional lower bound.

Finally, in the third part we prove tighter performance bounds for processes that disseminate a piece of information, either as quickly as possible (rumor spreading) or as anonymously as possible (cryptogenography).