Introduction to the ACM-SIAM Symposium on Discrete Algorithms (SODA) 2019 Special Issue

We are delighted to present a Special Issue of ACM Transactions on Algorithms, containing full versions of six articles that were presented at the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) 2019 in San Diego, California, on January 6–9, 2019. These articles, selected on the basis of their high ratings by the conference program committee, have been thoroughly reviewed according to the journal’s highest standards. In “Polynomial-time algorithm for Maximum Weight Independent Set on P6-free graphs”, Andrzej Grzesik, Tereza Klimošová, Marcin Pilipczuk, and Michał Pilipczuk study the classic problem of max-weight independent set. They provide the first polynomial-time algorithm to solve the problem for the class of P6-free graphs, i.e., all graphs that contain no path on six vertices as an induced subgraph. The approach is based on a careful enumeration of vertex subsets that inherently characterize all maximal independent sets. The article “I/O-Efficient Algorithms for Topological Sort and Related Problems” by Nairen Cao, Jeremy Fineman, Katina Russell, and Eugene Yang, introduces the first randomized algorithms for topological sorting and for identifying the strongly connected components of a directed graph in the I/O model. These algorithms have a runtime of O (sort (E) · poly(logV )), where sort (E) is the time needed for sorting E elements in the I/O-model. The algorithms are based on a recursive approach, which iteratively updates a random labeling until vertices in the same strongly connected component have the same label and the labeling represents a topological sorting of the strongly connected components. These new algorithms also imply efficient I/O-algorithms for various reachability and shortest paths problems in directed acyclic graphs. In the article “SETH-Based Lower Bounds for Subset Sum and Bicriteria Path”, by Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay, the authors introduce a reduction fromk-SAT to SUBSET SUM on dense instances. The new reduction proves that Bellman’s SUBSET SUM algorithm from 1969, which runs in O (Tn) time on n numbers and target T , cannot be improved to T 1−ε 2 (n) time for any constant ε > 0, unless the Strong Exponential Time Hypothesis (SETH) is false. The reduction is based on the results of Behrends (1946) about the existence of certain sequences of numbers, combined with clever partitioning and encoding. The authors also extend their techniques to prove SETH-based lower bounds for the BICRITERIA s, t-PATH problem. The article “Optimal Las Vegas Approximate Near Neighbors in p” by Alexander Wei presents Las Vegas data structures for solving approximate near neighbors in R under the p -norm. For 1 ≤ p ≤ 2, the author provides a data structure that matches the performance of optimal localitysensitive hashing. Moreover, using a locality-sensitive filter construction, the article gives the first