## Papers from the Theory Group Accepted to FOCS 2020

Professor Tim Roughgarden received a Test of Time award for his paper, How Bad is Selfish Routing?, published in 2000 and Runzhou Tao, a second-year PhD student, bagged a Best Paper award from the annual Foundations of Computer Science (FOCS) conference.

Edge-Weighted Online Bipartite Matching

Matthew Fahrbach *Google Research*, Zhiyi Huang *University of Hong Kong*, Runzhou Tao *Columbia University*, Morteza Zadimoghaddam *Google Research*

The online bipartite matching problem introduced by Richard Karp, Umesh Vazirani, and Vijay Vazirani in 1990 is one of the most important problems in the field of online algorithms. In this problem, only one side of the bipartite graph (called “offline nodes”) is given. The other side of the graph (called “online nodes”) is given one by one. Each time an online node arrives, the algorithm must decide whether and how it should be matched. This decision is irrevocable. The online bipartite matching problem has a wide range of applications, e.g., online ad allocation, in which we can see advertisers as offline nodes and users as online nodes.

This paper gives a positive answer for this 30-year open problem by giving an 0.508-competitive algorithm for the edge-weighted bipartite online matching problem. The algorithm uses a new subroutine called Online Correlated Selection, which takes a sequence of pairs as input and selects one element from each pair. By negatively correlating the selections, one can produce a better online matching algorithm.

This new technique will have further applications in the field of online algorithms.

An Adaptive Step Toward the Multiphase Conjecture and Lower Bounds on Nonlinear Networks

Young Kun Ko *New York University*, Omri Weinstein *Columbia University*

Consider the problem of maintaining a directed graph under edge addition/deletions, so that connectivity between any pair of vertices can be answered quickly. This basic problem has no efficient data structure, despite decades of research. In 2010, Patrascu proposed a communication problem (the “Multiphase Conjecture”), whose resolution would prove that problems like dynamic reachability indeed require slow (n^0.1) update or query time. We use information-theoretic tools to prove a weaker version of the Multiphase Conjecture, which implies a polynomial (~ \sqrt{n}) lower bound on “weakly-adaptive” dynamic data structures for the reachability problem. We also use this result to make progress on understanding the power of nonlinear gates in networks computing *linear* operators (x –> Ax).

Edit Distance in Near-Linear Time: It’s a Constant Factor

Alexandr Andoni *Columbia University*, Negev Shekel Nosatzki *Columbia University*

This paper resolves an open question about the complexity of estimating the edit distance up to a constant factor. Edit distance is a fundamental problem, and its exact quadratic-time algorithm is one of the most classic dynamic programming problems.

It was shown that under the SETH conjecture, no exact algorithm can resolve it in sub-quadratic, so the open question remained what is the best approximation one can obtain. A breakthrough result from 2018 showed the first trust sub-quadratic algorithm for Constant factor, and the question remained if it can be done in near-linear time. This paper resolved this question positively.

Polynomial Data Structure Lower Bounds in the Group Model

Alexander Golovnev *Harvard University*, Gleb Posobin *Columbia University*, Oded Regev *New York University*, Omri Weinstein *Columbia University*

Range-counting is one of the most omnipresent query spatial databases, computational geometry, and eCommerce (e.g. “Find all employees from countries X who have earned salary >X between years 2000-2018”). Fast data structures with linear space are known for various such problems, all of which use only additions and subtractions of pre-computed weighted sums (aka the “group model”). However, for general ranges (geometric shapes), no efficient data structures were known, yet proving > log(n)

Lower bounds in the group model remained a fundamental challenge since the early 1980s. The paper proves a *polynomial* (n^0.1) lower bound on the query time of linear-space data structures for an explicit range-counting problem of convex polygons in R^2.

On Light Spanners, Low-treewidth Embeddings, and Efficient Traversing in Minor-free Graphs

Vincent Cohen-Addad *Google Research*, Arnold Filtser *Columbia University*, Philip N. Klein *Brown University*, Hung Le *University of Victoria and University of Massachusetts at Amherst*

Fundamental routing problems such as the Traveling Salesman Problem (TSP) and the Vehicle Routing Problem have been widely studied since the 1950s. Given a metric space, the goal is to find a minimum-weight collection of tours (only one for TSP) so as to meet a prescribed demand at some points of the metric space. Both problems have been the source of inspiration for many algorithmic breakthroughs and, quite frustratingly, remain good examples of the limits of the power of algorithmic methods.

The paper studies the geometry of weighted minor free graphs, which is a generalization of planar graphs, where the graph is somewhat topologically restricted. The framework is this of metric embeddings, where we create a “small-complexity” graph that approximately preserves distances between pairs of points in the original graph. We have two such structural results:

1. Light subset spanner: given a set K of terminals, we construct a subgraph of the original graph that preserves all distances between terminals up to 1+\eps factor and have total weight only slightly larger than the Steiner tree: the minimal weight subgraph connecting all terminals.

2. Stochastic metric embedding into low treewidth graphs: treewidth is a graph parameter measuring how much a graph is “treelike”. Many hard problems become tractable on bounded treewidth graphs. We create a distribution over mapping of the graph into a bounded treewidth graph, such that the distance between every pair of points increases only by a small additive constant (in expectation).

The structural results are then used to obtain an efficient polynomial approximation scheme (EPTAS) for subset TSP in minor-free graphs, and a quasi-polynomial approximation scheme (QPTAS) for the vehicle routing problem in minor-free graphs.