# Tag: theory

## Xi Chen Wins Both 2021 Gödel Prize and Fulkerson Prize

Xi Chen is one of a small number of researchers to be awarded two highly prestigious honors—the 2021 Gödel Prize and the 2021 Fulkerson Prize—in one year. He and his long-time collaborator Jin-Yi Cai, a professor at the University of Wisconsin–Madison, won the prizes for their paper, Complexity of Counting CSP with Complex Weights.

## Henry Yuen Wins NSF CAREER Award

The theoretical computer scientist will use the award to push the boundaries of quantum information science.

## Tim Roughgarden Elected Fellow of the Society for the Advancement of Economic Theory

## Quantum Q&A with PhD Student Hamoon Mousavi

Hamoon Mousavi is a PhD student who moved to Columbia from the University of Toronto last year with Henry Yuen, whose research group studies theoretical computer science and the differences between classical and quantum computers.

## Untangling Quantum Information at Columbia

Meet Henry Yuen, a computer scientist exploring the boundaries between classical and quantum computers. Yuen joined Columbia Engineering as an assistant professor in January 2021.

## 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.

## Four Papers from the Theory Group Accepted to FOCS 2019

Papers from CS researchers were accepted to the 60th Annual Symposium on Foundations of Computer Science (FOCS 2019). The papers delve into population recovery, sublinear time, auctions, and graphs.

Finding Monotone Patterns in Sublinear Time

Omri Ben-Eliezer *Tel-Aviv University*, Clement L. Canonne *Stanford University*, Shoham Letzter *ETH-ITS, ETH Zurich*, Erik Waingarten *Columbia University*

The paper is about finding increasing subsequences in an array in sublinear time. Imagine an array of *n* numbers where at least 1% of the numbers can be arranged into increasing subsequences of length *k*. We want to pick random locations from the array in order to find an increasing subsequence of length *k*. At a high level, in an array with many increasing subsequences, the task is to find one. The key is to cleverly design the distribution over random locations to minimize the number of locations needed.

Roughly speaking, the arrays considered have a lot of increasing subsequences of length *k*; think of these as “evidence of existence of increasing subsequences”. However, these subsequences can be hidden throughout the array: they can be spread out, or concentrated in particular sections, or they can even have very large gaps between the starts and the ends of the subsequences.

“The surprising thing is that after a specific (and simple!) re-ordering of the “evidence”, structure emerges within the increasing subsequences of length *k*,” said Erik Waingarten, a PhD student. “This allows for design efficient sampling procedures which are optimal for non-adaptive algorithms.”

Beyond Trace Reconstruction: Population Recovery From the Deletion Channel

Frank Ban *UC Berkeley*, Xi Chen *Columbia University*, Adam Freilich *Columbia University*, Rocco A. Servedio *Columbia University*, Sandip Sinha *Columbia University*

Consider the problem of reconstructing the DNA sequence of an extinct species, given some DNA sequences of its descendant(s) that are alive today. We know that DNA sequences get modified through random mutations, which can be substitutions, insertions and deletions.

A mathematical abstraction of this problem is to recover an unknown source string **x** of length n, given access to independent samples of **x** that have been corrupted according to a certain noise model. The goal is to determine the minimum number of samples required in order to recover **x** with high confidence. In the special case that the corruption occurs via a deletion channel (i.e., each character in **x** is deleted independently with some probability, say 0.1, and the surviving characters are concatenated and transmitted), each sample is called a *trace*. The corresponding recovery problem is called *trace reconstruction*, and it has received significant attention in recent years.

The researchers considered a generalized version of this problem (known as *population recovery*) where there are multiple unknown source strings, along with an unknown distribution over them specifying the relative frequency of each source string. Each sample is generated by first drawing a source string with the associated probability, and then generating a trace from it via the deletion channel. The goal is to recover the source strings, along with the distribution over them (up to small error), from the mixture of traces.

For the main sample complexity upper bound, they show that for any population size s = o(log n / log log n), a population of s strings from {0,1}^n can be learned under deletion channel noise using exp(n^{1/2 + o(1)}) samples. On the lower bound side, we show that at least n^{\Omega(s)} samples are required to perform population recovery under the deletion channel when the population size is s, for all s <= n^0.49.

“I found it interesting that our work is based on certain mathematical results in which, at first glance, seem to be completely unrelated to the computational problem we consider,” said Sandip Sinha, a PhD student. In particular, they used constructions based on *Chebyshev polynomials*, a certain sequence of polynomials which are extremal for many properties, and is hence ubiquitous throughout theoretical computer science. Similarly, previous work on trace reconstruction rely on certain extremal results about complex-valued polynomials. Continued Sinha, “I think it is quite intriguing that complex analytic techniques yield useful results about a problem which is fundamentally about discrete structures (binary strings).”

Settling the Communication Complexity of Combinatorial Auctions with Two Subadditive Buyers

Tomer Ezra *Tel Aviv University*, Michal Feldman *Tel Aviv University*, Eric Neyman *Columbia University*, Inbal Talgam-Cohen *Technion*; S. Matthew Weinberg *Princeton University*

The paper is about the theory of combinatorial auctions. In a combinatorial auction, an auctioneer wants to allocate several items among bidders. Each bidder has a certain amount that they value each item; bidders also have values for combinations of items, and in a combinatorial auction a bidder might not value a combination of items as much as each item individually.

For instance, say that a pencil and a pen will be auctioned. The pencil is valued at 30 cents and the pen at 40 cents, but the pen and pencil together at only 50 cents (it may be that there isn’t any additional value from having both the pencil and the pen). Valuation functions with this property — that the value of a combination of items is less than or equal to the sum of the values of each item — are called *subadditive*.

In the paper, the researchers answered a longstanding open question about combinatorial auctions with two bidders who have subadditive valuation — roughly speaking, is it possible for an auctioneer to efficiently communicate with both bidders to figure out how to allocate the items between them to make the bidders happy?

The answer turns out to be no. In general, if the auctioneer wants to do better than just giving all of the items to one bidder or the other at random, the auctioneer needs to communicate a very large amount with the bidders.

The result itself was somewhat surprising, the researchers expected it to be possible for the auctioneer to do pretty well without having to communicate with the bidders too much. “Also, information theory was extensively used as part of proving the result,” said Eric Neyman, a PhD student. “This is unexpected, because information theory has not been used much in the study of combinatorial auctions.”

Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time

Soheil Behnezhad *University of Maryland*, Mahsa Derakhshan *University of Maryland*, Mohammad Taghi Hajiaghayi *University of Maryland*, Cliff Stein *Columbia University*, Madhu Sudan *Harvard University*

In a graph, an independent set is a set of vertices with the property that none are adjacent. For example, in the graph of Facebook friends, vertices are people and there is an edge between two people who are friends. An independent set would be a set of people, none of whom are friends with each other. A basic problem is to find a large independent set. The paper focuses on one type of large independent set known as a maximal independent set, that is, one that cannot have any more vertices added to it.

Graphs, such as the friends graph, evolve over time. As the graph evolves, the maximal independent set needs to be maintained, without recomputing one from scratch. The paper significantly decreases the time to do so, from time that is polynomial in the input size to one that is polylogarithmic.

A graph can have many maximal independent sets (e.g. in a triangle, each of the vertices is a potential maximal independent set). One might think that this freedom makes the problems easier. The researchers picked one particular kind of maximal independent set, known as a lexicographically first maximal independent set (roughly this means that in case of a tie, the vertex whose name is first in alphabetical order is always chosen) and show that this kind of set can be maintained more efficiently.

“Giving up this freedom actually makes the problems easier,” said Cliff Stein, a computer science professor. “The idea of restricting the set of possible solutions making the problem easier is a good general lesson.”

## New algorithms developed by CS researchers at FOCS 2018

Four papers were accepted to the Foundations of Computer Science (FOCS) symposium. CS researchers worked alongside colleagues from various organizations to develop the algorithms.

**Learning Sums of Independent Random Variables with Sparse Collective Support**

Authors: Anindya De *Northwestern University*, Philip M. Long *Google,* Rocco Servedio *Columbia University*

The paper is about a new algorithm for learning an unknown probability distribution given draws from the distribution.

A simple example of the problem that the paper considers can be illustrated with a penny tossing scenario: Suppose you have a huge jar of pennies, each of which may have a different probability of coming up heads. If you toss all the pennies in the jar, you’ll get some number of heads; how many times do you need to toss all the pennies before you can build a highly accurate model of how many pennies are likely to come up heads each time? Previous work answered this question, giving a highly efficient algorithm to learn this kind of probability distribution.

The current work studies a more difficult scenario, where there can be several different kinds of coins in the jar — for example, it may contain pennies, nickels and quarters. Each time you toss all the coins, you are told the total *value* of the coins that came up heads (but not how many of each type of coin came up heads).

“Something that surprised me is that when we go from only one kind of coin to two kinds of coins, the problem doesn’t get any harder,” said Rocco Servedio, a researcher from Columbia University. “There are algorithms which are basically just as efficient to solve the two-coin problem as the one-coin problem. But we proved that when we go from two to three kinds of coins, the problem provably gets much harder to solve.”

**Holder Homeomorphisms and Approximate Nearest Neighbors**

Authors: Alexandr Andoni *Columbia University*, Assaf Naor *Princeton University*, Aleksandar Nikolov *University of Toronto*, Ilya Razenshteyn

*Microsoft Research Redmond*, Erik Waingarten *Columbia University*

This paper gives new algorithms for the approximate near neighbor (ANN) search problem for general normed spaces. The problem is a classic problem in computational geometry, and a way to model “similarity search”.

For example, Spotify may need to preprocess their dataset of songs so that new users may find songs which are most similar to their favorite songs. While a lot of work goes into designing good algorithms for ANN, these algorithm work for specific metric spaces of interest to measure the distance between two points (such as Euclidean or Manhattan distance). This work is the first to give non-trivial algorithms for general normed spaces.

“One thing which surprised me was that even though the embedding appears weaker than established theorems that are commonly used, such as John’s theorem, the embedding is efficiently computable and gives more control over certain parameters,” said Erik Waingarten, an algorithms and computational complexity PhD student.

**Parallel Graph Connectivity in Log Diameter Rounds**

Authors: Alexandr Andoni *Columbia University, Zhao Song Harvard University & UT-Austin, Clifford Stein Columbia University, Zhengyu Wang Harvard University, Peilin Zhong Columbia University*

This paper is about designing fast algorithms for problems on graphs in parallel systems, such as MapReduce. The latter systems have been widely-successful in practice, and invite designing new algorithms for these parallel systems. While many classic “parallel algorithms” were been designed in the 1980s and 1990s (PRAM algorithms), predating the modern massive computing clusters, they had a different parallelism in mind, and hence do not take full advantage of the new systems.

The typical example is the problem of checking connectivity in a graph: given N nodes together with some connecting edges (e.g., a friendship graph or road network), check whether there’s a path between two given nodes. The classic PRAM algorithm solves this in “parallel time” that is logarithmic in N. While already much better than the sequential-time of N (or more), the researchers considered to question whether one can do even better in the new parallel systems a-la MapReduce.

While obtaining a much better runtime seems out of reach at the moment (some conjecture impossible), the researchers realized that they may be able to obtain faster algorithms when the graphs have a small diameter, for example if any two connected nodes have a path at most 10 hops. Their algorithm obtains a parallel time that is logarithmic in the diameter. (Note that the diameter of a graph is often much smaller than N.)

“Checking connectivity may be a basic problem, but its resolution is fundamental to understanding many other problems of graphs, such as shortest paths, clustering, and others,” said Alexandr Andoni, one of the authors. “We are currently exploring these extensions.”

**Non-Malleable Codes for Small-Depth Circuits**

Authors: Marshall Ball *Columbia University*, Dana Dachman-Soled *University of Maryland*, Siyao Guo *Northeastern University*, Tal Malkin *Columbia University*, Li-Yang Tan *Stanford University*

With this paper, the researchers constructed new non-malleable codes that improve efficiency over what was previously known.

“With non-malleable codes, any attempt to tamper with the encoding will do nothing and what an attacker can only hope to do is replace the information with something completely independent,” said Maynard Marshall Ball, a fourth year PhD student. “That said, non-malleable codes cannot exist for arbitrary attackers.”

The constructions were derived via a novel application of a powerful circuit lower bound technique (pseudorandom switching lemmas) to non-malleability. While non-malleability against circuit classes implies strong lower bounds for those same classes, it is not clear that the converse is true in general. This work shows that certain techniques for proving strong circuit lower bounds are indeed strong enough to yield non-malleability.