# 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.”