# New Sublinear Methods in the Struggle Against Classical Problems

PhD Thesis, Massachusetts Institute of Technology, September 2010.

We study the time and query complexity of approximation algorithms that access only a minuscule fraction of the input, focusing on two classical sources of problems: combinatorial graph optimization and manipulation of strings. The tools we develop find applications outside of the area of sublinear algorithms. For instance, we obtain a more efficient approximation algorithm for edit distance and distributed algorithms for combinatorial problems on graphs that run in a constant number of communication rounds.

Combinatorial Graph Optimization Problems: The graph optimization problems considered by us include vertex cover, maximum matching, and dominating set. A graph algorithm is traditionally called a constant-time algorithm if it runs in time that is a function of only the maximum vertex degree, and in particular, does not depend on the number of vertices in the graph.

We show a general local computation framework that allows for transforming many classical greedy approximation algorithms into constant-time approximation algorithms for the optimal solution size. By applying the framework, we obtain the first constant-time algorithm that approximates the maximum matching size up to an additive $\epsilon n$, where $\epsilon$ is an arbitrary positive constant, and $n$ is the number of vertices in the graph.

It is known that a purely additive $\epsilon n$ approximation is not computable in constant time for vertex cover and dominating set. We show that nevertheless, such an approximation is possible for a wide class of graphs, which includes planar graphs (and other minor-free families of graphs) and graphs of subexponential growth (a common property of networks). This result is obtained via locally computing a good partition of the input graph in our local computation framework.

The tools and algorithms developed for these problems find several other applications:

• Our methods can be used to construct local distributed approximation algorithms for some combinatorial optimization problems.
• Our matching algorithm yields the first constant-time testing algorithm for distinguishing bounded-degree graphs that have a perfect matching from those far from having this property.
• We give a simple proof that there is a constant-time algorithm distinguishing bounded-degree graphs that are planar (or in general, have a minor-closed property) from those that are far from planarity (or the given minor-closed property, respectively). Our tester is also much more efficient than the original tester of Benjamini, Schramm, and Shapira (STOC 2008).

Edit Distance: We study a new asymmetric query model for edit distance. In this model, the input consists of two strings $x$ and $y$, and an algorithm can access $y$ in an unrestricted manner (without charge), while being charged for querying every symbol of $x$.

We design an algorithm in the asymmetric query model that makes a small number of queries to distinguish the case when the edit distance between $x$ and $y$ is small from the case when it is large. Our result in the asymmetric query model gives rise to a near-linear time algorithm that approximates the edit distance between two strings to within a polylogarithmic factor. For strings of length $n$ and every fixed $\epsilon > 0$, the algorithm computes a $(\log n)^{O(1/\epsilon)}$ approximation in $n^{1+\epsilon}$ time. This is an exponential improvement over the previously known near-linear time approximation factor $2^{\tilde O(\sqrt{\log n})}$ (Andoni and Onak, STOC 2009; building on Ostrovsky and Rabani, J. ACM 2007). The algorithm of Andoni and Onak was the first to run in $O(n^{2-\delta})$ time, for any fixed constant $\delta >0$, and obtain a subpolynomial, $n^{o(1)}$, approximation factor, despite a sequence of papers.

We provide a nearly-matching lower bound on the number of queries. Our lower bound is the first to expose hardness of edit distance stemming from the input strings being “repetitive”, which means that many of their substrings are approximately identical. Consequently, our lower bound provides the first rigorous separation on the complexity of approximation between edit distance and Ulam distance.