Aaron Potechin’s research while affiliated with Institute for Advanced Study and other places

We separate monotone analogues of L and NL by proving that any monotone switching network solving directed connectivity on n vertices must have size at least nΩ(lg n).

In this paper, we show that while almost all functions require exponential size branching programs to compute, for all functions f there is a branching program computing a doubly exponential number of copies of f which has linear size per copy of f. This result disproves a conjecture about non-uniform catalytic computation, rules out a certain type of bottleneck argument for proving non-monotone space lower bounds, and can be thought of as a constructive analogue of Razborov's result that submodular complexity measures have maximum value O(n).

Space complexity is the study of how much space/memory it takes to solve problems. Unfortunately, proving general lower bounds on space complexity is notoriously hard. Thus, we instead consider the restricted case of monotone algorithms, which only make deductions based on what is in the input and not what is missing. In this thesis, we develop techniques for analyzing monotone space complexity via a model called the monotone switching network model. Using these techniques, we prove tight bounds on the minimal size of monotone switching networks solving the directed connectivity, generation, and k-clique problems. These results separate monotone analgoues of L and NL and provide an alternative proof of the separation of the monotone NC hierarchy first proved by Raz and McKenzie. We then further develop these techniques for the directed connectivity problem in order to analyze the monotone space complexity of solving directed connectivity on particular input graphs.

A set of q triangles sharing a common edge is a called a book of size q. Letting bk(G) denote the size of the largest book in a graph G, Erd\H{o}s and Rothschild \cite{erdostwo} asked what the minimal value of bk(G) is for graphs G with n vertices and a set number of edges where every edge is contained in at least one triangle. In this paper, we show that for any graph G with n vertices and $\frac{n^2}{4} - nf(n)$ edges where every edge is contained in at least one triangle, $bk(G) \geq \Omega\left(\min{\{\frac{n}{\sqrt{f(n)}}, \frac{n^2}{f(n)^2}\}}\right)$.

We prove tight size bounds on monotone switching networks for the NP-complete problem of k-clique, and for an explicit monotone problem by analyzing a pyramid structure of height h for the P-complete problem of generation. This gives alternative proofs of the separations of m-NC from m-P and of m-NCⁱ from m-NCⁱ⁺¹, different from Raz-McKenzie (Combinatorica 1999). The enumerative-combinatorial and Fourier analytic techniques in this paper are very different from a large body of work on circuit depth lower bounds, and may be of independent interest.

In this paper, we extend the work of our previous paper "Bounds on monotone switching networks for directed connectivity" by further analyzing the monotone space complexity of directed connectivity. In particular, we analyze the monotone space compexity of directed connectivity for a variety of input graphs, not just minimal YES-instances. We show that the monotone space complexity of input graphs with a very large number of lollipops (vertices v for which there is an edge from s to v or an edge from v to t) is low by giving a randomized monotone generalization of Savitch's algorithm which uses a parity argument. This gives us upper bounds for both monotone switching networks and monotone circuits. We then give lower bounds on monotone switching networks for directed connectivity which are tight whenever the input graph is acyclic and does not have any vertices v which are connected to a large number of other vertices by short paths.

In this paper, we prove almost tight bounds on the size of sound monotone switching networks accepting permutations sets of directed trees. This roughly corresponds to proving almost tight bounds bounds on the monotone memory efficiency of the directed ST-connectivity problem for the special case in which the input graph is guaranteed to have no path from s to t or be isomorphic to a specific directed tree.

We prove tight size bounds on monotone switching networks for the k-clique problem, and for an explicit monotone problem by analyzing the generation problem with a pyramid structure of height h. This gives alternative proofs of the separations of m-NC from m-P and of m-NCi from m-NCi+1, different from Raz-McKenzie (Combinatorica '99). The enumerative-combinatorial and Fourier analytic techniques in this work are very different from a large body of work on circuit depth lower bounds, and may be of independent interest.