Abstract. We investigate the s-t-connectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspace-reducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case. We also consider a previously-studied subclass of planar graphs known as grid graphs. We show that the directed planar s-t-connectivity problem reduces to the reachability problem for directed grid graphs. A special case of the grid-graph reachability problem where no edges are directed from right to left is known as the “acyclic grid graph reachability problem”. We show that this problem lies in the complexity class UL. 1

We show that the st-connectivity problem for directed planar graphs can be decided in unambiguous logarithmic space. 1.

Abstract. There has been a great deal of progress in the fifteen years that have elapsed since Wigderson published his survey on the complexity of the graph connectivity problem [Wig92]. Most significantly, Reingold solved the longstanding question of the complexity of the s-t connectivity problem in undirected graphs, showing that this is complete for logspace (L) [Rei05]. This survey talk will focus on some of the remaining open questions dealing with graph reachability problems. Particular attention will be paid to these topics: – Reachability in planar directed graphs (and more generally, in graphs of low genus) [ADR05,BTV07]. – Reachability in different classes of grid graphs [ABC + 06]. – Reachability in mangroves [AL98]. The problem of finding a path from one vertex to another in a graph is the first problem that was identified as being complete for a natural subclass of P; it was shown to be complete for nondeterministic logspace (NL) by Jones [Jon75]. Restricted versions of this problem were subsequently shown to be complete for other natural complexity

We study the complexity of restricted versions of s-t-connectivity, which is the standard complete problem for NL. In particular, we focus on different classes of planar graphs, of which grid graphs are an important special case. Our main results are: • Reachability in graphs of genus one is logspace-equivalent to reachability in grid graphs (and in particular it is logspace-equivalent to both reachability and non-reachability in planar graphs). • Many of the natural restrictions on grid-graph reachability (GGR) are equivalent under AC 0 reductions (for instance, undirected GGR, outdegree-one GGR, and indegree-one-outdegree-one GGR are all equivalent). These problems are all equivalent to the problem of determining whether a completed game position in HEX is a winning position, as well as to the problem of reachability in mazes studied by Blum and Kozen [BK78]. These problems provide natural examples of problems that are hard for NC 1 under AC 0 reductions but are not known to be hard for L; they thus give insight into the structure of L. • Reachability in layered planar graphs is logspace-equivalent to layered grid graph reachability (LGGR). We show that LGGR lies in UL (a subclass of NL). • Series-Parallel digraphs (on which reachability was shown to be decidable in logspace by Jakoby et al.) are a special case of single-source-single-sink planar directed acyclic graphs (DAGs); reachability for such graphs logspace reduces to single-source-single-sink acyclic grid graphs. We show that reachability on such grid graphs AC 0 reduces to undirected GGR. • We build on this to show that reachability for single-source multiple-sink planar DAGs is solvable in L. 1

. We locate the complexities of evaluating, of inverting, and of testing membership in the image of, morphisms h : \Sigma ! \Delta . By and large, we show these problems complete for classes within NL. Then we develop new properties of finite codes and of finite sets of words, which yield image membership subproblems that are closely tied to the unambiguous space classes found between L and NL. 1 Introduction Free monoid morphisms h : \Sigma ! \Delta , for finite alphabets \Sigma and \Delta, are an important concept in the theory of formal languages (e.g. [6, 11]), and they are relevant to complexity theory. Indeed, it is well known (e.g. [7]) that NP = Closure( AC 0 m ; HOM n:e: ) ae Closure( AC 0 m ; HOM) = R:E:; where AC 0 m denotes many-one AC 0 -reducibility, HOM (resp. HOM n:e: ) is the set of morphisms (resp. nonerasing morphisms), and Closure denotes the smallest class of languages containing a finite nontrivial language and closed under the relations speci...

Abstract. We investigate different variants of unambiguity in the context of computing multi-valued functions. We propose a modification to the standard computation models of Turing Machines and configuration graphs, which allows for unambiguity-preserving composition. We define a notion of reductions (based on function composition), which allows nondeterminism but controls its level of ambiguity. In light of this framework we establish reductions between different variants of path counting problems. We obtain improvements of results related to inductive counting.

We present a deterministic algorithm running in space O , log n= log log n # solving the connectivity problem on strongly unambiguous graphs. In addition, we presentanO#log n# time-bounded algorithm for this problem running on a parallel pointer machine.

We present a deterministic algorithm running in space O \Gamma log 2 n= log log n \Delta solving the connectivity problem on strongly unambiguous graphs. In addition, we present an O(logn) time-bounded algorithm for this problem running on a parallel pointer machine.

In this thesis, we deal largely with complexity theoretic aspects in planar restrictions and obliviousness. Our main motivation was to identify problems for which the planar restriction is much easier, computationally, than the unrestricted version. First, we study constant width polynomial-sized circuits of low (polylogarithmic) genus; we show how such circuits characterize exactly the well-known circuit complexity class ACC0 (given that the unrestricted version captures the whole of NC1). We also give a new circuit characterization of the class NC1. Shifting our focus from circuits to graphs, we look at different notions of connectivity. We investigate the directed planar graph reachability problem, as a possibly more tractable special case of the arbitrary graph reachability problem (which is NL-complete). We prove that this problem logspace-reduces to its complement, and also that reachability questions on genus 1 graphs reduce to that in planar graphs. We also prove that reachability in a particularly simple class of planar graphs (namely, grid graphs) is no easier than the general directed planar reachability question. We then proceed to isolate to several large classes of planar graphs for which the reachability questions are solvable in deterministic logspace. Counting the number of spanning trees in a graph is a useful extension of the task of determining

We consider the reachability problem for a certain class of directed acyclic graphs embedded on surfaces. Let G(m, g) be the class of directed acyclic graphs with m = m(n) source vertices embedded on a surface (orientable or non-orientable) of genus g = g(n). We give a log-space reduction that on input G, u, v where G ∈G(m, g) and u and v are two vertices of G, outputs G,u,v where G is directed graph, and u,v are vertices of G, so that (a) there is a directed path from u to v in G if and only if there is a directed path from u to v in G and (b) G has O(m + g) vertices. By a direct application of Savitch’s theorem on the reduced instance we get a deterministic O(log n + log 2 (m + g))-space algorithm for the reachability problem for graphs in G(m, g). By setting m and g to be 2O(√log n) we get that the reachability problem for directed acyclic graphs with 2O(√log n) O( sources embedded on surfaces of genus 2 √ log n) is in L (deterministic logarithmic