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

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.

A monotone planar circuit (MPC) is a Boolean circuit that can be embedded in a plane, and that has only AND and OR gates. Yang showed that the one-input-face monotone planar circuit value problem (MPCVP) is in NC 2, and Limaye et. al. improved the bound to LogCFL. Barrington et. al. showed that evaluating monotone upward stratified circuits, a restricted version of the oneinput-face MPCVP, is in LogDCFL. In this paper, we prove that the unrestricted one-input-face MPCVP is also in LogDCFL. We also show this problem to be L-hard under quantifier free projections.

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

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to monotone planar circuits MPCVP, is known to be in NC 3, and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we re-examine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that stratified cylindrical monotone circuits can be evaluated in LogDCFL, and arbitrary cylindrical monotone circuits can be evaluated in AC 1 (LogDCFL), while monotone circuits with one-input-face planar embeddings can be evaluated in LogCFL. For monotone circuits with focused embeddings, we show an upper bound of AC 1 (LogDCFL). We re-examine the NC 3 algorithm for general MPCVP, and note that it is in AC 1 (LogCFL) = SAC 2. Finally, we consider extensions beyond MPCVP. We show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. Also, special kinds of arbitrary genus circuits can also be evaluated in NC. We also show that planar non-monotone circuits with polylogarithmic negation-height can be evaluated in NC.

Abstract. A monotone planar circuit (MPC) is a Boolean circuit that can be embedded in a plane, and that has only AND and OR gates. Yang showed that the one-input-face monotone planar circuit value problem (MPCVP) is in NC 2, and Limaye et. al. improved the bound to LogCFL. Barrington et. al. showed that evaluating monotone upward stratified circuits, a restricted version of the one-input-face MPCVP, is in LogDCFL. In this paper, we prove that the unrestricted one-input-face MPCVP is also in LogDCFL. We also show this problem to be L-hard under quantifier free projections. Key Words: L, LogDCFL, monotone planar circuits. 1

Viewing the computation of the determinant and the permanent of integer matrices as combinatorial problems on associated graphs, we explore the restrictiveness of planarity on their complexities and show that both problems remain as hard as in the general case, i.e. GapL and #P complete. On the other hand, both bipartite planarity and bimodal planarity bring the complexity of permanents down (but no further) to that of determinants. The permanent or the determinant modulo 2 is complete for ⊕L, and we show that parity of paths in a layered grid graph (which is bimodal planar) is also complete for this class. We also relate the complexity of grid graph reachability to that of testing existence/uniqueness of a perfect matching in a planar bipartite graph.

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and for the special case of upward stratified circuits, it is known to be in LogDCFL. In this paper we re-examine the complexity of MPCVP, with special attention to circuits with cylindrical embeddings. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that stratified cylindrical monotone circuits can be evaluated in LogDCFL, and arbitrary cylindrical monotone circuits can be evaluated in AC 1 (LogDCFL), while monotone circuits with one-input-face planar embeddings can be evaluated in LogCFL. For monotone circuits with focused embeddings, we show an upper bound of AC 1 (LogDCFL). We re-examine the NC 3 algorithm for general MPCVP, and note that it is in AC 1 (LogCFL) = SAC 2. Finally, we consider extensions beyond MPCVP. We show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC. Also, special kinds of arbitrary genus circuits can also be evaluated in NC. We also show that planar non-monotone circuits with polylogarithmic negation-height can be evaluated in NC.