Electronic Colloquium on Computational Complexity, Report No. 149 (2005) Grid Graph Reachability Problems Eric Allender∗ David A. Mix Barringtony Tanmoy Chakrabortyz Samir Dattax Sambuddha Roy{ Abstract reductions (for instance, undirected GGR, out- degree-one GGR, and indegree-one-outdegree- We study the complexity of restricted versions of st- one GGR are all equivalent). These problems are connectivity, which is the standard complete problem all equivalent to the problem of determining if a for NL. Grid graphs are a useful tool in this regard, completed game position in HEX is a winning po- since sition, as well as to the problem of reachability in mazes studied by Blum and Kozen [5]. • reachability on grid graphs is logspace- equivalent to reachability in general planar • Series-Parallel digraphs are a special case of digraphs, and single-source-single-sink planar dags; reachabil- ity for such graphs logspace reduces to single- • reachability on certain classes of grid graphs source-single-sink acyclic grid graphs. We show gives natural examples of problems that are hard that reachability on such grid graphs AC0 re- for NC1 under AC0 reductions but are not known duces to undirected GGR. to be hard for L; they thus give insight into the structure of L. • We build on this to show that reachability for single-source multiple-sink planar dags is solv- In addition to explicating the structure of L, another of able in L. our goals is to expand the class of digraphs for which connectivity can be solved in logspace, by building on the work of Jakoby et al. [11], who showed that reach- 1 Introduction ability in series-parallel digraphs is solvable in L. Our main results are: Graph reachability problems have long played a fundamental role in complexity theory. The general st- • Many of the natural restrictions on grid-graph connectivity problem in directed graphs is the standard reachability (GGR) are equivalent under AC0 complete problem for NL, while the st-connectivity ∗Department of Computer Science, Rutgers, the State Univer- problems for directed graphs of outdegree 1 [9, 7] and sity of NJ. Supported in part by NSF Grant CCF-0514155. email: undirected graphs [13] are complete for L. It follows [email protected]. from [3] that reachability in directed graphs of width y Computer Science Dept., University of Massachusetts O(1) (or even width five, with outdegree 1) is com- Amherst. Supported in part by NSF Grant CCR-9988260. e-mail: NC1 [email protected]. plete for . zChennai Mathematical Institute, Chennai, India. e-mail: Grid graphs are a special class of planar graphs [email protected]. whose vertices are located on grid points, and whose x Chennai Mathematical Institute, Chennai, India. e-mail: vertices are adjacent only to their immediate horizon- [email protected] . tal or vertical neighbors. Barrington et al. showed {Department of Computer Science, Rutgers, the State Univer- sity of NJ. Supported in part by NSF Grant CCF-0514155. email: [4] that st-connectivity in width k (directed or undi- [email protected]. rected) graphs is complete for depth k AC0 under first- 1 ISSN 1433-8092 order projections. In this paper we study grid graphs • Single-Source Multiple-Sink Planar DAGs without any width restrictions. The general grid-graph (SMPDs): the class of DAGs having one vertex reachability problem (GGR) is equivalent to the st- of indegree zero. Reachability in such graphs connectivity problem in directed planar graphs (and is clearly equivalent to reachability in Multiple- graphs of genus one) under logspace reducibility [1]. Source Single Sink DAGs (MSPDs) by simply The best upper bound known for GGR is NL, although reversing all of the edges. a slightly better upper bound is known for so-called “layered” grid graphs (LGGR): LGGR 2 UL \ coUL • Multiple-Source Multiple-Sink Planar DAGs [1]. (MMPD). This is simply the class of all planar Our focus in this paper is primarily on classes of DAGs. grid graphs whose reachability problem is solvable in logspace. Reachability in undirected grid graphs We show that the SMPD reachability problem (and (UGGR) was studied by Blum and Kozen [5]; they hence also that for MSPD) lies in logspace. In addi- showed that UGGR is solvable in logspace (which tion, reachability in SSPDs, restricted to grid graphs, was superseded a quarter-century later by the work of is reducible to UGGR. Our algorithmic approach for Reingold [13]). Buss has studied UGGR in connec- SMPD extends to certain classes of graphs that are not tion with tautologies arising from the game of HEX acyclic. This is discussed in more detail in Section 5. [6] (namely, the tautology that every completed game The rest of the paper is organized as follows. In board of HEX has a winner); he credits Barrington Section 2 we introduce the various grid graph prob- with the observation that UGGR is equivalent to the lems that we will be considering, and present reduc- problem of determining if a given completed HEX tions showing how these problems relate to each other. board position is a win for one player. Reachability In Section 3 we present a generic reduction showing in grid graphs of outdegree one (1GGR) is another re- that, for many of the problems we consider, it is no striction on GGR that is clearly solvable in logspace. loss of generality to assume that s and t appear on the One of our theorems is that UGGR and 1GGR are external boundary of the graph. Our hardness results equivalent under AC0 reductions (and even under first- are presented in Section 4. Our logspace algorithms order projections). We show that these problems are for SSPD and SMPD are presented in Section 5. We hard for NC1, and thus this gives a cluster of natural conclude with open questions in Section 6. problems that are candidates for having complexity in- termediate between NC1 and L, since even the general 2 Versions of the GGR Problem GGR problem is not known to be hard for L under AC0 reductions. We begin by defining and exploring a number of Jakoby, Liskiewicz, and Reischuk showed that special cases of the GGR problem, based on a variety reachability in series-parallel digraphs is solvable in of restrictions on the grid graphs and on the vertices s logspace [11], thus solving the reachability question and t. for an important subclass of planar directed graphs. Series-parallel digraphs are a special case of planar di- 2.1 Classes and Reductions rected acyclic graphs having a single source and single sink. Motivated by a desire to solve the reachability We assume familiarity with the following important problem for a larger class of planar DAGs, we intro- subclasses of nondeterministic logspace (NL): L, NC1, duce the following three classes of DAGs: TC0, and AC0. When defining notions of reducibility • Single-Source Single-Sink Planar DAGs and completeness in order to investigate the structure (SSPDs): the class of DAGs having one vertex of of such small complexity classes, some form of AC0 indegree zero and one vertex of outdegree zero. reducibility is usually employed. We will frequently Reachability in SSPDs generalizes the problem make use of the terminology and notation employed of reachability in series-parallel digraphs studied by Immerman [10], which exploits the close connec- in [11]. tions between AC0 and first-order logic. In particular, 2 0 AC0 AC -Turing reducibility (≤T ) to a set A can be de- GGR fined equivalently in terms of AC 0 circuits augmented with “oracle gates” for A, or in terms of first-order for- mulae with A as a built-in predicate symbol applied to a structure defined in first-order. For details refer 1-GGR GGR-B AC0 to [10]. For this reason, we sometimes refer to ≤T reductions as FO reductions. The class of problems AC0 ≤T reducible to A is sometimes denoted as FO+A. 11GGR 1GGR-B LGGR Immerman also gives good motivation for study- AC0 ing a restricted form of ≤m reductions called first- order projections (≤FO ). These can be visualized as proj 11GGR-B 1LGGR many-one reductions computed by first-order uniform circuits having no gates (other than NOT gates); thus each bit of the output is either a constant or is a copy (or a negated copy) of one bit of the input. 11LGGR Figure 1. Nine GGR problems. 2.2 Nine Problems We first consider two restrictions on the global both degrees 1) give us nine special cases of the GGR structure of a GGR problem, and two local restrictions: problem: GGR, 1GGR, 11GGR, GGR-B, 1GGR-B, 11GGR-B, LGGR, 1LGGR, and 11LGGR. Even the • The problem GGR-B is the set of tuples (G; s; t) easiest of these problems, 11LGGR, is non-trivial, as where G is a directed grid graph, s and t are ver- we will show in Section 4 that it is hard for the class tices on the boundary of G, and there is a path TC0. from s to t in G. There are other natural ways to define a layered • The problem LGGR is the set of tuples (G; s; t) graph. We could forbid only one of the four directions where G is a layered directed grid graph, having of edges rather than two. Or we could allow diago- only east and south edges, and there is a path nal edges but force them to go only northeast, east, or from s to t. southeast, making each north-south column a layer ac- cording to the standard definition. But it is an easy ex- • The problem 1GGR is the set of tuples (G; s; t) ercise to construct a first-order projection from a graph where G is a directed grid graph of out-degree at satisfying any one of these restrictions to one satisfy- most 1 and there is a path from s to t.