### DIMACS Theoretical Computer Science Seminar

Title: Grid Graph Reachability Problems

Speaker: **Eric Allender**, Rutgers University

Date: February 7, 2006 2:00-3:00pm

Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

We study the complexity of restricted versions of st-connectivity,
which is the standard complete problem for NL. Grid graphs are a
useful tool in this regard, since
* reachability on grid graphs is logspace-equivalent to reachability
in general planar digraphs, and
* reachability on certain classes of grid graphs gives natural examples
of problems that are hard for NC^1 under AC^0 reductions but are not
known to be hard for DLOG; they thus give insight into the structure of
DLOG.
In addition to explicating the structure of DLOG, another of 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.
who showed that reachability in series-parallel digraphs is solvable in DLOG.
Our main results are:
* Many of the natural restrictions on grid-graph reachability (GGR) are
equivalent under AC^0 reductions (for instance, undirected GGR,
out-degree-one GGR, and indegree-one-outdegree-one GGR are all
equivalent). These problems are all equivalent to the problem of
determining if 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.
* Series-Parallel digraphs are a special case of single-source-single-sink
planar 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 DLOG.
This is joint work with David A. Mix Barrington, Tanmoy Chakraborty,
Samir Datta, and Sambuddha Roy.