Balanced ST-Connectivity

Today’s post is about a new open problem arising from my recent paper  (available on ECCC). The problem is as follows :

Let $G(V,E)$ be a directed graph. Let $G'(V,E')$ be the underlying undirected graph of $G$. Let $P$ be a path in $G'$. Let $e = (u,v)$ be an edge along the path $P$. Edge $e$ is called neutral edge if both $(u,v)$ and $(v,u)$ are in $E$. Edge $e$ is called forward edge if $(u,v) \in E$ and $(v,u) \notin E$. Edge $e$ is called backward edge if $(u,v) \notin E$ and $(v,u) \in E$.

A path (say $P$) from $s \in V$ to $t \in V$ in $G'(V,E')$ is called balanced if the number of forward edges along $P$ is equal to the number of backward edges along $P$. A balanced path might have any number of neutral edges. By definition, if there is a balanced path from $s$ to $t$ then there is a balanced path from $t$ to $s$. The path $P$ may not be a simple path. We are concerned with balanced paths of length at most $n$.

Balanced ST-Connectivity : Given a directed graph $G(V,E)$ and two distinguished nodes $s$ and $t$, decide if there is balanced path (of length at most $n$) between $s$ and $t$.

In my paper, I proved that SGSLOGCFL, a generalization of Balanced ST-Connectivity, is contained in DSPACE(lognloglogn). Details about SGSLOGCFL are in my paper.

Theorem 1 : SGSLOGCFL is in DSPACE(lognloglogn).

Open Problem : Is $SGSLOGCFL \in L$ ?

Cash Prize : I will offer \$100 for a proof of $SGSLOGCFL \in L$. I have spent enough sleepless nights trying to prove it. In fact, an alternate proof of Theorem 1 (or even any upper bound better than $O({\log}^2n)$) using zig-zag graph product seems to be a challenging task.

Usually people offer cash prizes for a mathematical problem when they are convinced that :

• it is a hard problem.
• it is an important problem worth advertising.
• the solution would be beautiful, requires new techniques and sheds new light on our understanding of related problems.

My reason is “All the above”. Have Fun solving it !!

A cute puzzle : In Balanced ST-Connectivity we are only looking for paths of length at most $n$. There are directed graphs where the only balanced st-path is super-linear. The example in the following figure shows an instance of Balanced ST-Connectivity where the only balanced path between $s$ and $t$ is of length $\Theta(n^2)$. The directed simple path from $s$ to $t$ is of length $n/2$. There is a cycle of length $n/2$ at the vertex $v$. All the edges (except $(v,u)$) on this cycle are undirected. The balanced path from $s$ to $t$ is obtained by traversing from $s$ to $v$, traversing the cycle clockwise for $n/2$ times and then traversing from $v$ to $t$.

Puzzle : Are there directed graphs where every balanced st-path is of super-polynomial size ?

Update : The above puzzle is now solved.

Open Problems

• Is $SGSLOGCFL \in L$ ?
• Are there directed graphs where every balanced st-path is of super-polynomial size ? (solved)
• More open problems are mentioned in my paper.