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I was wondering what the complexity of the following problem is:

Given: A flow network $N$ with a source $s$, sink $t$ and a number $k$.
Question: Is there an $s$-$t$ cut of capacity at most $k$?

Obviously, the problem is in P by standard methods. When trying to logspace reduce the P-complete LP-optimization problem, I encountered the following problems:

  • If the LP problem has no feasible solution, then $t$ should not be reachable from $s$. In this case, checking whether an LP problem has a feasible solution would be quite simple (don't know whether this is true).
  • If the LP problem is unbounded, then the min-cut should be unbounded as well. This could possibly be the case if $s$ and $t$ are identical. Again, checking whether an LP has an unbounded solution would be even easier.

For the lower bound, NL-hardness is quite easy: For a given directed graph $G$ with nodes $s$ and $t$, just take $G$ as a flow network and assign capacity 1 to each edge. Then ask if there is an $s$-$t$ cut of capacity at most 0. $t$ is reachable from $s$ if and only if the answer to this question is no. Since coNL=NL, we are done.

Moreover, the problem is NL-complete for outerplanar graphs. For this, $s$ and $t$ are joined by an edge of infinite capacity. This edge creates the two new faces $f_1$ and $f_2$. Now, answer the question whether there is a path of length at most $k$ from $f_1$ to $f_2$ in the dual graph.

I do not see how to generalize this to general graphs nor how to overcome the problems described when reducing LP optimization.

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    $\begingroup$ Have you tried reducing a known P-complete problem to it, e.g. LP optimisation? What else have you tried (but searching)? $\endgroup$ – Raphael Jul 24 '14 at 17:55
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The described problem is problem A.4.4 of the book "Limits to Parallel Computation: P-Completeness Theory" and is P-complete. It is proven via a reduction from Alternating Monotone Fanin 2, Fanout 2 CVP (problem A.1.4 of the same book).

The reduction described in this book originates from the 1990 Theoretical Computer Science paper "The Binary Network Flow Problem is Logspace Complete for P" (T. Lengauer, K. Wagner).

The reduction gets an Alternating Monotone Fanin 2, Fanout 2 CVP instance (a circuit $C$) and creates a flow network $N$ that allows a max flow of value $D 4^{k-1}$ where D is the number of input gates in $C$, iff $C$ evaluates to 1.

The idea is to construct the network by creating two nodes $(i,x,0)$ and $(i,x,1)$ for each gate $(i,x)$ (gate $x$ on the $i$th layer in $C$, one for each possible value 0 and 1). The input layer is layer 0. The source of $N$ is connected to $(0,x,0)$ ($(0,x,1)$) with capacity $4^k$ iff $(0,x)$ is a constant 0 (1) in $C$. Each edge $((i,x),(i+1,y))$ of $C$ is then replaced by a small subnetwork which sends 3/4 of the flow directly to the sink. The remaining quarter is sent to $(i+1,y,0) ((i+1,y,1))$, if $x$ in $C$ values to 0 (1). (At least the max flow needs to take these routes. Hence these small subnetworks simulate the gates.)

Now, the output gate $(k,x)$ is copied as well, but only the $(k,x,1)$ copy is connected to the edge. That is: Iff $C$ evaluates to 0, then exactly 1 flow unit cannot be sent to the sink. Hence, the flow in $N$ is $D 4^{k-1}-1$. If $C$ evaluates to 1, then the full $D 4^{k-1}$ flow units can reach the sink. This completes the reduction.

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