# Shortest path with permutations and fixed dimension

I'm thinking of extensions of the shortest path problem which are solvable in polynomial time. One way to do this is to consider the shortest path problem on a weighted directed graph with weights on $$\mathbb{Q}$$ as the problem of minimizing a rational value $$x$$ which is updated at each edge of the graph with the label $$v$$ of the edge. This we can write as $$x:=x+v$$. The shortest path problem then becomes the problem of minimising the value of $$x$$. This is known to be solvable in polynomial time.

Now suppose we generalise this to a constant number of variables $$\vec{x}$$. We consider at each step an update of a vector $$\vec{x} = A \vec{x} + \vec{b}$$ where $$A \in \mathbb{Q}^{n \times n}, \vec{b} \in \mathbb{Q}^{n \times 1}$$. The matrix $$A$$ and the vector $$\vec{b}$$ may be different at each edge. At the target node one computes an affine function $$o = \vec{c} \vec{x} + d$$ where $$\vec{c} \in \mathbb{Q}^{n \times 1}$$ and $$d \in \mathbb{Q}$$. This target function may be different for each final node. The optimisation problem is to determine the minimal value of $$o$$.

If we don't restrict the values of $$A$$ then the problem is strongly $$NP$$-hard (Shortest path with affine updates and fixed dimension). I'm interested in the way to solve simpler problems. For instance,

• $$A = I$$ the identity matrix.
• $$A$$ a 0/1-matrix with exactly one "1" per row.
• $$A$$ a 0/1-matrix with exactly one "1" per column.

Is it known whether any of these problems for fixed $$n$$ is solvable in polynomial time?

## 2 Answers

Summary:

• A dimension restriction is necessary. Lemma 2 below observes that if arbitrary dimension is allowed the problem (even restricted to permutation matrices) is at least as hard as Graph Isomorphism (GI-hard).

• OP's problem (as restricted by OP to instances having constant dimension and matrices that either (i) all have at most one 1 per row, or (ii) all have at most one 1 per column) has a poly-time algorithm.

The upper bound we show is actually slightly stronger. Given an instance of OP's problem, let $$\mathcal A$$ denote the set of matrices labeling the edges. Let $$\mathcal A^*\vec c$$ be the set of vectors obtainable as products of the form $$M_1 M_2 \cdots M_k \vec c$$ for any $$k\ge 0$$, with each $$M_i\in \mathcal A$$.

Lemma 1. The problem has a poly($$|G|+|\mathcal A^*\vec c|)$$-time algorithm (with no restriction on the matrices or dimension).

For example, the following special cases are in P:

• If every edge has the identity matrix $$I$$, then $$\mathcal A^*\vec c=\{\vec c\}$$, so $$|\mathcal A^*\vec c| = 1$$. (This is the result shown in D.W.'s answer.)

• If the matrices are (0,1) matrices, either all having at most one 1 in each column, or all having at most one 1 in each row, then $$|\mathcal A^*\vec c|\le |\mathcal A^*| \le n^n$$. So when the dimension $$n$$ is constant (or even $$O(\log(|G|)/\log\log |G|)$$) the algorithm runs in time poly$$(|G|)$$.

Proof sketch for Lemma 1. Given the digraph $$G=(V,E)$$, the algorithm computes $$\mathcal A^*\vec c$$. It then constructs a bigger edge-weighted digraph $$G'$$ with a vertex $$q(v, y)$$ for every pair $$(v, y)$$ in $$V \times \mathcal A^*\vec c$$. For every pair of vertices $$q(v, y)$$ and $$q(v', y')$$, there is an edge from the former to the latter iff

• $$(v, v')$$ is in $$E$$, and
• $$y = Ay'$$, where $$A$$ is the matrix labeling edge $$(v, v')$$.

If there is such an edge, it is given (scalar) weight $$\vec b\, \vec y'$$.

The algorithm then uses any standard shortest-path algorithm to find the shortest path from any starting vertex in $$G'$$ to the ending vertex in $$G'$$, where the starting vertices in $$G'$$ are those of the form $$q(u, y)$$ where $$u$$ is the starting vertex for OP's instance and $$y\in\mathcal A^*\vec c$$, and the ending vertex is $$q(v, \vec c)$$, where $$v$$ is the ending vertex for OP's instance.

The idea for correctness is that any path $$q(v_1, y_1), q(v_2, y_2), \ldots, q(v_k, y_k=\vec c)$$ in $$G'$$ from a starting vertex to the ending vertex represents the path $$(v_1, v_2, \ldots, v_k)$$ in $$G$$ from the starting vertex $$v_1$$ to the ending vertex $$v_k$$ in $$G$$. The conditions for edges in $$G'$$ guarantee that after this path traverses an edge $$(v_i, v_{i+1})$$ labeled with some matrix $$A$$ and vector $$\vec b$$, it is the case that the product $$M_{i+1} M_{i+2} \cdots M_k \vec c$$ along the remainder of the path will equal $$y_{i+1}$$. Hence, (in $$G)$$ the vector $$\vec b$$ of the edge $$(v_i, v_{i+1})$$ will contribute $$\vec b y_{i+1}$$ to the final cost, just as the corresponding edge (with weight $$\vec b y_{i+1}$$) on the path in $$G'$$ contributes to that path's final cost. Thus, the cost of the path in $$G'$$ will equal the cost of the corresponding path in $$G$$. $$~~~~\Box$$

Lemma 2. The problem (restricted to permutation matrices, but not constant dimension) is GI-hard.

Proof sketch for Lemma 2. The proof is by a reduction from Graph Isomorphism (GI). Given a GI instance $$(G_1, G_2)$$ of two undirected graphs, each (WLOG) with vertex set $$[n]$$, and with the same number of edges, the reduction outputs the instance of OP's problem defined as follows. (Note that here $$n$$ is the number of vertices in $$G_1$$ and $$G_2$$, not the dimension of an instance of OP's problem. The instance of OP's problem produced by the reduction will have dimension $$n\choose 2$$.)

Let $$U$$ denote the set of possible edges (i.e. unordered pairs of distinct vertices) from vertex set $$[n]$$. So $$|U| = {n\choose 2}$$ and, for example, the edge sets of $$G_1$$ and $$G_2$$ are subsets of $$U$$.

Let $$\{0,1\}^U$$ represent the indicator vectors for such subsets of $$U$$ in the standard way: a given $$x\in\{0,1\}^U$$ is identified with the edge set $$E$$ such that $$x_e = 1$$ iff $$e\in E$$.

For $$(u,v)\in U$$, let $$\pi(u, v)$$ denote the permutation induced on $$U$$ by exchanging the (names of the) vertices $$u$$ and $$v$$. That is, for every vertex $$w\not\in \{u, v\}$$, $$\pi(u,v)$$ swaps $$(u, w) \in U$$ with $$(v, w)\in U$$, leaving all other possible edges unchanged.

Let $$\Pi = \{\pi(u, v) : (u,v)\in U\}$$ denote the set of such permutations. Note $$|\Pi| = {n\choose 2}$$.

Now $$G_1=([n], E_1)$$ is isomorphic to $$G_2=([n], E_2)$$ if and only if there is a sequence of permutations in $$\Pi$$ such that applying the sequence of permutations to (the indicator vector $$x(E_1) \in \{0,1\}^U$$ of) $$E_1$$ yields (the indicator vector $$x(E_2) \in \{0,1\}^U$$ of) $$E_2$$.

Furthermore, there is a constant $$c>0$$ such that there is such a sequence if and only if there is such a sequence of length at most $$cn \log n$$.

The reduction outputs a multi-digraph $$D$$ with vertex set $$[N]$$ where $$N=\lceil c n\log n\rceil$$ and, for each $$i\in [N-1]$$, there are $$|\Pi|$$ multi-edges from $$i$$ to $$i+1$$, with each such edge being labeled with the matrix for a (distinct) permutation in $$\Pi$$.

Thus, the paths from $$1$$ to $$N$$ in $$D$$ correspond to the sequences of permutations in $$\Pi$$. The start vertex $$1$$ is labeled with (the starting) vector $$x(E_1)$$. The target vertex $$N$$ is labeled with the vector $$\vec c = -x(E_2)$$. Then, the cost of taking a path from $$1$$ to $$N$$ is (at most) $$-|E_1| = -|E_2|$$ iff the sequence of permutations induced by the path, applied to $$x(E_1)$$, yields $$x(E_2)$$. Thus, this cost is achievable if and only if $$G_1$$ and $$G_2$$ are isomorphic.

(Note that if desired the multi-digraph $$D$$ can be converted into an equivalent digraph using the standard idea of replacing each edge $$(i, j)$$ by a path $$(i, v, j)$$, where $$v$$ is a new artificial vertex, unique to the edge.) $$~~~\Box$$

• why do you multiply $M_{i+1}\ldots M_k c$ should it be $c M_k \ldots M_{i+1}$? Commented Feb 28 at 18:08
• I see, yes, as you've defined $c$ that is probably right. I was assuming the final cost would be calculated as $x c$ (as a matrix product, considering the two vectors as matrices, so equivalent to the the dot product), but I've probably got the dimensions of $c$ backwards. Commented Feb 28 at 19:44
• p.s. If I were going to implement this I would not construct the graph $G'$ explicitly, rather I would use an implicit definition (constructing vertices as encountered) and I would look for the path backwards in $G'$ (using the single target vertex in $G'$ as the source). I would expect that to be more efficient in practice. Commented Feb 28 at 19:47

The usual rule is to ask only one question per post. I'll answer the first, about $$A=\text{Id}$$. For that special case, the shortest path problem is easy to solve in polynomial time. I'll describe how to compute the (length of the) shortest path from a source node $$s$$ to a destination node $$t$$ in polynomial time. You can then repeat this for all pairs $$s,t$$ if you wish.

Let the affine function at the target $$t$$ be $$o= \vec{c} \vec{x} + d$$. Note that $$d$$ is irrelevant (it does not affect which path is shortest) so WLOG we can take $$d=0$$. For any $$\vec{x}$$, define $$y=\vec{c} \vec{x}$$. Then an update $$\vec{x} := A \vec{x} + \vec{b}$$ simplifies to $$\vec{x} := \vec{x} + \vec{b}$$ if $$A=\text{Id}$$, and the effect on $$y$$ becomes $$y := y + \vec{c} \vec{b}$$.

It follows that it suffices to track the effect on $$y$$. Thus, we have a standard shortest-path problem. We replace each edge with update-operation $$\vec{x} := \vec{x} + \vec{b}$$ with an edge of length $$\vec{c} \vec{b}$$. Then, we look for the shortest path in the resulting transformed graph. The shortest path in the transformed graph will also be the shortest path in the original graph. Computing the shortest path in the transformed graph runs in polynomial time, so we obtain a polynomial-time solution to your problem when $$A=\text{Id}$$.