\(
\def\<#1>{\left<#1\right>}
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% an undirected version of \rightarrow:
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\newcommand*{\su}{\mathbin{\ensuremath{\oplus}}}
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\)
Is Bellman-Ford-Moore single source shortest paths (min,+) circuit optimal?
The single-source shortest paths problem $\SSSP$ is,
given an assignment of nonnegative weights $x_{i,j}$ to the edges $\{i,j\}$ of the complete graph $K_n$ on $\{1,\ldots,n\}$, to simultaneously compute
the minimum weights of paths from a single fixed source node $s\in\{1,\ldots,n\}$ to all remaining $n-1$ nodes.
In the all-pairs shortest paths problem $\APSP$,
the goal is to simultaneously compute, for all pairs $\{i,j\}$ of
nodes, the minimum weight of a path between $i$ and $j$.
The $(\min,+)$ circuit (with $n-1$ output gates) resulting from the Bellman-Ford-Moore pure DP algorithm solves $\SSSP$ using only $O(n^3)$ gates (Example 1.7 in the book).
So, by taking $n$ copies of the $(\min,+)$ circuit of
Bellman-Ford-Moore, the $\APSP$ problem can be solved by a $(\min,+)$ circuit using $O(n^4)$ gates. But the $(\min,+)$ circuit ) resulting from the Floyd-Warshall-Roy pure DP algorithm solves the $\APSP$ problem using only $O(n^3)$ gates (Example 1.8 in the book).
On the other hand, Corollary 2.3 in the book shows that the $\APSP$ problem requires $(\min,+)$ circuits of size $\Omega(n^3)$. Thus, the Floyd-Warshall-Roy pure DP algorithm for $\APSP$ is optimal in the class of all pure DP algorithms: every
$(\min,+)$ circuit solving this problem must have $\Omega(n^3)$ gates.
Note that $\APSP$ actually solves $n$ copies of $\SSSP$
for all possible source nodes $s\in\{1,\ldots,n\}$. And this property
($n$ distinct source nodes) was essential in the proof (in Corollary 2.3) of the lower bound
$\Omega(n^3)$ for the tropical $(\min,+)$ circuit complexity of the $\APSP$ problem.
But what about the $\SSSP$ problem itself: is the Bellman-Ford-Moore pure DP algorithm optimal for this problem?
Problem:
Does the $\SSSP$ problem requires $(\min,+)$ circuits of size $\Omega(n^3)$?
We have proved the $\Omega(n^3)$ lower bound for the $\APSP$ problem by reducing it to the
tropical matrix product problem $\MP{n}$: given a circuit for the $\APSP$ problem, we can obtain a circuit of the same size solving the $\MP{n/3}$ problem.
By Schnorr's theorem (Theorem 2.1 i the book), $\MP{n}$ requires(1) $(\min,+)$ circuits of size $\Omega(n^3)$.
(As mentioned in the book, this lower bound for $\MP{n}$ was earlier proved by Kerr [1, Theorem 6.1] using different
arguments.)
However, such a simple reduction does not seem to work for the $\SSSP$ problem (just because here we have only one source node, instead of $n$ as in $\APSP$).
Whether the Bellman-Ford-Moore $(\min,+)$ branching program
is optimal for the $\SSSP$ problem also remains open; see Remark 1 in
this comment.
Footnotes:
(1)
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