Finding a Shortest $M$-link Path in a Monge Directed Acyclic Graph

A Monge directed acyclic graph (DAG) $G$ on the nodes $1,2,\cdots,N$ has edges $\left( i,j\right) $ for $1\leq i<j\leq N$ carrying submodular edge-lengths. Finding a shortest $M$-link path from $1$ to $N$ in $G$ for any given $1<M<N-1$ has many applications. In this paper, we give a contract-and-conquer algorithm for this problem which runs in $O\left( \sqrt{NM\left( N-M\right) \log\left( N-M\right) }\right) $ time and $O\left( N\right) $ space. It is the first $o\left( NM\right) $-time algorithm with linear space complexity, and its time complexity decreases with $M$ when $M\geq N/2$. In contrast, all previous strongly polynomial algorithms have running time growing with $M$. For both $O\left( poly\left( \log N\right) \right) $ and $N-O\left( poly\left( \log N\right) \right) $ regimes of $M$, our algorithm has running time $O\left( N\cdot poly\left( \log N\right) \right) $, which partially answers an open question rased in \cite{AST94} affirmatively.