This is a solution to the 2024 PACE challenge, by @mjdv and @RagnarGrootKoerkamp.

In particular, we solve the one sided crossing minimization problem:

• Given is a bipartite graph $(A,B)$, that is drawn in the plane with straight lines, with vertices in $A$ at $(0,i)$ and vertices in $B$ at $(1,j)$.
• The order of $A$ is fixed. The goal is to find an order of $B$ that minimizes the number of crossings.

This is an exact solver.

• On the exact track, it solves around 75% of cases (given half an hour per case).
• On the parameterized tack, is solves all 100 cases on optil.io in around 3 seconds total.

Our method uses branch and bound.

• First, we find an initial solution using some simple local search (swapping adjacent elements and optimally inserting elements elsewhere).
• The branch and bound works from left to right. At each step, we try (a subset of) all remaining nodes as the next node in the order.
  • We track the score of the prefix $P$ fixed so far, and the number of crossings between $P$ and the remaining tail $T$.
  • We also use the 'trivial' lower bound for the number of crossings in the tail.
  • We prune branches where the score is higher than the best score seen so far.

Some of the more interesting optimizations:

• Graph simplification: We merge identical edges, drop empty nodes from both components, and sort the nodes of $B$ by their position in the initial solution, for more efficient memory accesses. We also split 'disjoint' components.

• Caching: For each B&B state we cache (among other things) the best lower bound found so far, or the score of the best solution if known.

  • If a tail has not been seen yet, use the bound for the longest suffix of the tail that has been seen.

• Optimal insert: In the B&B, at each step we don't just append the next node, but insert it in the optimal position in the prefix.

• Dominating pairs: For some pairs of nodes $(u,v)$ in $B$, we can prove that $u$ must always come before $v$. We pre-compute these pairs, and never try $v$ as the next node when $u$ hasn't been chosen yet.

• Practical dominating pairs: A stronger variant of the above that not only considers $u$ and $v$ 'in isolation', but in context of the other nodes in $B$. Sometimes $v$ could 'in theory' come before $u$, but we can prove that in practice this can never be optimal (or at least, not strictly better than having $u$ before $v$).

  (Note: this is disabled for the parameterized track as it takes a lot of time to precompute for little gain.)

• Local dominating pairs: Same as practical dominating pairs above, but recomputed in every new B&B state.

• On-the-fly gluing: If there is a vertex $u$ in the tail such that all other $v$ in the tail are better (not worse) after $u$, then just glue $u$ onto the prefix.

There are some more optimizations that are disabled by default. See notes.md.

We do a number of algorithmic optimizations to implement everything efficiently. Mostly we ensure that the tail always remains sorted, so that loops over it can often be reduced to only a small subset of it.

This solver is written in Rust.

• First, install Rust, e.g. using rustup.
• Then build the binary for the 'submit' profile.
  • For the parameterized track (the default):

    RUSTFLAGS="-Ctarget-cpu=native" cargo build --profile submit
    
  • For the exact track:

    RUSTFLAGS="-Ctarget-cpu=native" cargo build --profile submit --features exact
    
    This turns on some optimizations that require more heavy precomputation.
• Find the binary at target/submit/ocmu64.
• Run as ocmu64 < input > output.