# PACE 2024

This year’s challenge is about the one-sided crossing minimization problem (OCM). This problem involves arranging the nodes of a bipartite graph on two layers (typically horizontal), with one of the layers fixed, aiming to minimize the number of edge crossings. OCM is one of the basic building block used for drawing hierarchical graphs. It is NP-hard, even for trees, but admits good heuristics, can be constant-factor approximated and solved in FPT time. For an extended overview, see Chapter 13.5 of the Handbook of Graph Drawing.

### Challenge Description

The goal of this year’s challenge is to compute a 2-layered drawing of a bipartite graph, where one side is fixed, with a minimum number of crossings. More precisely:

Input: A bipartite graph $G=((A\dot\cup B),E)$, and a linear order of $A$.
Output: A linear order of $B$.
Measure: The number of edge crossings in a straight-line drawing of $G$ with $A$ and $B$ on two parallel lines, following their linear order.

See an example here with two different linear orders for $B$. ### Tracks

This year, we will have three tracks:

• one focusing on Exact algorithms for instances with “few” crossings
• one for Heuristic algorithms for instances that might require many crossings
• one for exact Parameterized algorithms for instances with small cutwidth.

#### The Exact Track

The task is to compute an optimal solution for each given graph, that is, a linear order of $B$ that minimizes the number of crossings. For each instance, the solver has to output a solution within a time limit of 30 minutes and a memory limit of 8 GB.

Instances in this track have solutions where the number of crossings is smaller than the number of vertices, but we do not give explicit bounds.

Submissions should be based on provably optimal algorithms, however, this is not a formal requirement. Submissions that output an incorrect solution or a solution that is known to be non-optimal will be disqualified. Besides dedicated algorithms, we also encourage submissions based on other paradigms such as SAT, MaxSAT, or ILPs.

#### The Heuristic Track

In this track, the solver shall compute a good solution quickly. The solver will be run on each instance for 5 minutes and receives the Unix signal SIGTERM afterwards. When receiving this signal, the process has to output a linear order of $B$ immediately to the standard output and terminate. If the program does not halt in a reasonable time after reserving the signal, it will be stopped via SIGKILL. In this case the instance is counted as time limit exceeded. Information on how to handle Unix signals in various programming languages can be found on the optil.io webpage. The memory limit for this track is 8 GB as well.

For this track solutions do not have to be optimal. However, solvers that produce incorrect solution will be disqualified.

#### Parameterized Track

This track has the same rules as the Exact Track. However, the instances here can require a large number of crossings, but they have small cutwidth: there is an ordering of the vertices of the graph, such that every cut obtained by partitioning the vertices into earlier and later subsets of the ordering is crossed by at most $k$ edges. Note that in this order the vertices of $A$ and $B$ might be interleaved. An ordering that achieves minimum cutwidth will be provided together with the graph.

### Benchmark Sets

There will be four benchmark set:

1. A tiny set for debugging that contains graphs together with their one-sided crossing number.
2. The exact set containing 200 instances divided into 100 public instances for development and 100 private instances used for evaluation.
3. The heuristic set containing 200 instances divided into 100 public instances for development and 100 private instances used for evaluation.
4. The parameterized set containing 200 instances divided into 100 public instances for development and 100 private instances used for evaluation.

### Timeline

• September 2023: Announcement of the Challenge
• November 2023: Definition of the input and output format.
• December 2023: Announcement of the ranking methods and additional information about the submission process.
• A repository for an autotester (like a JUnit test) and public test instances will be provided
• January 2024: Public instances and details about the benchmark set get published.
• March 2024: Submission opens with public leaderboard.
• May 2024: The public leaderboard gets frozen.