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PACE Solver Description: Exact Solution of the One-sided Crossing Minimization Problem by the MPPEG Team

Michael Jünger, Paul J. Jünger, Petra Mutzel, Gerhard Reinelt

TL;DR

The paper addresses the one-sided crossing minimization problem for a bipartite graph $G=(T\dot\cup B,E)$ with a fixed top order $\pi_T$ and seeks a bottom order $\pi_B$ that minimizes crossings. It reduces OSCM to a linear ordering problem on $B$ with binary variables $x_{ij}$ for $i<j$ and objective $z=\sum_{i<j} a_{ij}x_{ij}$ plus a constant, enforcing feasibility via cycle constraints including longer Möbius-ladder inequalities, solved by a branch-and-cut with decomposition and variable fixing. The authors implement oscm, a practical solver, combining LP relaxations, hierarchical separation for cycles, problem decomposition via intervals $I_v$, partial order propagation, and a Kernighan–Lin–inspired initial solution. On PACE 2024, oscm achieves top results, placing first in the exact track (199 of 200 instances solved in about 5682 seconds) and third in the parameterized track (all 200 solved in about 25 seconds), illustrating the method's effectiveness for exact and parameterized OSCM.

Abstract

This is a short description of our solver OSCM submitted by our team MPPEG to the PACE 2024 challenge both for the exact track and the parameterized track, available at https://github.com/pauljngr/PACE2024 and https://doi.org/10.5281/zenodo.11546972.

PACE Solver Description: Exact Solution of the One-sided Crossing Minimization Problem by the MPPEG Team

TL;DR

The paper addresses the one-sided crossing minimization problem for a bipartite graph with a fixed top order and seeks a bottom order that minimizes crossings. It reduces OSCM to a linear ordering problem on with binary variables for and objective plus a constant, enforcing feasibility via cycle constraints including longer Möbius-ladder inequalities, solved by a branch-and-cut with decomposition and variable fixing. The authors implement oscm, a practical solver, combining LP relaxations, hierarchical separation for cycles, problem decomposition via intervals , partial order propagation, and a Kernighan–Lin–inspired initial solution. On PACE 2024, oscm achieves top results, placing first in the exact track (199 of 200 instances solved in about 5682 seconds) and third in the parameterized track (all 200 solved in about 25 seconds), illustrating the method's effectiveness for exact and parameterized OSCM.

Abstract

This is a short description of our solver OSCM submitted by our team MPPEG to the PACE 2024 challenge both for the exact track and the parameterized track, available at https://github.com/pauljngr/PACE2024 and https://doi.org/10.5281/zenodo.11546972.

Paper Structure

This paper contains 3 sections, 3 theorems, 4 equations.

Table of Contents

  1. Method
  2. Algorithm and Implementation
  3. Performance

Key Result

Lemma 1

For each node $v\in B$, we define the open interval $I_v=]l_v,r_v[$, where $l_v$ is the position of the leftmost and $r_v$ the position of the rightmost neighbor of $v$ in $\pi_T$. The union of the intervals $I_v$ induces a partition $B_1,B_2,\ldots,B_k$ of $B$ such that every $I_{B_i}=\bigcup_{v\in

Theorems & Definitions (3)

  • Lemma 1: Decomposition
  • Lemma 2: Variable fixing DBLP:journals/algorithmica/DujmovicW04
  • Lemma 3: Arbitrary ordering