University of Potsdam's Gasse Cuts Graph Coloring Run Time to Θ(n log n)
A gray-box mutation operator beats exponential black-box search by orders of magnitude.
Johanna Gasse's bachelor's thesis, submitted to the Digital Engineering Faculty of the University of Potsdam, tackles the NP-hard vertex coloring problem through the lens of local search. Focused on bipartite graphs, the research first identifies structural properties that determine whether local search landscapes contain only global optima or also non-global local optima. It shows that on general bipartite graphs, local search can return arbitrarily bad results, revealing fundamental limitations of black-box approaches.
To overcome these, Gasse introduces a gray-box mutation operator that uses additional information—specifically, it removes less frequent colors with higher probability. On complete bipartite graphs, this operator finds an optimal coloring in expected Θ(n log n) time, a stark contrast to the exponential run time of the standard black-box Random Local Search. This result demonstrates how domain-specific knowledge in mutation operators can dramatically improve optimization performance, with potential applications in scheduling, register allocation, and combinatorial optimization.
- Gray-box mutation operator removes less frequent colors with higher probability, achieving Θ(n log n) expected run time on complete bipartite graphs.
- Black-box Random Local Search requires exponential time for the same problem, highlighting the power of informed mutation.
- Paper identifies bipartite graph structures that ensure global optima vs. those that permit arbitrarily bad local optima.
Why It Matters
Proves that simple gray-box heuristics can transform exponential-time optimization into polynomial time on key graph coloring problems.