Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus

TL;DR: This paper addresses two longstanding questions about finding good separators in graphs of bounded genus and degree by showing that a simple spectral algorithm finds separators of cut ratio O(g/n) and vertex bisectors of size O(√gn) in these graphs, both of which are optimal.

Abstract: In this paper, we address two longstanding questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [12] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case.We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O(√g/n) and vertex bisectors of size O(√gn) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng. While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces.