Eigenvalue bounds, spectral partitioning, and metrical deformations via flows

Abstract

We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every n-vertex graph of genus g and maximum degree d satisfies). This recovers the O( ), but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that λ 2 (G) = O( dh 6 log h n ) whenever G is K h -minor free. This shows, in particular, that spectral partitioning can be used to recover O( √ n)-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on λ 2 for graphs which exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find small separators in a general class of geometric graphs. Moreover, while the standard "sweep" algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the result of Alon, Seymour, and Thomas that every excluded-minor family of graphs has O( √ n)-node balanced separators.

On the Fiedler value of large planar graphs

“…Consider the class of K h -minor-free graphs on n vertices. Biswal et al [4] proved that for bounded-degree K h -minor-free graphs,…”

A PTAS for subset TSP in minor-free graphs

“…(The subdivision step does not introduce bigger clique minors.) Finally, techniques developed for solving problems in H-minor-free graphs are often very different from techniques for the same problems in planar and bounded genus graphs -representative example problems are padded decompositions with strong diameter [2], eigenvalue bounds [8] and light spanners [14], and that the techniques for H-minor-free graphs often find applications in different contexts. This holds for our technique as well (see Section 1.3).…”

Spectral concentration and greedy k-clustering

“…For example, in their seminal work Spielman and Teng gave the first rigorous analysis of the performance of spectral clustering methods which use the second eigenvector of the matrix D − A on bounded degree planar graphs and finite element meshes [28]. This result was further generalized to graphs with bounded degree and bounded genus by Kelner [15] and excluded-minor graphs by Biswal, Lee and Rao [8]. Our result holds for arbitrary bounded degree graphs that admit a good quality k-partition and demonstrates the effectiveness of the spectral clustering algorithms that use only the first k eigenvectors (cf.…”

“…For example, it is generally not clear for which classes of graphs spectral clustering works well, or what the structure of the subgraph induced by vertices that correspond to embedded points from the same cluster is. Although the case for k = 2 (two clusters) is well understood, the case of general k is not yet settled and a growing body of work seeks to address the practical success of spectral clustering methods [7,8,15,20,27,33].…”