An SDP-based algorithm for linear-sized spectral sparsification

ABSTRACT

For any undirected and weighted graph G=(V,E,w) with n vertices and m edges, we call a sparse subgraph H of G, with proper reweighting of the edges, a (1+ε)-spectral sparsifier if

(1-ε)x^TL_Gx≤x^T L_H x≤(1+ε)x^TL_Gx

holds for any xΕℝⁿ, where L_G and L_H are the respective Laplacian matrices of G and H. Noticing that Ω(m) time is needed for any algorithm to construct a spectral sparsifier and a spectral sparsifier of G requires Ω(n) edges, a natural question is to investigate, for any constant ε, if a (1+ε)-spectral sparsifier of G with O(n) edges can be constructed in Ο(m) time, where the Ο notation suppresses polylogarithmic factors. All previous constructions on spectral sparsification require either super-linear number of edges or m^1+Ω(1) time.

In this work we answer this question affirmatively by presenting an algorithm that, for any undirected graph G and ε>0, outputs a (1+ε)-spectral sparsifier of G with O(n/ε²) edges in Ο(m/ε^O(1)) time. Our algorithm is based on three novel techniques: (1) a new potential function which is much easier to compute yet has similar guarantees as the potential functions used in previous references; (2) an efficient reduction from a two-sided spectral sparsifier to a one-sided spectral sparsifier; (3) constructing a one-sided spectral sparsifier by a semi-definite program.

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Index Terms

• Mathematics of computing

  • Probability and statistics

    • Probabilistic algorithms

• Theory of computation

  • Design and analysis of algorithms

    • Graph algorithms analysis

      • Sparsification and spanners