[Paper Review] Node Expansions and Cuts in Gromov-hyperbolic Graphs.

Bhaskar DasGupta, Marek Karpiński+2 more|arXiv (Cornell University)|Oct 29, 2015

Geometric and Algebraic Topology23 references3 citations

TL;DR

This paper establishes constructive, polynomial-time computable bounds on node expansions and cut-sizes in Gromov-hyperbolic graphs, proving that small cuts and low-expansion structures exist when source and target nodes are logarithmically separated. It enables efficient algorithms for hitting sets of size-constrained cuts and solves a small-set expansion problem originally posed by Arora, Barak, and Steurer in polynomial time.

ABSTRACT

Gromov-hyperbolic graphs (or, hyperbolic graphs for short) are a non-trivial interesting classes of non-expander graphs. Originally conceived by Gromov in 1987 in a different context while studying fundamental groups of a Riemann surface, the hyperbolicity measure for graphs has recently been a quite popular measure in the network science community in quantifying curvature and closeness to a tree topology for a given network, and many real-world networks have been empirically observed to be hyperbolic. In this paper, we provide constructive non-trivial bounds on node expansions and cut-sizes for hyperbolic graphs, and show that witnesses for such non-expansion or cut-size can in fact be computed efficiently in polynomial time. We also provide some algorithmic consequences of these bounds and their related proof techniques for a few problems related to cuts and paths for hyperbolic graphs, such as the existence of a large family of s-t cuts with small number of cut-edges when s and t are at least logarithmically far apart, efficient approximation of hitting sets of size-constrained cuts, and a polynomial-time solution for a type of small-set expansion problem originally proposed by Arora, Barak and Steurer.

Motivation & Objective

• To establish non-trivial, constructive bounds on node expansions and cut-sizes in Gromov-hyperbolic graphs.
• To demonstrate that witnesses for low expansion or small cuts can be computed efficiently in polynomial time.
• To derive algorithmic consequences for problems involving s-t cuts, hitting sets, and small-set expansion in hyperbolic graphs.
• To resolve a small-set expansion problem originally proposed by Arora, Barak, and Steurer for hyperbolic graphs.
• To provide theoretical foundations for efficient computation in networks with tree-like curvature, such as real-world networks.

Proposed method

• Leveraging the hyperbolicity measure of graphs to characterize structural properties related to tree-likeness and curvature.
• Using the logarithmic separation between source and target nodes (s and t) as a key parameter to derive bounds on cut-sizes and expansions.
• Designing a polynomial-time algorithm to compute witnesses for small cuts and low-expansion subgraphs.
• Applying proof techniques rooted in hyperbolic geometry and graph-theoretic decomposition to bound expansion and cut-size parameters.
• Reducing the small-set expansion problem to a structure-aware decomposition that exploits hyperbolicity for efficient solution.
• Formulating and solving a size-constrained hitting set problem for cuts using the derived structural bounds.

Experimental results

Research questions

• RQ1 Can non-trivial, constructive bounds on node expansion and cut-size be established for Gromov-hyperbolic graphs?
• RQ2 Can witnesses for small cuts or low-expansion subgraphs be computed in polynomial time in hyperbolic graphs?
• RQ3 What algorithmic consequences arise from the structural bounds on cuts and expansions in hyperbolic graphs?
• RQ4 Does the existence of logarithmically separated s-t pairs guarantee a large family of small cuts?
• RQ5 Can the small-set expansion problem proposed by Arora, Barak, and Steurer be solved in polynomial time for hyperbolic graphs?

Key findings

• Non-trivial, constructive bounds on node expansion and cut-sizes are established for Gromov-hyperbolic graphs.
• Witnesses for small cuts and low-expansion subgraphs can be computed in polynomial time.
• When s and t are at least logarithmically far apart, a large family of s-t cuts with few cut-edges exists.
• An efficient approximation algorithm is provided for hitting sets of size-constrained cuts in hyperbolic graphs.
• A polynomial-time solution is achieved for the small-set expansion problem originally posed by Arora, Barak, and Steurer in the context of hyperbolic graphs.
• The structural properties of hyperbolic graphs enable efficient computation for several fundamental graph problems related to cuts and paths.