Testing Small Set Expansion in General Graphs

Abstract

We consider the problem of testing small set expansion for general graphs. A graph G is a (k,\phi)-expander if every subset of volume at most k has conductance at least \phi. Small set expansion has recently received significant attention due to its close connection to the unique games conjecture, the local graph partitioning algorithms and locally testable codes.

We give testers with two-sided error and one-sided error in the \adjacency list model that allows degree and neighbor queries to the oracle of the input graph. The testers take as input an n-vertex graph G, a volume bound k, an expansion bound \phi and a distance parameter \varepsilon>0. For the two-sided error tester, with probability at least 2/3, it accepts the graph if it is a (k,\phi)-expander and rejects the graph if it is \varepsilon-far from any (k^*,\phi^*)-expander, where k^*=\Theta(k\varepsilon) and \phi^*=\Theta(\frac{\phi^4}{\min\{\log(4m/k),\log n\}\cdot(\ln k)}). The query complexity and running time of the tester are \widetilde{O}(\sqrt{m}\phi^{-4}\varepsilon^{-2}), where m is the number of edges of the graph. For the one-sided error tester, it accepts every (k,\phi)-expander, and with probability at least 2/3, rejects every graph that is \varepsilon-far from (k^*,\phi^*)-expander, where k^*=O(k^{1-\xi}) and \phi^*=O(\xi\phi^2) for any 0<\xi<1. The query complexity and running time of this tester are \widetilde{O}(\sqrt{\frac{n}{\varepsilon^3}}+\frac{k}{\varepsilon \phi^4}).

We also give a two-sided error tester in the \textit{rotation map} model that allows \textit{(neighbor, index)} queries and degree queries. This tester has asymptotically almost the same query complexity and running time as the two-sided error tester in the adjacency list model, but has a better performance: it can distinguish any $(k,\phi)$-expander from graphs that are $\varepsilon$-far from $(k^*,\phi^*)$-expanders, where $k^*=\Theta(k\varepsilon)$ and $\phi^*=\Theta(\frac{\phi^2}{\min\{\log(4m/k),\log n\}\cdot(\ln k)})$.

In our analysis, we introduce a new graph product called \textit{non-uniform replacement product} that transforms a general graph into a bounded degree graph, and approximately preserves the expansion profile as well as the corresponding spectral property.