Testing Small Set Expansion in General Graphs

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 with smaller gap between $\phi^*$ and $\phi$ in the rotation map model that allows (neighbor, index) queries and degree queries.Comment: 23 pages; STACS 201

Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments

A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdos on packing Hamilton cycles in random tournaments. As corollaries to the main result, we also obtain several results on packing Hamilton cycles in undirected graphs, giving e.g. the best known result on a conjecture of Nash-Williams. We also apply our result to solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust decomposition lemma' for application in subsequent paper