A sublinear bipartiteness tester for bounded degree graphs

We present a sublinear-time algorithm for testing whether a bounded degree grapli is bipartite or far from being bipartite. Graphs are represented by incidence lists of bounded length d, and the testing algorithm can perform queries of the form: "who is the itIi neighbor of vertex v". The tester should determine with high probability whether the graph is bipartite or ∈-far from bipartite for any given distance parameter ∈. Distance between graphs is defined to be the fraction of entries on which the graphs differ in their incidence-lists representation. Our testing algorithm has query complexity and running time poly((logN)/∈) · √N where N is the number of graph vertices. It was shown before that Ω(√N) queries are necessary (for constant ∈), and hence the performance of our algorithm is tight (in its dependence on N), up to polylogarithmic factors. In our analysis we use techniques that were previously applied to prove fast convergence of random walks on expander graphs. Here we use the contrapositive statement by which slow convergence implies small cuts in the graph, and further show that these cuts have certain additional properties. This implication is applied in showing that for any graph, the graph vertices can be divided into disjoint subsets such that: (1) the total number of edges between the different subsets is small; and (2) each subset itself exhibits a certain mixing property that is useful in our analysis.