Distance Approximation in Bounded-Degree and General Sparse Graphs

We address the problem of approximating the distance of sparse graphs from having some predefined property P. The distance of G from having P is εP(G) if the minimum number of edge modifications needed in order to enforce P on G is εP(G) · m̄, where m̄ is an upper bound on the number of edges in the graph. When dealing with bounded-degree graphs of maximal degree d we take m̄ = dn, where n is the number of vertices in the graph. For general sparse graphs of average degree d̄, we take m̄ = d̄n. We say that A is an (α, δ)-distance approximation algorithm if with high probability its estimate ε̂P satisfies εP (G) − δ ≤ ε̂P ≤ α · εP (G) + δ . We say that it is a δ-distance approximation algorithm if |ε̂P − εP(G)| ≤ δ . We present the following distance approximation algorithms: • Subgraph-Freeness. For any fixed subgraph H, we give an (mH , δ)-distance approximation algorithm for H-freeness in bounded-degree graphs, where mH is the number of edges in H. Its query and time complexity is dO(øH ·m 2 H ·log(dH/δ)), where øH is the diameter of H and dH is the number of H-subgraphs that an edge in G can belong to. In particular, for triangle-freeness this implies a (3, δ)-distance approximation algorithm whose query and time complexity is dO(log(d/δ)). • k-Edge-Connectivity. A δ-distance approximation algorithm for k-edgeconnectivity in general sparse graphs whose query and time complexity is O ( ( k δd̄ )6 log( k δd̄ ) ) . • Eulerian. A δ-distance approximation algorithm for being Eulerian in general sparse graphs whose query and time complexity is O(δ−4d̄−4). • Cycle-Freeness. A δ-distance approximation algorithm for cycle-freeness in bounded-degree graphs whose query and time complexity is O(δ−3d−2). Our subgraph-freeness distance approximation algorithm can be executed distributively, achieving a multiplicative approximation ratio of (mH + δ) to the minimum subgraph cover of a graph, that is, to the minimum set of edges whose removal results in a subgraph free graph. The number of rounds needed is O(mH log(dH/δ)). In [KMW06], a more general algorithm was presented, but for this task it achieves the same approximation ratio using O((mH/δ) 3 · log dH) rounds. Further modifications