Constant time algorithms in sparse graph model

Abstract

We focus on constant-time algorithms for graph problems in bounded degree model. We introduce several techniques to design constant-time approximation algorithms for problems such as Vertex Cover, Maximum Matching, Maximum Weighted Matching, Maximum Independent Set and Set Cover. Some of our techniques can also be applied to design constant-time testers for minor-closed properties. In Chapter 1, we show how to construct a simple oracle that provides query access to a fixed Maximal Independent Set (MIS) of the input graph. More specifically, the oracle gives answers to queries of the form "Is v in the MIS?" for any vertex v in the graph. The oracle runs in constant-time, i.e., the running time for the oracle to answer a single query, is independent to the size of the input graph. Combining this oracle with a simple sampling scheme immediately implies an approximation algorithm for size of the minimum vertex cover. The second technique, called oracle hierarchy, transforms classical approximation algorithms into constant-time algorithms that approximate the size of the optimal solution. The technique is applicable to a certain subclass of algorithms that compute a solution in a constant number of phases. In the transformation, oracle hierarchy uses the MIS oracle to simulates each phase. The problems amenable to these techniques include Maximum Matching, Maximum Weight Matching, Set Cover, and Minimum Dominating Set. For example, for Maximum Matching, we give the first constant-time algorithm that for the class of graphs of degree bounded by d, computes the maximum matching size to within en, for any e > 0, where n is the number of vertices in the graph. The running time of the algorithm is independent of n, and only depends on d and e. In Chapter 2, we introduce a new tool called partitioning oracle which provides query access to a fixed partition of the input graph. In particular, the oracle gives answers to queries of the form "Which part in the fixed partition contains v?" for any vertex v in the graph. We develop methods for constructing a partitioning oracle for any class of bounded-degree graphs with an excluded minor. For any e > 0, our partitioning oracle provides query access to a fixed partition of the input constant-degree minor-free graph, in which every part has size 0(1/ 2 ), and the number of edges removed is at most en. We illustrate the power of this technique by using it to extend and simplify a number of previous approximation and testing results for sparse graphs, as well as to provide new results that were unachievable with existing techniques. For instance: " We give constant-time approximation algorithms for the size of the minimum vertex cover, the minimum dominating set, and the maximum independent set for any class of graphs with an excluded minor. * We show a simple proof that any minor-closed graph property is testable in constant time in the bounded degree model. Finally, in Chapter 3, we construct a more efficient partitioning oracle for graphs with constant treewidth. Although the partitioning oracle in Chapter 2 runs in time independent of the size of the input graph, it has to make 2POlY(1/E)) queries to the input graph to answer a query about the partition. Our new partitioning oracle improves this query complexity to poly(1/E) for graphs with constant treewidth. The new oracle can be used to test constant treewidth in poly(1/E) time in the bounded-degree model. Another application is a poly(1/E)-time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive en in bounded treewidth graphs.