Approximations via Partitioning

We consider the approximation of weighted maximum subgraph problems by partitioning the input graph into easier subproblems. In particular, we obtain efficient approximations of the weighted independent set problem with performance ratios of O(n(log log n/ log n)) and (∆ + 2)/3, with the latter improving on a ∆/2 ratio of Hochbaum for ∆ ≥ 5. We also obtain a O(n/ log n) performance ratio for various maximization problems where a subset of a solution is also a solution. 1 Partitioning and hereditary induced subgraph problems A property of graphs is hereditary if whenever it holds for a graph it also holds for its induced subgraphs. For a hereditary property, the associated subgraph problem is that of finding a subgraph of maximum weight satisfying the property. We say that a problem is approximable within f(n) if there is a polynomial time algorithm that on graphs with n vertices returns a feasible solution within f(n) factor of optimal. Hereditary can be generalized to other discrete structures. A property is hereditary if whenever it holds for a subset X of the instance, it also holds for any subset of X. The main purpose of this note is to study the implications of the following lemma. We state it in the language of graphs, while it can also be applied to other hereditary problems. Proposition 1 Let Π be the problem of finding a maximum induced subgraph satisfying a hereditary property π. If we can partition the graph G into subgraphs G1, G2, . . . Gt and solve Π optimally on each Gi, then we can approximate Π on G within t. Our results We present several applications of this approach in Section 2. The first is to partition the input into small bits, each of which can be searched exhaustively for an optimal solution. This suffices to obtain a performance ratio of O(n/ log n) for various problems for which a subset of any solution is also a solution, including hereditary induced subgraph problems, and the problems Longest Common Subsequence and Maximum Satisfying Linear Subsystem (see [8] for statements of these problems and references). Here, n is the total number of items in the instance, as given in the measure. This also holds for weighted versions of these problems. We strengthen this approximation slightly for unweighted versions to show how to find a solution of size log2n/OPT n, where OPT is the size of the optimal solution. We then obtain a stronger performance ratio of O(n(log log n/ log n)2) for the Weighted Independent Set problem, and other hereditary problems with a forbidden clique or independent set. This is obtained by partitioning the graph into subgraphs that are either independent sets or unions of at most log n/ log log n cliques. We then consider graphs of maximum degree ∆. We present an efficient algorithmic proof of an obscure lemma of Lovász, and use it to approximate weighted hereditary induced subgraph