Many polynomial-time solvable combinatorial optimization problems become NP-hard if an additional complicating constraint is added to restrict the set of feasible solutions. In this paper, we consider two such problems, namely maximumweight matching and maximum-weight matroid intersection with one additional budget constraint. We present the first polynomial-time approximation schemes for these problems. Similarly to other approaches for related problems, our schemes compute two solutions to the Lagrangian relaxation of the problem and patch them together to obtain a near-optimal solution. However, due to the richer combinatorial structure of the problems considered here, standard patching techniques do not apply. To circumvent this problem, we crucially exploit the adjacency relations on the solution polytope and, surprisingly, the solution to an old combinatorial puzzle.
In this article, we consider k -way vertex cut: the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of components. Our main contribution is the derivation of an efficient polynomial-time approximation scheme for the problem on planar graphs. Also, we show that k -way vertex cut is polynomially solvable on graphs of bounded treewidth and fixed-parameter tractable on planar graphs with the size of the separator as the parameter.
Abstract. We present new approximation schemes for various classical problems of finding the minimum-weight spanning subgraph in edge-weighted undirected planar graphs that are resistant to edge or vertex removal. We first give a PTAS for the problem of finding minimum-weight 2-edge-connected spanning subgraphs where duplicate edges are allowed. Then we present a new greedy spanner construction for edge-weighted planar graphs, which augments any connected subgraph A of a weighted planar graph G to a (1 + ε)-spanner of G with total weight bounded by weight(A)/ε. From this we derive quasi-polynomial time approximation schemes for the problems of finding the minimum-weight 2-edge-connected or biconnected spanning subgraph in planar graphs. We also design approximation schemes for the minimum-weight 1-2-connectivity problem, which is the variant of the survivable network design problem where vertices have non-uniform (1 or 2) connectivity constraints. Prior to our work, for all these problems no polynomial or quasi-polynomial time algorithms were known to achieve an approximation ratio better than 2.
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