For an undirected n-vertex planar graph G with non-negative edge-weights, we consider the following type of query: given two vertices s and t in G, what is the weight of a min st-cut in G? We show how to answer such queries in constant time with O(n log⁵ n) preprocessing time and O(n log n) space. We use a Gomory-Hu tree to represent all the pairwise min cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Since all-pairs min cut and the minimum cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum cycle basis in O(n log⁵n) time and O(n log n) space and an explicit representation with additional O(C) time and space where C is the size of the basis. These results require that shortest paths be unique. We deterministically remove this assumption with an additional log²n factor in the running time.

To appear at FOCS 2010.

[ arXiv ]

It was shown in [Indyk-Sidiropoulos 07] that any orientable graph of genus g can be probabilistically embedded into a graph of genus g-1 with constant distortion. Removing handles one by one gives an embedding into a distribution over planar graphs with distortion 2^O(g). By removing all g handles at once, we present a probabilistic embedding with distortion O(g²) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of [Erickson-HarPeled 04] has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma from [Lee-Sidiropoulos 08].

Proceedings of the Annual Symposium on Computational Geometry (SoCG), Aarhus, Denmark, 2009.

[ doi ] [ journal version available ]

We present the first polynomial-time approximation schemes (PTASes) for the following subset-connectivity problems in edge-weighted graphs of bounded genus: Steiner tree, low-connectivity survivable-network design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and non-orientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu (2007 and 2006) from planar graphs to bounded-genus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to bounded-genus graphs.

Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS), Freiburg, Germany, 2009.

[ publishers link ] [ pdf ]

We give a randomized O(n² log n)-time approximation scheme for the Steiner forest problem in the Euclidean plane. For every fixed ε > 0 and given any n pairs of terminals in the plane, our scheme finds a (1 + ε)- approximation to the minimum-length forest that connects every pair of terminals.

Symposium of Foundations of Computer Science (FOCS), Philadelphia, Pennsylvania, 2008.

[ doi ] [ pdf: fixes a bug in the FOCS proceedings ]

Consider the following problem: given a graph with edge weights and a subset Q of vertices, find a minimum-weight subgraph in which there are two edge-disjoint paths connecting every pair of vertices in Q. The problem is a failure-resilient analog of the Steiner tree problem, and arises in telecommunications applications. A more general formulation, also employed in telecommunications optimization, assigns a number (or requirement) rv ∈ {0, 1, 2} to each vertex v in the graph; for each pair u, v of vertices, the solution network is required to contain min{ru, rv} edge-disjoint u-to-v paths.
We address the problem in planar graphs, considering a popular relaxation in which the solution is allowed to use multiple copies of the input-graph edges (paying separately for each copy). The problem is SNP-hard in general graphs and NP-hard in planar graphs. We give the first polynomial-time approximation scheme in planar graphs. The running time is O(n log n).

Under the additional restriction that the requirements are in {0, 2}
for vertices on the boundary of a single face of a planar graph, we give a
linear-time algorithm to find the optimal solution.

Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP) Reykjavik, Iceland, 2008.

[ doi ] [ pdf ]

We give an algorithm that, for any ε > 0, any undirected planar graph G, and any set S of nodes of G, computes a (1+ε)-optimal Steiner tree in G that spans the nodes of S. The algorithm takes time O(2^poly(1/ε)n log n).

Proceedings of the Workshop on Algorithms and Data Structures (WADS), Halifax, Nova Scotia, 2007.

[ doi ] [ pdf ]

We give an O(n log n) approximation scheme for Steiner tree in planar graphs.

Proceedings of the Symposium on Discrete Algorithms (SODA), New Orleans, Louisiana, 2007.

[ publishers link ] [ pdf ]

We give the first correct O(n log n) algorithm for finding a maximum st-flow in a directed planar graph. After a preprocessing step that consists in finding single-source shortest-path distances in the dual, the algorithm consists of repeatedly saturating the leftmost residuals-to-t path.

Proceedings of the Symposium of Discrete Algorithms (SODA), Miami, Florida, 2006.

[ doi ] [ pdf for full version ]