Glencora Borradaile » Other

Approximation algorithms for constrained knapsack problems

Glencora — Tue, 06 Oct 2009 03:36:12 +0000

Glencora Borradaile, Brent Heeringa, Gordon Wilfong

We study constrained versions of the knapsack problem in which dependencies between items are given by a graph. In one version, an item can be selected only if one of its neighbours is also selected. In the other version, an item can be selected only when all its neighbours are also selected. These problems generalize and unify several problems including the prize collecting and budgeted maximum coverage problems. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.

[ arXiv ]

Planarity testing

Glencora — Wed, 04 Jun 2008 23:31:27 +0000

Glencora Borradaile

Encyclopedia of Algorithms, invited entry, 2008.

[ doi ] [ pdf ]

Exploiting Planarity for Network Flow and Connectivity Problems

Glencora — Tue, 04 Dec 2007 23:25:31 +0000

Glencora Borradaile

By restricting the input to a problem, it often becomes possible to design more accurate or more eﬃcient algorithms to solve that problem. In this thesis we restrict our attention to planar graphs and achieve both these goals. Planar graphs exhibit many structural and combinatorial properties that enable the design of good algorithms. These properties include: corresponding to every planar graph there is a dual planar graph; the dual of the complement of the edges of a spanning tree form a spanning tree of the dual graph; a set of edges is a cycle if and only if the dual edges form a cut; cycles can be said to enclose edges, faces and vertices in the planar embedding; paths can be compared as to their relative embedding.

We capitalize on these properties to design (a) faster algorithms for polynomial-time-solvable network ﬂow problems and (b) algorithms with better approximation guarantees for NP-hard connectivity problems. We give a conceptually simple O(n log n)-time algorithm for ﬁnding the maximum st-ﬂow in a directed planar graph, proving a theorem that was incorrectly claimed over a decade ago. We also show how to compute the minimum cut between all pairs of vertices on a common face of a planar graph in linear time. We give the ﬁrst polynomial-time approximation schemes for the Steiner-tree and 2-edge-connected subgraph problems. Both schemes are NP-hard in planar graphs and admit no PTAS in general graphs. Our schemes run in O(n log n) time.

Doctoral Dissertation, Brown University, 2007.

[ pdf ]