Versions of this talk have been given at the

• School of EECS Colloquium, Oregon State University (November 2, 2009) to an audience dominated by first year EE and CS graduate students.
• Combinatorial Potlatch, Vancouver, BC (November 21, 2009) to a raucous group of discrete mathematicians.

Joint work with Brent Heeringa (Williams College) and Gordon Wilfong (Bell Labs)

Suppose you are planning a party, but you can only invite 20 people, (caviar is expensive).  You can’t invite more than 20 people, but you want to maximize the number (or perhaps worth) of people that will show up.  But an attendee will only come to your party if one of their friends is also attending.  How do you find a set of friends to invite?

More specifically, we are given a knapsack of capacity K and a graph representing constraints (i.e. friendships) between the nodes; each node has a weight and a profit.  The goal is to find a maximum-profit set of nodes of total weight at most K such that if node x is selected, then a neighbour of x is also selected.

It turns out that this knapsack problem models several interesting problems (albeit less important than party planning) such as: network formation, tool magazine management, database storage and strategical investing.  In this talk I will take you on a tour of greedy algorithms, approximation guarantees and hardness of approximation in solving the graph-constrained knapsack problem.

Graph algorithms abound: telecommunications networks can be designed by finding low-cost subgraphs connecting terminals and analyzed using maximum flows and minimum cuts; logistics in road maps may amount to shortest paths or travelling salesperson tours; image segmentations are graph cuts. In these examples, not only can we model the problem as a graph problem, but often as a planar graph problem. Can we exploit this planar structure to design better, faster, more accurate algorithms for such problems?

Consider a two-dimensional drawing, or embedding, of a planar graph.  We can use this embedding as a guide for scanning through the graph, building a solution to a problem as we go. Alternatively, we can decompose the graph into small pieces (also planar), solve an easier problem in each piece, and combine the pieces together to build a solution for the original graph. Both techniques are inspired by geometric counterparts. We use the former to efficiently solve network-flow problems (such as maximum flow), and the latter to approximate NP-hard problems (such as Steiner tree). While planar graphs do arise in applications, we may occasionally face a few non-planarities: for example, bridges and tunnels in road maps. We can often turn such nearly-planar graphs into planar graphs and then proceed using planar techniques.

This talk will highlight these techniques, illustrating how they have led to faster algorithms than currently achievable without using planarity and more accurate algorithms than possible (unless P=NP) for general graphs.

[ keynote ] [ powerpoint - may contain errors due to conversion from keynote ] [ video ]