# Algorithms for Optimization Problems in Planar Graphs

## Motivation

There is a long tradition of research in algorithms for optimization problems in graphs, including work on many classical problems, both polynomial-time solvable problems and NP-hard problems, e.g. shortest paths, maximum flow and minimum cut, matching, T-joins, disjoint paths, traveling salesman, Steiner tree, graph bisection, vehicle routing, facility location, k-center, and maximum cut. One theme of such research addresses the complexity of these problems when the input graph is required to be a planar graph or a graph embedded on a low-genus surface.

There are three reasons for this theme. First, optimization problems in planar graphs arise in diverse application areas. Second, researchers have discovered that, by exploiting the planarity of the input, much more effective algorithms can be developed - algorithms that are faster or more accurate than those that do not exploit graph structure. Third, the study of algorithms for surface-embedded graphs drives the development of interesting algorithmic techniques.

One source of applications for planar-graph algorithms is geographic problems. Road maps are nearly planar, for example, so distances in planar graphs can model, e.g., travel times in road maps. Network design in planar graphs can be used to model scenarios in which cables must be run under roads. Planar graphs can also be used to model metrics on the earth's surface that reflect physical features such as terrain; this aspect of planar graphs has been used in studying wildlife corridors. Another source of applications is image processing. Some algorithms for problems such as image segmentation and stereo involve finding minimum cuts in a grid in which each vertex represents a pixel. Sometimes an aggregation technique (superpixels) coalesces regions into vertices, turning the grid into an arbitrary planar graph. A third example application is VLSI.

Algorithmic exploitation of a planar embedding goes back at least to the introduction of maximum flow by Ford and Fulkerson in 1956. Current research can be divided in three parts. For polynomial-time-solvable problems, such as maximum flow, shortest paths, matching, and min-cost circulation, researchers seek planarity-exploiting algorithms whose running times beat those of general-graph algorithms, ideally algorithms whose running times are linear or nearly linear. For NP-hard problems, there are two strategies: fixed-parameter algorithms and approximation algorithms.

In all three research subareas, there has recently been significant progress. However, many researchers are expert in only one or two subareas. A key benefit of this Dagstuhl Seminar is to bring together researchers from the different subareas, to introduce them to techniques from subareas that might be unfamiliar, and to and foster collaboration across the subareas. The seminar will thus help to spur further advances in this active and growing area.