Petersen's theorem is a classic result in matching theory from 1891, stating that every 3-regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3-regular graphs. We have developed an O(n log^4 n)-time algorithm for perfect matching in a 3-regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)-time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.

In the Art Gallery problem a polygon is given and the goal is to place as few guards as possible so that the entire area of the polygon is covered. We address a closely related problem: how to place a fixed number of guards on the vertices or the edges of a simple polygon so that the total guarded area inside the polygon is maximized. Recall that an optimization problem is called APX-hard, if there exists an ɛ> 0 such that an approximation ratio of 1 + ɛ cannot be guaranteed by any polynomial time approximation algorithm, unless P = NP. We prove that our problem is APX-hard and we present a polynomial time algorithm achieving constant approximation ratio. Finally we extend our results for the case where the guards are required to cover valued items inside the polygon. The valued items or “treasures ” are modeled as simple closed polygons. 1