On the Polygon-Search Conjecture

The polygon search problem is the problem of searching a mobile intruder in a polygonal region by a mobile searcher with various visibility. The k-searcher is a mobile guard whose visibility is limited to k rays emanating from his position, where the directions of the rays can be changed continuously with bounded angular rotation speed. It has been conjectured (see [9]) that the 2-searcher has the same search capability as the searcher with 360 vision, called the ∞-searcher. In this paper, we show a slightly weak variant of this conjecture: If the ∞-searcher can search a polygon in such a way that one point moving along the boundary is always visible from the ∞-searcher throughout the search, the 2-searcher can also search it. A special case of our proof shows that if the searcher is restricted to move along the boundary throughout the search, the 1-searcher and the ∞-searcher have the same search capability.