Counting Circles Without Computing Them

In this paper we engineer a fast algorithm to count the number of triangles defined by three lines out of a set of n lines whose circumcircle contains the origin. The trick is not to compute any triangles or circles

Solitaire Clobber

Clobber is a new two-player board game. In this paper, we introduce the one-player variant Solitaire Clobber where the goal is to remove as many stones as possible from the board by alternating white and black moves. We show that a checkerboard configuration on a single row (or single column) can be reduced to about n/4 stones. For boards with at least two rows and two columns, we show that a checkerboard configuration can be reduced to a single stone if and only if the number of stones is not a multiple of three, and otherwise it can be reduced to two stones. We also show that in general it is NP-complete to decide whether an arbitrary Clobber configuration can be reduced to a single stone.Comment: 14 pages. v2 fixes small typ

An Algorithmic Analysis of the Honey-Bee Game

The Honey-Bee game is a two-player board game that is played on a connected hexagonal colored grid or (in a generalized setting) on a connected graph with colored nodes. In a single move, a player calls a color and thereby conquers all the nodes of that color that are adjacent to his own current territory. Both players want to conquer the majority of the nodes. We show that winning the game is PSPACE-hard in general, NP-hard on series-parallel graphs, but easy on outerplanar graphs. In the solitaire version, the goal of the single player is to conquer the entire graph with the minimum number of moves. The solitaire version is NP-hard on trees and split graphs, but can be solved in polynomial time on co-comparability graphs.Comment: 20 pages, 9 figure

06421 Abstracts Collection -- Robot Navigation

From 15.10.06 to 20.10.06, the Dagstuhl Seminar 06421 ``Robot Navigation\u27\u27generate automatically was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Traveling salesmen in the presence of competition

AbstractWe propose the “competing salesmen problem” (CSP), a two-player competitive version of the classical traveling salesman problem. This problem arises when considering two competing salesmen instead of just one. The concern for a shortest tour is replaced by the necessity to reach any of the customers before the opponent does.In particular, we consider the situation where players take turns, moving along one edge at a time within a graph G=(V,E). The set of customers is given by a subset VC⊆V of the vertices. At any given time, both players know of their opponent's position. A player wins if he is able to reach a majority of the vertices in VC before the opponent does.We prove that the CSP is PSPACE-complete, even if the graph is bipartite, and both players start at distance 2 from each other. Furthermore, we show that the starting player may not be able to avoid losing the game, even if both players start from the same vertex. However, for the case of bipartite graphs, we show that the starting player always can avoid a loss. On the other hand, we show that the second player can avoid to lose by more than one customer, when play takes place on a graph that is a tree T, and VC consists of leaves of T. It is unclear whether a polynomial strategy exists for any of the two players to force this outcome. For the case where T is a star (i.e., a tree with only one vertex of degree higher than two) and VC consists of n leaves of T, we give a simple and fast strategy which is optimal for both players. If VC consists not only of leaves, we point out that the situation is more involved

Competitive Online Searching for a Ray in the Plane

We consider the problem of a searcher that looks, for example, for a lost flashlight in a dusty environment. The searcher finds the flashlight as soon as it crosses the ray emanating from the flashlight. In order to pick it up, the searcher moves to the origin of the light beam. We compare the length of the path of the searcher to the shortest path to the goal. First, we give a search strategy for a special case of the ray search---the window shopper problem---, where the ray we are looking for is perpendicular to a known ray. Our strategy achieves a competitive factor of $1.059ldots$, which is optimal. Then, we consider rays in arbitrary position in the plane. We present an online strategy that achieves a factor of $22.513ldots$, and give a lower bound of $2pi,e=17.079ldots$