Tillmann Miltzow

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In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is(More)
Given a graph G = (V,E) with V = {1, . . . , n}, we place on every vertex a token T1, . . . , Tn. A swap is an exchange of tokens on adjacent vertices. We consider the algorithmic question of finding a shortest sequence of swaps such that token Ti is on vertex i. We are able to achieve essentially matching upper and lower bounds, for exact algorithms and(More)
Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The point-guard art gallery problem asks for a minimum set S such that every point in P is visible from a point in S. Assuming integer coordinates and a special general position assumption, we present the(More)
In this paper we study the art gallery problem, which is one of the fundamental problems in computational geometry. The objective is to place a minimum number of guards inside a simple polygon so that the guards together can see the whole polygon. We say that a guard at position x sees a point y if the line segment xy is contained in the polygon. Despite an(More)
In an instance of the house allocation problem, two sets A and B are given. The set A is referred to as applicants and the set B is referred to as houses. We denote by m and n the size of A and B respectively. In the house allocation problem, we assume that every applicant a ∈ A has a preference list over the set of houses B. We call an injective mapping τ(More)
Let X2k be a set of 2k labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of X2k. Two such matchings, M and M ′, are disjoint compatible if they do not have common edges, and no edge of M crosses an edge of M ′. Denote by DCMk the graph whose vertices correspond to such matchings, and two(More)
Given n red and n blue points in general position in the plane, it is well-known that there is a perfect matching formed by non-crossing line segments. We characterize the bichromatic point sets which admit exactly one non-crossing matching. We give several geometric descriptions of such sets, and find an O(n log n) algorithm that checks whether a given(More)