Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a, b ∈ P can see each other if the line segment seg(a, b) is contained in P. We denote by V (P) the family of all minimum guard placements. The Hausdorff distance makes V (P) a metric space and thus a topological space. We show homotopy-universality, that is, for every semi-algebraic set S there is a polygon P such that V (P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T , we describe instances I of the art gallery problem such that V (I ) is homeomorphic to T .

In the Exact Matching Problem (EM), we are given a graph equipped with a fixed coloring of its edges with two colors (red and blue), as well as a positive integer k. The task is then to decide whether the given graph contains a perfect matching exactly k of whose edges have color red. EM generalizes several important algorithmic problems such as perfect matching and restricted minimum weight spanning tree problems. When introducing the problem in 1982, Papadimitriou and Yannakakis conjectured EM to be NP-complete. Later however, Mulmuley et al. presented a randomized polynomial time algorithm for EM, which puts EM in RP. Given that to decide whether or not RP=P represents a big open challenge in complexity theory, this makes it unlikely for EM to be NP-complete, and in fact indicates the possibility of a deterministic polynomial time algorithm. EM remains one of the few natural combinatorial problems in RP which are not known to be contained in P, making it an interesting instance for testing the hypothesis RP=P. Despite EM being quite well-known, attempts to devise deterministic polynomial algorithms have remained illusive during the last 40 years and progress has been lacking even for very restrictive classes of input graphs. In this paper we finally push the frontier of positive results forward by proving that EM can be solved in deterministic polynomial time for input graphs of bounded independence number, and for bipartite input graphs of bounded bipartite independence number. This generalizes previous positive results for complete (bipartite) graphs which were the only known results for EM on dense graphs.

In 1982, Papadimitriou and Yannakakis introduced the Exact Matching (EM) problem where given an edge colored graph, with colors red and blue, and an integer k, the goal is to decided whether or not the graph contains a perfect matching with exactly k red edges. Although they conjectured it to be NP-complete, soon after it was shown to be solvable in randomized polynomial time in the seminal work of Mulmuley et al. placing it in the complexity class RP. Since then, all attempts at finding a deterministic algorithm for EM have failed, thus leaving it as one of the few natural combinatorial problems in RP but not known to be contained in P, and making it an interesting instance for testing the hypothesis RP=P. Progress has been lacking even on very restrictive classes of graphs despite the problem being quite well known as evidenced by the number of works citing it. In this paper we aim to gain more insight into the problem by considering two directions of study. In the first direction, we study EM on bipartite graphs with a relaxation of the color constraint and provide an algorithm where the output is required to be a perfect matching with a number of red edges differing from k by at most k/2. We also introduce an optimisation problem we call Top-k Perfect Matching (TkPM) that shares many similarities with EM. By virtue of being an optimization problem, it is more natural to approximate so we provide approximation algorithms for it. In the second direction, we look at the parameterized algorithms. Here we introduce new tools and FPT algorithms for the study of EM and TkPM.