Seminars in July 2018

  • Tillmann Miltzow, The Art Gallery Problem is ∃R-complete

    The Art Gallery Problem is ∃R-complete
    Tillmann Miltzow
    Computer Science Department, Université Libre de Bruxelles, Brussels
    2018/7/24 Tue 5PM-6PM (Room 3434, Bldg. E6-1)
    We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational geometry. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p∈P is seen by at least one guard g∈G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃R. The class ∃R consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP⊆∃R. We prove that the art gallery problem is ∃R-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃R. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many geometric approaches to the problem. This is joint work with Mikkel Abrahamsen and Anna Adamaszek.
  • (Intensive Lecture Series) Ron Aharoni, Lectures on topological methods in combinatorics

    Intensive Lecture Series

    Lectures on topological methods in combinatorics
    Ron Aharoni
    Department of Mathematics, Technion, Israel
    2018/7/17-19 2PM-5PM (Room 3434, Bldg. E6-1)
    The lectures will give an introduction to the application of topological methods in matching theory, graph theory, and combinatorics.
    Topics that will be covered:</p>

    – A topological extension of Hall’s theorem
    – combinatorial applications of the nerve theorem
    – d-Leray complexes and rainbow matchings
    – Matroid complexes and applications
    – Open problems

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