Inclusion-exclusion formulas from independent complexes

Using inclusion-exclusion, we can write the indicator function of a union of finitely many balls as an alternating sum of indicator functions of common intersections of balls. We exhibit abstract simplicial complexes that correspond to minimal inclusion-exclusion formulas. They include the dual complex, as defined in [2], and are characterized by the independence of their simplices and by geometric realizations with the same underlying space as the dual complex. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—Geometrical problems and computations, Computations on discrete structures; G.2.1 [Discrete Mathematics]: Combinatorics—Counting problems General Terms Theory, Algorithms Keywords Combinatorial topology, discrete geometry, dual complexes, balls, spheres, indicator functions

• Type: Conference
• Year: 2005
• Where: COMPGEOM
• Authors: Dominique Attali, Herbert Edelsbrunner
• Keywords: Abstract Simplicial Complexes, COMPGEOM 2005, Discrete Geometry, Dual Complex, Indicator Functions