Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, July 26, 2018, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Uli Wagner (IST Austria)

Shellability is NP-complete

Consider a pure 2-dimensional simplicial complex X (in other words, a 3-uniform hypergraph). A shelling of X is an inductive procedure of building X by adding the triangles of X one at a time, in such a way that each new triangle (except the first one) intersects the previously constructed complex in a connected 1-dimensional piece (consisting of one, two, or all three edges on the boundary of the triangle); X is called shellable if has a shelling. The definition naturally generalizes to higher-dimensional complexes.

Shellings and shellability are fundamental concepts in a variety of areas including polytope theory (the boundary of every convex polytope is shellable, by a theorem of Brugesser and Mani), piecewise-linear topology, topological combinatorics, and others. One reason for its importance is that shellability – a purely combinatorial property – has strong topological consequences. For instance if a d-dimensional complex X is shellable and moreover a pseudomanifold (i.e., if every (d-1)-simplex of X contained in exactly two d-simplices, which is easily checked) then X is homeomorphic to a d-dimensional sphere – a property that is algorithmically undecidable for d\geq 5.

From a computational viewpoint, it is a natural question (raised at least as early as in the 1970’s by Danaraj ad Klee) whether one can decide efficiently whether a given complex is shellable. We settle this in the negative (modulo P\neq NP) and show that shellability of d-dimensional complexes is NP-complete for every d\geq 2.

Joint work with Xavier Goaoc, Pavel Paták, Zuzana Patáková, and Martin Tancer

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