Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, October 23, 2008, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Uli Wagner

Hardness of Embedding Simplicial Complexes in R^d (Part II)

It is well known that one can test in linear time whether a given graph is planar. We consider the higher-dimensional generalization of this problem: Given a k-dimensional simplicial complex K and a target dimension d, does K embed into R^d? Surprisingly, rather little seems to be known about the algorithmic aspects of this problem.

In Part 1 of the talk, we will review and explain the relevant topological notions, at least on an intuitive level, and discuss some background, known results, and related problems. For instance, known results easily imply that the embeddability problem is solvable in polynomial time if k=d=2 or d=2k \geq 6. Moreover, we observe that the problem is algorithmically undecidable for k=d-1 and d \geq 5. This follows from a famous result of Novikov on the unsolvability of recognizing the 5-sphere.

In Part 2, we will prove our main result: It is at least NP-hard to decide if a 2-dimensional simplicial complex embeds into R^4. More generally, the problem is NP-hard if d \geq 4 and d \geq k \geq (2d-2)/3. These dimensions fall outside the so-called metastable range of a theorem of Haefliger and Weber, which characterizes embeddability in terms of the so-called deleted product obstruction. Our reductions are based on examples, due to Segal, Spiez, Freeman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range, the deleted product obstruction is insufficient.

Joint work with Jiří Matoušek and Martin Tancer.

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