Algorithmic aspects of embedding simplicial complexes in R d

Transcription

1 Algorithmic aspects of embedding simplicial complexes in R d Jiří Matoušek Charles University, Prague and ETH Zurich joint work with Martin Čadek, Marek Krčál, Eric Sedgwick, Francis Sergeraert, Martin Tancer, Lukáš Vokřínek, and Uli Wagner

2 Čadek, Vokřínek Brno Krčál Prague Vienna M. Prague & Zurich Tancer Prague Stockholm Vienna Sergeraert Grenoble Wagner Zurich Lausanne Vienna Sedgwick Chicago

3 The embedding problem k d fixed positive integers EMBED k d is the following algorithmic problem: Input: A simplicial complex K of dimension at most k. Question: Is K embeddable (piecewise linearly) into R d?

4 The embedding problem k d fixed positive integers EMBED k d is the following algorithmic problem: Input: A simplicial complex K of dimension at most k. Question: Is K embeddable (piecewise linearly) into R d? Embedding of K in R d : want to place a copy of K into R d, possibly deformed but no self-intersections. EMBED 1 2 : also known as GRAPH PLANARITY.

5 The embedding problem Embeddings of spaces studied a lot for over a century... see, e.g., [Skopenkov 2006], [Repovš and Skopenkov 1996].

6 The embedding problem Embeddings of spaces studied a lot for over a century... see, e.g., [Skopenkov 2006], [Repovš and Skopenkov 1996]. But apparently, ours is the first systematic study of computational complexity of the embedding problem.

7 Simplicial complex? What? k-dimensional geometric simplex k = 0 k = 1 k = 2 k = 3 A (geometric) simplicial complex: a collection of simplices in some R m glued together along faces. GOOD BAD

8 Simplicial complex? What? k-dimensional geometric simplex k = 0 k = 1 k = 2 k = 3 A (geometric) simplicial complex: a collection of simplices in some R m glued together along faces. GOOD BAD A purely combinatorial way of specifying a topological space. A 1-dimensional simplicial complex = a (simple) graph.

9 What kind of embedding? linear piecewise linear (PL) topological For graphs, all equivalent (only one notion of planarity).

10 What kind of embedding? linear piecewise linear (PL) topological For graphs, all equivalent (only one notion of planarity). But for higher dimensions, PL linear

11 What kind of embedding? linear piecewise linear (PL) topological For graphs, all equivalent (only one notion of planarity). But for higher dimensions, PL linear and also topological PL in some cases (e.g., 4 5); mind-boggling but true.

12 What kind of embedding? linear piecewise linear (PL) topological For graphs, all equivalent (only one notion of planarity). But for higher dimensions, PL linear and also topological PL in some cases (e.g., 4 5); mind-boggling but true. Linear embeddability always in PSPACE (solvability of polynomial inequalities in real variables). We consider PL embeddability entirely different and more complex picture.

13 What we know: the complexity of EMBED k d d k P 2 P Dec NP 3 Dec NP NP P 4 NP Un NP NP P 5 Un Un NP NP P P 6 Un Un NP NP NP P P 7 Un Un NP NP NP P P P Un = algorithmically undecidable [M., Tancer, Wagner] NP = NP-hard [M., Tancer, Wagner] Dec = decidable [M.,Sedgwick, Tancer, Wagner] P = polynomial-time: planarity; [Van Kampen] P = polynomial-time [Čadek, Krčál, M., Vokřínek, Wagner]

14 Embedding in R 3 Very different from the other cases: 3D topology. Our approach: reduce EMBED 2 3, possibly with exponential-time overhead, to the following question: Does a given a compact 3-manifold M with boundary embed in S 3? We = [M.,Sedgwick,Tancer,Wagner].

15 Embedding in R 3 Very different from the other cases: 3D topology. Our approach: reduce EMBED 2 3, possibly with exponential-time overhead, to the following question: Does a given a compact 3-manifold M with boundary embed in S 3? We = [M.,Sedgwick,Tancer,Wagner]. The boundary of an embeddable M must be a disjoint union of spheres with handles. Wanted: fill the holes = glue handlebodies to the boundary components so that S 3 is obtained.

16 Embedding a 3-manifold in S 3 Does a given a compact 3-manifold M with boundary embed in S 3? Simplest nontrivial case: M is a torus, need to glue solid torus to it. Only parameter: p, the number of twists (like in Dehn surgery). This case known to be decidable [Jaco, Sedgwick 2003].

17 Embedding a 3-manifold in S 3 Does a given a compact 3-manifold M with boundary embed in S 3? The general case Use normal surfaces (like Haken s and other unknot algorithms, S 3 recognition, etc.) Lots of additional 3D topology tools; had to extend/modify some a mess.

18 Embedding a 3-manifold in S 3 The general case More or less standard reductions M irreducible, incompressible boundary. Main claim: if such an M embeds in S 3, then it has an embedding with a short meridian (short w.r.t. given triangulation). Meridian γ: a disk D S 3 \ M, D = γ. X outside X outside γ Algorithm: for each candidate for such a meridian γ, glue D, thicken it simpler M ; recurse.

19 Now we consider... d k P 2 P Dec NP 3 Dec NP NP P 4 NP Un NP NP P 5 Un Un NP NP P P 6 Un Un NP NP NP P P 7 Un Un NP NP NP P P P... the P (newly polynomial) cases: d 6, k 2 3 d 1. Six recent papers, 200+ pages. :-(

20 Haefliger & Weber tell us... Theorem (Haefliger Weber) If K is a k-dimensional simplicial complex, where k 2 3 d 1, then K embeds in R d iff there is an equivariant map f : K 2 S d 1.

21 Haefliger & Weber tell us... Theorem (Haefliger Weber) If K is a k-dimensional simplicial complex, where k 2 3d 1, then K embeds in R d iff there is an equivariant map f : K 2 S d 1. Dictionary: K 2 := {(x, y) K K : x y} is the deleted product of K. S d 1 the unit sphere in R d. Equivariant here means that f is continuous and f (y, x) = f (x, y).

22 Haefliger & Weber tell us... Theorem (Haefliger Weber) If K is a k-dimensional simplicial complex, where k 2 3d 1, then K embeds in R d iff there is an equivariant map f : K 2 S d 1. Dictionary: K 2 := {(x, y) K K : x y} is the deleted product of K. S d 1 the unit sphere in R d. Equivariant here means that f is continuous and f (y, x) = f (x, y). easy; the hard part. fails for k > 2 3 d 1, and we have NP-hardness (or more??) there [MTW].

23 Algorithm wanted: Does an equivariant map exist? X, Y spaces with antipodality (in particular, Y = S d 1 ). Decide the existence of an equivariant map X Y! Borsuk Ulam theorem no equivariant map S d S d 1.

24 Algorithm wanted: Does an equivariant map exist? X, Y spaces with antipodality (in particular, Y = S d 1 ). Decide the existence of an equivariant map X Y! Borsuk Ulam theorem no equivariant map S d S d 1. Ask the experts?

25 Algorithm wanted: Does an equivariant map exist? X, Y spaces with antipodality (in particular, Y = S d 1 ). Decide the existence of an equivariant map X Y! Borsuk Ulam theorem no equivariant map S d S d 1. Ask the experts? Topologist A: This problem is probably solved by obstruction theory and characteristic classes.

26 Algorithm wanted: Does an equivariant map exist? X, Y spaces with antipodality (in particular, Y = S d 1 ). Decide the existence of an equivariant map X Y! Borsuk Ulam theorem no equivariant map S d S d 1. Ask the experts? Topologist A: This problem is probably solved by obstruction theory and characteristic classes. Topologist B: People tried to compute higher obstructions and nobody has managed. This is probably unsolvable.

27 Algorithm wanted: Does an equivariant map exist? X, Y spaces with antipodality (in particular, Y = S d 1 ). Decide the existence of an equivariant map X Y! Borsuk Ulam theorem no equivariant map S d S d 1. Ask the experts? Topologist A: This problem is probably solved by obstruction theory and characteristic classes. Topologist B: People tried to compute higher obstructions and nobody has managed. This is probably unsolvable. The situation: Such things studied mainly in the 1950s-1970s. Apparently most authors retired or dead. A powerful (and complicated) machinery developed at that time. But algorithmic aspects of a programme outlined, e.g., in [Steenrod 1972] never realized, as far as we could find.

28 The extension problem Decide: equivariant f : X Y? A generally interesting problem, not only for embeddings. E.g., lower bounds for chromatic number and other Borsuk Ulam style applications of topology.

29 The extension problem Decide: equivariant f : X Y? A generally interesting problem, not only for embeddings. E.g., lower bounds for chromatic number and other Borsuk Ulam style applications of topology. More generally, the equivariant extension problem: Given X, Y, some A X, and an equivariant map f : A Y, can it be extended equivariantly to all of X?

30 The extension problem Decide: equivariant f : X Y? A generally interesting problem, not only for embeddings. E.g., lower bounds for chromatic number and other Borsuk Ulam style applications of topology. More generally, the equivariant extension problem: Given X, Y, some A X, and an equivariant map f : A Y, can it be extended equivariantly to all of X? We start with a simpler setting: drop equivariance. Extension problem: A f A Y f X X?? Y Absolutely basic question in algebraic topology.

31 The extension problem Extension problem: A f Y X??

32 The extension problem Extension problem: A f Y X?? A = S 1, X = D 2... undecidable (triviality of an element of the fundamental group of Y ).

33 The extension problem Extension problem: A f Y X?? A = S 1, X = D 2... undecidable (triviality of an element of the fundamental group of Y ). Theorem (Čadek Krčál M. Sergeraert Vokřínek Wagner) If dim X 2d 1 and Y is (d 1)-connected, d 2, then the extension problem can be solved, even in polynomial time for d fixed.

34 The extension problem Extension problem: A f Y X?? A = S 1, X = D 2... undecidable (triviality of an element of the fundamental group of Y ). Theorem (Čadek Krčál M. Sergeraert Vokřínek Wagner) If dim X 2d 1 and Y is (d 1)-connected, d 2, then the extension problem can be solved, even in polynomial time for d fixed. (d 1)-connected... no holes in dimension d 1 or smaller (= every map f : S k Y is homotopic to a constant map, 0 k d 1). Basic example: Y = S d.

35 The extension problem Theorem (Čadek Krčál M. Sergeraert Vokřínek Wagner) If dim X 2d 1 and Y is (d 1)-connected, d 2, then the extension problem can be solved, even in polynomial time for d fixed. Can test homotopy of two given maps as well, and compute a representation of [X, Y ], all maps X Y up to homotopy. (Unexpected) application: [Franek, Krčál] certifying a robust zero of a map R m R n.

36 Complementary hardness Theorem (Čadek Krčál M. Sergeraert Vokřínek Wagner) If dim X 2d 1 and Y is (d 1)-connected, d 2, then the extension problem can be solved, even in polynomial time for d fixed. Theorem (Čadek Krčál M. Vokřínek Wagner) For dim X = 2d and Y (d 1)-connected, d 2, the extension problem is undecidable. For d = 2k, even for Y = S d.

37 Complementary hardness Theorem (Čadek Krčál M. Sergeraert Vokřínek Wagner) If dim X 2d 1 and Y is (d 1)-connected, d 2, then the extension problem can be solved, even in polynomial time for d fixed. Theorem (Čadek Krčál M. Vokřínek Wagner) For dim X = 2d and Y (d 1)-connected, d 2, the extension problem is undecidable. For d = 2k, even for Y = S d. Reduce from undecidability of Diophantine equations.

38 Tools Topology (1960s): simplicial sets, Eilenberg MacLane spaces, Postnikov systems, obstruction theory, a bit of exact sequences ( Homotopy Theory I, but sometimes in a less usual form... ). In particular, [Brown 1957].

39 Tools Topology (1960s): simplicial sets, Eilenberg MacLane spaces, Postnikov systems, obstruction theory, a bit of exact sequences ( Homotopy Theory I, but sometimes in a less usual form... ). In particular, [Brown 1957]. Making all steps algorithmic; effective algebraic topology. In particular, objects with effective homology [Sergeraert, Rubio, Dousson, Romero].

40 Tools Topology (1960s): simplicial sets, Eilenberg MacLane spaces, Postnikov systems, obstruction theory, a bit of exact sequences ( Homotopy Theory I, but sometimes in a less usual form... ). In particular, [Brown 1957]. Making all steps algorithmic; effective algebraic topology. In particular, objects with effective homology [Sergeraert, Rubio, Dousson, Romero]. Among others, we need to compute a Postnikov system of Y... follows by known methods but never written down carefully.

41 Tools Topology (1960s): simplicial sets, Eilenberg MacLane spaces, Postnikov systems, obstruction theory, a bit of exact sequences ( Homotopy Theory I, but sometimes in a less usual form... ). In particular, [Brown 1957]. Making all steps algorithmic; effective algebraic topology. In particular, objects with effective homology [Sergeraert, Rubio, Dousson, Romero]. Among others, we need to compute a Postnikov system of Y... follows by known methods but never written down carefully. Analogous polynomial-time homology worked out. Many technical steps needed; but our main contribution perhaps in a synthesis.

42 The equivariant case and beyond The equivariant extension problem solved as well [Čadek, Krčál, Vokřínek]. Heavier topological tools. Polynomial time for fixed d, too. Gives polynomiality for EMBED k d in the metastable range (the P s). Turns out the natural problem to solve (for induction) is lifting-extension: A f Y ι?? X g B Even for the existence of an equivariant map X Y. ϕ

43 Papers J. Matoušek, M. Tancer, U. Wagner. Hardness of embedding simplicial complexes in R d. J. Eur. Math. Soc., 13 (2011), NP-hardness and undecidability for EMBED k d. M. Čadek, M. Krčál, J. Matoušek, F. Sergeraert, L. Vokřínek, U. Wagner. Computing all maps into a sphere. arxiv: , SODA 2012, to appear in JACM. All maps X Y up to homotopy. M. Čadek, M. Krčál, J. Matoušek, L. Vokřínek, U. Wagner. Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension. arxiv: Polynomiality for d fixed. M. Krčál, J. Matoušek, F. Sergeraert. Polynomial-time homology for simplicial Eilenberg MacLane spaces. arxiv: , to appear in Foundat. Comput. Math. A key step for polynomiality: handle K(Z, 1). M. Čadek, M. Krčál, J. Matoušek, L. Vokřínek, U. Wagner. Extendability of continuous maps is undecidable. arxiv: , to appear in DCG. Undecidability for the extension problem. M. Čadek, M. Krčál, J. Matoušek, L. Vokřínek, U. Wagner. Extending continuous maps: polynomiality and undecidability. STOC A summary.

44 Papers M. Čadek, M. Krčál, L. Vokřínek. Algorithmic solvability of the lifting-extension problem. arxiv: The equivariant case and lifting-extension. L. Vokřínek. Constructing homotopy equivalences of chain complexes of free ZG-modules, arxiv: A crucial step for equivariant computations. J. Matoušek. Computing higher homotopy groups is W [1]-hard. arxiv: More on the hardness of computing higher homotopy. J. Matoušek, E. Sedgwick, M. Tancer, U. Wagner. Untangling two systems of noncrossing curves. Graph Drawing A spinoff of the S 3 embedding project concerning curves. Application in the Robertson Seymour graph minors theory. J. Matoušek, E. Sedgwick, M. Tancer, U. Wagner. Embeddability in the 3-sphere is decidable. Manuscript, EMBED 2 3, EMBED 3 3 decidable.