Efficient Edge-Skeleton Computation for Polytopes Defined by Oracles

1. Efficient edge-skeleton computation for polytopes defined by oracles Vissarion Fisikopoulos Joint work with I.Z. Emiris (UoA), B. Gärtner (ETHZ) Dept. of Informatics & Telecommunications, University of Athens ACAC, 22.Aug.2013

2. Outline Polytopes & Oracles Algorithms for polytopes given by oracles

4. Classical Polytope Representations A convex polytope P ⊆ Rd can be represented as the 1. convex hull of a pointset {p1, . . . , pn} (V-representation) 2. intersection of halfspaces {h1, . . . , hm} (H-representation) convex hull problem vertex enumeration problem These problems are equivalent by polytope duality.

5. Algorithmic Issues For general dimension d there is no polynomial algorithm for the convex hull (or vertex enumeration) problem since m can be O n d/2 [McMullen’70]. It is open whether there exist a total poly-time algorithm for the convex hull (or vertex enumeration) problem, i.e. runs in poly-time in n, m, d.

6. Polytope Oracles Implicit representation for a polytope P ⊆ Rd. OPTP: Given direction c ∈ Rd return the vertex v ∈ P that maximizes cT v. SEPP: Given point y ∈ Rd, return yes if y ∈ P otherwise a hyperplane h that separates y from P. c v P P h y

7. Well-described polytopes and oracles Deﬁnition A rational polytope P ⊆ Rd is well-described (with a parameter ϕ) if there exists an H-representation for P in which every inequality has encoding length at most ϕ. The encoding length of P is P = d + ϕ. Proposition (Gr¨otschel et al.’93) For a well-described polytope, we can compute OPTP from SEPP (and vice versa) in oracle polynomial-time. The runtime (polynomially) depends on d and ϕ.

8. Why oracles? Polynomial time algorithms for combinatorial optimization problems using the ellipsoid method [Gr¨otschel-Lov´asz-Schrijver’93] Randomized polynomial-time algorithm for approximating the volume of convex bodies [Dyer-Frieze-Kannan ’90]

9. Our Motivation Resultant, Discriminant, Secondary polytopes Vertices → triangulations of a pointset’s convex hull OPTP is available via a triangulation computation [Emiris-F-Konaxis-Pe˜naranda ’12] Applications in Computational Algebraic Geometry, Geometric Modelling, Combinatorics

11. Main problems Vertex enumeration in the oracle model Given OPTP for P ⊆ Rd, compute the vertices of P. Vertex enumeration (in the oracle model) with edge-directions Given OPTP and a superset D of the edge directions D(P) of P ⊆ Rd, compute the vertices of P. Remark Edge-skeleton = V-representation + edges Thus, edge-skeleton computation subsumes vertex enumeration.

12. Vertex enumeration in the oracle model Algorithm sketch [Emiris-F-Konaxis-Peñaranda ’12] first compute d + 1 aff. independent vertices of P and compute their convex hull at each step call OPTP with the outer normal vector of a halfspace and → either validate this halfspace → or add a new vertex to the convex hull N(R) Q

13. Vertex enumeration in the oracle model Algorithm sketch [Emiris-F-Konaxis-Peñaranda ’12] first compute d + 1 aff. independent vertices of P and compute their convex hull at each step call OPTP with the outer normal vector of a halfspace and → either validate this halfspace → or add a new vertex to the convex hull Q N(R)

14. Vertex enumeration in the oracle model Algorithm sketch [Emiris-F-Konaxis-Peñaranda ’12] first compute d + 1 aff. independent vertices of P and compute their convex hull at each step call OPTP with the outer normal vector of a halfspace and → either validate this halfspace → or add a new vertex to the convex hull N(R) Q w

34. Vertex enumeration in the oracle model Algorithm sketch [Emiris-F-Konaxis-Peñaranda ’12] first compute d + 1 aff. independent vertices of P and compute their convex hull at each step call OPTP with the outer normal vector of a halfspace and → either validate this halfspace → or add a new vertex to the convex hull Complexity Given P ⊆ Rd, H-, V-repr. & triang. T of P can be computed in O(d5 ns2 ) arithmetic operations + O(n + m) calls to OPTP s is the number of cells of T.

35. Vertex enumeration in the oracle model Algorithm sketch [Emiris-F-Konaxis-Peñaranda ’12] first compute d + 1 aff. independent vertices of P and compute their convex hull at each step call OPTP with the outer normal vector of a halfspace and → either validate this halfspace → or add a new vertex to the convex hull Complexity Given P ⊆ Rd, H-, V-repr. & triang. T of P can be computed in O(d5 ns2 ) arithmetic operations + O(n + m) calls to OPTP s is the number of cells of T. BUT: s can be O n d/2

36. Vertex enumeration with edge-directions Given OPTP and a superset D of the edge directions D(P) of P ⊆ Rd, compute the vertices P. Proposition (Rothblum-Onn ’07) Let P ⊆ Rd given by OPTP, and D ⊇ D(P). All vertices of P can be computed in O(|D|d−1 ) calls to OPTP + O(|D|d−1 ) arithmetic operations.

37. The edge-skeleton algorithm Input: OPTP Edge vec. P (dir. & len.): D Output: Edge-skeleton of P P c Sketch of Algorithm: Compute a vertex of P (x = OPTP(c) for arbitrary cT ∈ Rd)

38. The edge-skeleton algorithm Input: OPTP Edge vec. P (dir. & len.): D Output: Edge-skeleton of P P Sketch of Algorithm: Compute a vertex of P (x = OPTP(c) for arbitrary cT ∈ Rd) Compute segments S = {(x, x + d), for all d ∈ D}

39. The edge-skeleton algorithm Input: OPTP Edge vec. P (dir. & len.): D Output: Edge-skeleton of P P Sketch of Algorithm: Compute a vertex of P (x = OPTP(c) for arbitrary cT ∈ Rd) Compute segments S = {(x, x + d), for all d ∈ D} Remove from S all segments (x, y) s.t. y /∈ P (OPTP → SEPP)

40. The edge-skeleton algorithm Input: OPTP Edge vec. P (dir. & len.): D Output: Edge-skeleton of P P Sketch of Algorithm: Compute a vertex of P (x = OPTP(c) for arbitrary cT ∈ Rd) Compute segments S = {(x, x + d), for all d ∈ D} Remove from S all segments (x, y) s.t. y /∈ P (OPTP → SEPP) Remove from S the segments that are not extreme

41. The edge-skeleton algorithm Input: OPTP Edge vec. P (dir. & len.): D Output: Edge-skeleton of P P Sketch of Algorithm: Compute a vertex of P (x = OPTP(c) for arbitrary cT ∈ Rd) Compute segments S = {(x, x + d), for all d ∈ D} Remove from S all segments (x, y) s.t. y /∈ P (OPTP → SEPP) Remove from S the segments that are not extreme Can be altered to work with edge directions only

42. Runtime of the edge-skeleton algorithm Theorem Given OPTP and a superset of edge directions D of a well- described polytope P, the edge skeleton of P can be computed in oracle total polynomial-time O n |D|O( P + D ) + LP(4d3 |D|( P + D )) , D is the binary encoding length of the vector set D, n the number of vertices of P, O( P ) : runtime of oracle conversion algorithm for P, LP( A + b + c ) runtime of max cT x over {x : Ax ≤ b}.

43. Applications Corollary The edge skeleton of resultant, secondary and discriminant polytopes (under some genericity assumption) can be computed in oracle total polynomial-time. Convex combinatorial optimization: generalization of linear combinatorial optimization. [Rothblum-Onn ’04] Convex integer programming: maximize a convex function over the integer hull of a polyhedron. [De Loera et al. ’09]

44. Conclusions New & simple algorithm for vertex enumeration of a polytope given by an oracle and known edge directions Remove the exponential dependence on the dimension First total polynomial time algorithms for resultant, discriminant polytopes (under some genericity assumption) Future work Remove the assumption on the knowledge of edge directions Volume computation for polytopes given by optimization oracles

45. Thank you!

Efficient Edge-Skeleton Computation for Polytopes Defined by Oracles

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Efficient Edge-Skeleton Computation for Polytopes Defined by Oracles