What is a cone? Anastasia Chavez. Field of Dreams Conference President s Postdoctoral Fellow NSF Postdoctoral Fellow UC Davis

Transcription

1 What is a cone? Anastasia Chavez President s Postdoctoral Fellow NSF Postdoctoral Fellow UC Davis Field of Dreams Conference 2018

2 Roadmap for today 1 Cones 2 Vertex/Ray Description 3 Hyperplane Description 4 An Application

3 Intuitive idea of a Cone "Set of vectors closed under positive combinations"

4 Example Intuitive idea of a Cone "Set of vectors closed under positive combinations" For V = {( 1, 1 2), (1, 2), (2, 1), ( 1 2, 3 4 )}

5 Example Intuitive idea of a Cone "Set of vectors closed under positive combinations" For V = {( 1, 1 2), (1, 2), (2, 1), ( 1 2, 3 4)}, the cone of V is C(V) = ( {a 1 1, 1 ) ( 1 + a 2 (1, 2) + a 3 (2, 1) + a 4 2 2, 3 ) } a i R 0 4

6 Vertex/Ray Description The space generated by a finite set of vertices/rays" Let V = {v 1, v 2,..., v i, r i+1,..., r m } be a set of vertices and rays in R n. The cone generated by V is C(V) = {λ 1 v λ m r m λ i R n 0 }.

7 Vertex/Ray Description The space generated by a finite set of vertices/rays" Let V = {v 1, v 2,..., v i, r i+1,..., r m } be a set of vertices and rays in R n. The cone generated by V is C(V) = {λ 1 v λ m r m λ i R n 0 }.

8 Hyperplane Description The intersection of halfspaces" H 1 = { (x 1, x 2 ) R 2 1 } 2 x 1 x 2 0 Definition A hyperplane H is the set {x R n a(x) = 0}, for linear map a over R n. A closed halfspace H is choosing a side" of H: {x R n a(x) 0}.

9 Hyperplane Description The intersection of halfspaces" H 1 = { (x 1, x 2 ) R 2 1 } 2 x 1 x 2 0 H 2 = { (x 1, x 2 ) R 2 2x 1 x 2 0 } Definition A hyperplane H is the set {x R n a(x) = 0}, for linear map a over R n. A closed halfspace H is choosing a side" of H: {x R n a(x) 0}.

10 Hyperplane Description The intersection of halfspaces" Definition A convex cone C is a collection of closed halfspaces A, such that C = {x R n Ax 0}.

11 Cones from vectors/rays and hyperplanes Theorem (Weyl Minkowski Theorem) A convex polyhedral cone has both a vertex/ray and hyperplane description, which are equivalent.

12 Where Cones Commonly Show Up Solvability of a general system of linear equations (Farka s lemma) Integer point enumeration, Ehrhart Theory Discrete optimization, linear programming, feasibility problems Computational Complexity Where else might they show up?

13 Definition Using Cones to Understand Graphs A graph G = (V, E) is a set of vertices and edges. A cycle of G is a set of edges forming a path that returns to itself only once. Example

14 Using Cones to Understand Graphs We can describe all the cycles of a graph using vectors! Let c {0, 1} n be the indicator vector of a cycle of graph G, where c i = 1 if e i E and 0 if not. Example Cycle in G = (1, 0, 1, 0, 1, 1)

15 Using Cones to Understand Graphs We can describe all the cycles of a graph using vectors! Let c {0, 1} n be the indicator vector of a cycle of graph G, where c i = 1 if e i E and 0 if not. Example C = Every column is a cycle and rows are indexed by edges.

16 Using Cones to Understand Graphs Using the set of cycles of G, we can generate the cone C G over all cycles of G: C G = {λ 1 c λ n c n λ R n } Example C G = λ λ λ λ i R 7

17 Using Cones to Understand Graphs Why use cones? For a new perspective! CDC conjecture: For any graph G, there exists a set of cycles covering the edges of G so that every edge is in exactly 2 cycles.

18 Using Cones to Understand Graphs Why use cones? For a new perspective! CDC conjecture: For any graph G, there exists a set of cycles covering the edges of G so that every edge is in exactly 2 cycles. { }

19 Using Cones to Understand Graphs Why use cones? For a new perspective! CDC conjecture: For any graph G, there exists a set of cycles covering the edges of G so that every edge is in exactly 2 cycles. { }

20 Using Cones to Understand Graphs Why use cones? For a new perspective! CDC conjecture: For any graph G, there exists a set of cycles covering the edges of G so that every edge is in exactly 2 cycles. { }

21 Using Cones to Understand Graphs Why use cones? For a new perspective! CDC conjecture: For any graph G, there exists a set of cycles covering the edges of G so that every edge is in exactly 2 cycles. { } Via Cones: The integral cone of cycles of G always contains (2, 2,..., 2).

22 Using Cones to Understand Graphs Why use cones? For a new perspective! CDC conjecture: For any graph G, there exists a set of cycles covering the edges of G so that every edge is in exactly 2 cycles. { } Via Cones: The integral cone of cycles of G always contains (2, 2,..., 2). In general: Given vector u = (u 1, u 2,..., u n ), is there a set of cycles so that edge i is covered u i many times?

23 Using Cones to Understand Graphs Why use cones? For a new perspective! CDC conjecture: For any graph G, there exists a set of cycles covering the edges of G so that every edge is in exactly 2 cycles. { } Via Cones: The integral cone of cycles of G always contains (2, 2,..., 2). In general: Given vector u = (u 1, u 2,..., u n ), is there a set of cycles so that edge i is covered u i many times? Does the integral cone of cycles of G contain u?

24 References > Beck and Robins, Computing the Continuous Discretely, Springer, > De Loera, Hemmecke, and Köppe, Algebraic and Geometric Ideas in the Theory of Discrete Optimization, Society for Industrial and Applied Mathematics, > Ziegler, Lectures on Polytopes, Springer, 1995.

25 Cones Vertex/Ray Description Hyperplane Description About me ct Instin Play time! Commun ity Suppo rt Allies and Cheerleaders An Application

26 Cones Vertex/Ray Description Hyperplane Description Thank you! ct Instin Play time! Commun ity Suppo rt Allies and Cheerleaders An Application