Lower Bounds on Extended Formulations Noah Fleming University of Toronto Supervised by Toniann Pitassi

Linear Programming Max/Min w T x← Objective Function Subject To: Ax ≤ b← Linear Constraints x ≥ 0← Nonnegativity Constraints Size: The number of constraints in the linear program

Approximate Vertex Cover Input: A graph G= (V,E) Output: The minimum set of vertices such that each e ∈ E is adjacent to at least one v ∈ V Min [1,...,1] T x Subject To: x i + x j ≥ 1 ∀ (i, j) ∈ E x i ≥ 0 ∀ i ∈ V

Geometric Representation Max w T x Polytope: n-dimensional shape with flat sides Facets of polytope defined by inequalities in Ax ≤ b Feasible solution: Any point satisfying all the constraints of the linear program

Extended Formulation Add extra variables and constraints to decrease the size of the polytope An extended formulation Ex + Fy = b expresses Ax ≤ b if for every x that satisfies Ax ≤ b, there exists a y such that (x,y) satisfies Ex + Fy = b Extended Formulation Original polytope

Approximate Vertex Cover Input: A graph G= (V,E) Output: The minimum set of vertices such that each e ∈ E is adjacent to at least one v ∈ V Min [1,...,1] T x Subject To: x i + x j ≥ 1 ∀ (i, j) ∈ E x i ≥ 0 ∀ i ∈ V

Extended Formulation Add extra variables and constraints to decrease the size of the polytope An extended formulation Ex + Fy = b expresses Ax ≤ b if for every x that satisfies Ax ≤ b, there exists a y such that (x,y) satisfies Ex + Fy = b Extended Formulation Original polytope

Lower Bounds on Polytopes Slack variable: let x be a vertex, and a i x ≤ b i be a constraint in the system, y is the slack for this constraint if a i x + y = b i Slack matrix SM[i,j] = b i - a i x j

Lower Bounds on Polytopes Factorization Theorem (Yannakakis): Let r be the smallest number such that S M can be written as the product of two nonnegative matrices F and V of dimension f x r and r x v. The minimum size of any linear program expressing the polytope is θ(n + r). Rank + (S M ):= r

Communication Complexity Alice P(x,y) = ? Nondeterministic Communication Complexity N(P): Number of bits needed to represent the proof plus the minimum Number of bits of communication needed to compute P(x,y). Bob Communication Matrix P(x,y)

Lower bounds on polytopes by communication FV(x,y) = 1 if vertex x does not lie on facet y FV(x,y) = 0 otherwise F(x,y) 0/1 S M SMSM Lemma: N(FV) ≤ log 2 (rank + (S M ))

Proof Complexity Propositional Proof System: A polynomial time algorithm P such that for all x, x ∈ TAUT iff there exists a string π such that P(x,π) = 1 Complexity: the smallest function f:→ such that x ∈ TAUT iff there exists a proof π of size |π|  f(|x|) such that P(x, π)=1 Theorem: There exists a propositional proof system with complexity bounded above by a polynomial iff NP=coNP