Lower Bounds on the Sizes of Integer Programs Without Additional ...
Lower Bounds on the Sizes of Integer Programs Without Additional Variables Volker Kaibel & Stefan Weltge
Aussois, 2014
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once.
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T )
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn }
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn } = min {hc, xi : x ∈ conv(TSPn )}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn } = min {hc, xi : x ∈ conv(TSPn )} = min {hc, xi : Ax ≤ b}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn } = min {hc, xi : x ∈ conv(TSPn )} = min {hc, xi : Ax ≤ b} Ax ≤ b is very large and complicated
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn } = min {hc, xi : x ∈ conv(TSPn )} = min {hc, xi : Ax ≤ b} Ax ≤ b is very large and complicated Why not use additional variables? Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2)
min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2) Theorem (Fiorini, Pokutta et al. 2012) There is no polynomial size system Ax + By ≤ b such that min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b} holds for all c ∈ RE .
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2) Theorem (Fiorini, Pokutta et al. 2012) There is no polynomial size system Ax + By ≤ b such that min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b} holds for all c ∈ RE . But there are several polynomial size integer programming formulations of the form min hc, xi : Ax + By ≤ b, x ∈ {0, 1}E , y ∈ Zk for solving the TSP (Miller-Tucker-Zemlin, flow based, ...)
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2) Theorem (Fiorini, Pokutta et al. 2012) There is no polynomial size system Ax + By ≤ b such that min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b} holds for all c ∈ RE . But there are several polynomial size integer programming formulations of the form min hc, xi : Ax + By ≤ b, x ∈ {0, 1}E , y ∈ Zk for solving the TSP (Miller-Tucker-Zemlin, flow based, ...)
Theorem For any family of sets Xd ⊆ {0, 1}d with the property that the problem “Given x ∈ Zd , is x ∈ Xd ?” is in NP,
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2) Theorem (Fiorini, Pokutta et al. 2012) There is no polynomial size system Ax + By ≤ b such that min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b} holds for all c ∈ RE . But there are several polynomial size integer programming formulations of the form min hc, xi : Ax + By ≤ b, x ∈ {0, 1}E , y ∈ Zk for solving the TSP (Miller-Tucker-Zemlin, flow based, ...)
Theorem For any family of sets Xd ⊆ {0, 1}d with the property that the problem “Given x ∈ Zd , is x ∈ Xd ?” is in NP, we have polynomial size systems Ax + By ≤ b such that Xd = x : Ax + By ≤ b, x ∈ {0, 1}d , y ∈ Zk . Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Motivating question Is there a polynomial size system Ax ≤ b such that min {hc, xi : x ∈ TSPn } = min hc, xi : Ax ≤ b, x ∈ ZE holds for all c ⊆ RE ?
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Motivating question Is there a polynomial size system Ax ≤ b such that min {hc, xi : x ∈ TSPn } = min hc, xi : Ax ≤ b, x ∈ ZE holds for all c ⊆ RE ?
Definition For a set X ⊆ Zd , a polyhedron P ⊆ Rd is called a relaxation for X if P ∩ Zd = X .
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Motivating question Is there a polynomial size system Ax ≤ b such that min {hc, xi : x ∈ TSPn } = min hc, xi : Ax ≤ b, x ∈ ZE holds for all c ⊆ RE ?
Definition For a set X ⊆ Zd , a polyhedron P ⊆ Rd is called a relaxation for X if P ∩ Zd = X . The relaxation complexity of X is the smallest number of facets of any relaxation for X . (short: rc(X )) Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Motivating question Is rc(TSPn ) polynomial in n?
Definition For a set X ⊆ Zd , a polyhedron P ⊆ Rd is called a relaxation for X if P ∩ Zd = X . The relaxation complexity of X is the smallest number of facets of any relaxation for X . (short: rc(X ))
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn Rnsub has exponentially many facets
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn Rnsub has exponentially many facets But: We can optimize over Rnsub in polynomial time!
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn Rnsub has exponentially many facets But: We can optimize over Rnsub in polynomial time!
Theorem For any family of sets Xd ⊆ {0, 1}d with the property that the problem “Given x ∈ Zd , is x ∈ Xd ?” is in P,
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn Rnsub has exponentially many facets But: We can optimize over Rnsub in polynomial time!
Theorem For any family of sets Xd ⊆ {0, 1}d with the property that the problem “Given x ∈ Zd , is x ∈ Xd ?” is in P, there are relaxations Rd for Xd and we can optimize over Rd in polynomial time. Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Cube
Basics
Lower bounds
Conclusion
Clearly: rc({0, 1}d ) ≤ 2d
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
5 / 15
Motivation
Cube
Basics
Lower bounds
Conclusion
Clearly: rc({0, 1}d ) ≤ 2d
Theorem
rc({0, 1}d ) = d + 1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
5 / 15
Motivation
Cube
Basics
Lower bounds
Conclusion
Clearly: rc({0, 1}d ) ≤ 2d
Theorem
rc({0, 1}d ) = d + 1
{0, 1}d =
x ∈ Zd : 0 ≤ x1 +
d X 1 xi 2i i=2
xk ≤ 1 +
d X 1 x for k = 1, . . . , d i 2i
i=k+1 Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
5 / 15
Lower bounds
Motivation
Basics
Lower bounds
Conclusion
Definition Let X ⊆ Zd . A set H ⊆ Zd \ X is called a hiding set for X if: H ⊆ aff(X ) For any two distinct points a, b ∈ H: conv({a, b}) ∩ conv(X ) 6= ∅
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
6 / 15
Motivation
Lower bounds
Basics
Lower bounds
Conclusion
Definition Let X ⊆ Zd . A set H ⊆ Zd \ X is called a hiding set for X if: H ⊆ aff(X ) For any two distinct points a, b ∈ H: conv({a, b}) ∩ conv(X ) 6= ∅
Proposition H hiding set for X Stefan Weltge
⇒ rc(X ) ≥ |H|
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
6 / 15
Hiding set for TSP
Stefan Weltge
Motivation
Basics
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Lower bounds
Conclusion
Aussois, 2014
7 / 15
Proof strategy
Stefan Weltge
Motivation
Basics
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Lower bounds
Conclusion
Aussois, 2014
8 / 15
Motivation
Proof strategy
1 2·
Basics
Lower bounds
Conclusion
+ 12 · =
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
8 / 15
Motivation
Proof strategy
1 2·
Basics
Lower bounds
Conclusion
+ 12 · =
1 2·
Stefan Weltge
+ 12 ·
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
8 / 15
Motivation
Basics
Lower bounds
Conclusion
Hiding set for TSP (2)
1 2
·
+
1 2
·
= Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
9 / 15
Motivation
Basics
Lower bounds
Conclusion
Hiding set for TSP (2)
1 2
·
+
1 2
·
= Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
9 / 15
Motivation
Basics
Lower bounds
Conclusion
Hiding set for TSP (2)
1 2
·
+
1 2
Stefan Weltge
·
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
10 / 15
Motivation
Basics
Lower bounds
Conclusion
Hiding set for TSP (2)
1 2
·
+
1 2
·
∈ conv(TSPn ) Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
10 / 15
Results for TSP
Motivation
Basics
Lower bounds
Conclusion
Theorem The asymptotical growth of rc(TSPn ) is 2Θ(n) .
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
11 / 15
Results for TSP
Motivation
Basics
Lower bounds
Conclusion
Theorem The asymptotical growth of rc(TSPn ) is 2Θ(n) . The subtour elimination relaxation is asymptotically smallest possible.
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
11 / 15
Results for TSP
Motivation
Basics
Lower bounds
Conclusion
Theorem The asymptotical growth of rc(TSPn ) is 2Θ(n) . The subtour elimination relaxation is asymptotically smallest possible.
Theorem The asymptotical growth of rc(?) is 2Θ(n) , where ? is the set of characteristic vectors of ... spanning trees arborescences forests branchings connected edge sets for the complete (undirected/directed) graph on n nodes.
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
11 / 15
Motivation
Further Results
( EVENn :=
n
n X
n
i=1 n X
x ∈ {0, 1} : (
ODDn :=
x ∈ {0, 1} :
Basics
Lower bounds
Conclusion
) xi even ) xi odd
i=1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
12 / 15
Motivation
Further Results
( EVENn :=
n
n X
n
i=1 n X
x ∈ {0, 1} : (
ODDn :=
x ∈ {0, 1} :
Basics
Lower bounds
Conclusion
) xi even ) xi odd
i=1
Theorem (Jeroslow 1973) ODDn is a hiding set for EVENn .
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
12 / 15
Motivation
Further Results
( EVENn :=
n
n X
n
i=1 n X
x ∈ {0, 1} : (
ODDn :=
x ∈ {0, 1} :
Basics
Lower bounds
Conclusion
) xi even ) xi odd
i=1
Theorem (Jeroslow 1973) ODDn is a hiding set for EVENn .
Corollary rc(EVENn ) ≥ 2n−1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
12 / 15
Motivation
Basics
Lower bounds
Conclusion
Further Results (2) For T ⊆ V with n, |T | even: T -JOINSn := set of characteristic vectors of T -joins in (V , E )
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
13 / 15
Motivation
Basics
Lower bounds
Conclusion
Further Results (2) For T ⊆ V with n, |T | even: T -JOINSn := set of characteristic vectors of T -joins in (V , E )
Theorem n
rc(T -JOINSn ) ≥ 2 4 −1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
13 / 15
Motivation
Basics
Lower bounds
Conclusion
Further Results (2) For T ⊆ V with n, |T | even: T -JOINSn := set of characteristic vectors of T -joins in (V , E )
Theorem n
rc(T -JOINSn ) ≥ 2 4 −1
DIFFm,n := x ∈ {0, 1}m×n : x has pairwise distinct rows
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
13 / 15
Motivation
Basics
Lower bounds
Conclusion
Further Results (2) For T ⊆ V with n, |T | even: T -JOINSn := set of characteristic vectors of T -joins in (V , E )
Theorem n
rc(T -JOINSn ) ≥ 2 4 −1
DIFFm,n := x ∈ {0, 1}m×n : x has pairwise distinct rows
Theorem rc(DIFF2,n ) ≥ 2n
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
13 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: - Acyclicity
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: - Acyclicity - Connectivity
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: - Acyclicity - Connectivity - Parity
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: -
Stefan Weltge
Acyclicity Connectivity Parity Distinctness
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: -
Acyclicity Connectivity Parity Distinctness
Everything becomes easy with additional variables!
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: -
Acyclicity Connectivity Parity Distinctness
Everything becomes easy with additional variables!
Projection is a powerful tool!
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Let ∆d := {O, e1 , . . . , ed }
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Let ∆d := {O, e1 , . . . , ed } Clearly: rc(∆d ) ≤ d + 1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Let ∆d := {O, e1 , . . . , ed } Clearly: rc(∆d ) ≤ d + 1
Is there a polyhedron P with less than d + 1 facets such that P ∩ Zd = ∆d ?
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Let ∆d := {O, e1 , . . . , ed } Clearly: rc(∆d ) ≤ d + 1
Is there a polyhedron P with less than d + 1 facets such that P ∩ Zd = ∆d ? (If yes, P must be unbounded and hence irrational!)
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
Aussois, 2014
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once.
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T )
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn }
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn } = min {hc, xi : x ∈ conv(TSPn )}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn } = min {hc, xi : x ∈ conv(TSPn )} = min {hc, xi : Ax ≤ b}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn } = min {hc, xi : x ∈ conv(TSPn )} = min {hc, xi : Ax ≤ b} Ax ≤ b is very large and complicated
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Motivation
Basics
Lower bounds
Conclusion
Traveling Salesman Problem Find a shortest cycle in a complete graph (V , E ) that hits each node exactly once. Natural approach: (binary) variable xij for each edge {i, j} identify each tour T with its characteristic vector χ(T ) ∈ {0, 1}E (χ(T )e = 1 ⇐⇒ e ∈ T ) TSPn := {χ(T ) : T hamiltonian tour} given weights cij , we would like to solve: min {hc, xi : x ∈ TSPn } = min {hc, xi : x ∈ conv(TSPn )} = min {hc, xi : Ax ≤ b} Ax ≤ b is very large and complicated Why not use additional variables? Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
1 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2)
min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b}
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2) Theorem (Fiorini, Pokutta et al. 2012) There is no polynomial size system Ax + By ≤ b such that min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b} holds for all c ∈ RE .
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2) Theorem (Fiorini, Pokutta et al. 2012) There is no polynomial size system Ax + By ≤ b such that min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b} holds for all c ∈ RE . But there are several polynomial size integer programming formulations of the form min hc, xi : Ax + By ≤ b, x ∈ {0, 1}E , y ∈ Zk for solving the TSP (Miller-Tucker-Zemlin, flow based, ...)
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2) Theorem (Fiorini, Pokutta et al. 2012) There is no polynomial size system Ax + By ≤ b such that min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b} holds for all c ∈ RE . But there are several polynomial size integer programming formulations of the form min hc, xi : Ax + By ≤ b, x ∈ {0, 1}E , y ∈ Zk for solving the TSP (Miller-Tucker-Zemlin, flow based, ...)
Theorem For any family of sets Xd ⊆ {0, 1}d with the property that the problem “Given x ∈ Zd , is x ∈ Xd ?” is in NP,
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (2) Theorem (Fiorini, Pokutta et al. 2012) There is no polynomial size system Ax + By ≤ b such that min {hc, xi : x ∈ TSPn } = min {hc, xi : Ax + By ≤ b} holds for all c ∈ RE . But there are several polynomial size integer programming formulations of the form min hc, xi : Ax + By ≤ b, x ∈ {0, 1}E , y ∈ Zk for solving the TSP (Miller-Tucker-Zemlin, flow based, ...)
Theorem For any family of sets Xd ⊆ {0, 1}d with the property that the problem “Given x ∈ Zd , is x ∈ Xd ?” is in NP, we have polynomial size systems Ax + By ≤ b such that Xd = x : Ax + By ≤ b, x ∈ {0, 1}d , y ∈ Zk . Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
2 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Motivating question Is there a polynomial size system Ax ≤ b such that min {hc, xi : x ∈ TSPn } = min hc, xi : Ax ≤ b, x ∈ ZE holds for all c ⊆ RE ?
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Motivating question Is there a polynomial size system Ax ≤ b such that min {hc, xi : x ∈ TSPn } = min hc, xi : Ax ≤ b, x ∈ ZE holds for all c ⊆ RE ?
Definition For a set X ⊆ Zd , a polyhedron P ⊆ Rd is called a relaxation for X if P ∩ Zd = X .
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Motivating question Is there a polynomial size system Ax ≤ b such that min {hc, xi : x ∈ TSPn } = min hc, xi : Ax ≤ b, x ∈ ZE holds for all c ⊆ RE ?
Definition For a set X ⊆ Zd , a polyhedron P ⊆ Rd is called a relaxation for X if P ∩ Zd = X . The relaxation complexity of X is the smallest number of facets of any relaxation for X . (short: rc(X )) Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Basics
Lower bounds
Conclusion
Motivation (3) Observation All such polynomial size formulations for the TSP use auxiliary variables.
Motivating question Is rc(TSPn ) polynomial in n?
Definition For a set X ⊆ Zd , a polyhedron P ⊆ Rd is called a relaxation for X if P ∩ Zd = X . The relaxation complexity of X is the smallest number of facets of any relaxation for X . (short: rc(X ))
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
3 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn Rnsub has exponentially many facets
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn Rnsub has exponentially many facets But: We can optimize over Rnsub in polynomial time!
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn Rnsub has exponentially many facets But: We can optimize over Rnsub in polynomial time!
Theorem For any family of sets Xd ⊆ {0, 1}d with the property that the problem “Given x ∈ Zd , is x ∈ Xd ?” is in P,
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Separation
Basics
Lower bounds
Conclusion
Subtour elimination polytope Rnsub := x ∈ [0, 1]E : x(δ(v )) = 2 for all v ∈ V x(δ(S)) ≥ 2 for all ∅ = 6 S ⊂V
Rnsub is a relaxation for TSPn Rnsub has exponentially many facets But: We can optimize over Rnsub in polynomial time!
Theorem For any family of sets Xd ⊆ {0, 1}d with the property that the problem “Given x ∈ Zd , is x ∈ Xd ?” is in P, there are relaxations Rd for Xd and we can optimize over Rd in polynomial time. Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
4 / 15
Motivation
Cube
Basics
Lower bounds
Conclusion
Clearly: rc({0, 1}d ) ≤ 2d
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
5 / 15
Motivation
Cube
Basics
Lower bounds
Conclusion
Clearly: rc({0, 1}d ) ≤ 2d
Theorem
rc({0, 1}d ) = d + 1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
5 / 15
Motivation
Cube
Basics
Lower bounds
Conclusion
Clearly: rc({0, 1}d ) ≤ 2d
Theorem
rc({0, 1}d ) = d + 1
{0, 1}d =
x ∈ Zd : 0 ≤ x1 +
d X 1 xi 2i i=2
xk ≤ 1 +
d X 1 x for k = 1, . . . , d i 2i
i=k+1 Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
5 / 15
Lower bounds
Motivation
Basics
Lower bounds
Conclusion
Definition Let X ⊆ Zd . A set H ⊆ Zd \ X is called a hiding set for X if: H ⊆ aff(X ) For any two distinct points a, b ∈ H: conv({a, b}) ∩ conv(X ) 6= ∅
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
6 / 15
Motivation
Lower bounds
Basics
Lower bounds
Conclusion
Definition Let X ⊆ Zd . A set H ⊆ Zd \ X is called a hiding set for X if: H ⊆ aff(X ) For any two distinct points a, b ∈ H: conv({a, b}) ∩ conv(X ) 6= ∅
Proposition H hiding set for X Stefan Weltge
⇒ rc(X ) ≥ |H|
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
6 / 15
Hiding set for TSP
Stefan Weltge
Motivation
Basics
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Lower bounds
Conclusion
Aussois, 2014
7 / 15
Proof strategy
Stefan Weltge
Motivation
Basics
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Lower bounds
Conclusion
Aussois, 2014
8 / 15
Motivation
Proof strategy
1 2·
Basics
Lower bounds
Conclusion
+ 12 · =
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
8 / 15
Motivation
Proof strategy
1 2·
Basics
Lower bounds
Conclusion
+ 12 · =
1 2·
Stefan Weltge
+ 12 ·
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
8 / 15
Motivation
Basics
Lower bounds
Conclusion
Hiding set for TSP (2)
1 2
·
+
1 2
·
= Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
9 / 15
Motivation
Basics
Lower bounds
Conclusion
Hiding set for TSP (2)
1 2
·
+
1 2
·
= Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
9 / 15
Motivation
Basics
Lower bounds
Conclusion
Hiding set for TSP (2)
1 2
·
+
1 2
Stefan Weltge
·
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
10 / 15
Motivation
Basics
Lower bounds
Conclusion
Hiding set for TSP (2)
1 2
·
+
1 2
·
∈ conv(TSPn ) Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
10 / 15
Results for TSP
Motivation
Basics
Lower bounds
Conclusion
Theorem The asymptotical growth of rc(TSPn ) is 2Θ(n) .
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
11 / 15
Results for TSP
Motivation
Basics
Lower bounds
Conclusion
Theorem The asymptotical growth of rc(TSPn ) is 2Θ(n) . The subtour elimination relaxation is asymptotically smallest possible.
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
11 / 15
Results for TSP
Motivation
Basics
Lower bounds
Conclusion
Theorem The asymptotical growth of rc(TSPn ) is 2Θ(n) . The subtour elimination relaxation is asymptotically smallest possible.
Theorem The asymptotical growth of rc(?) is 2Θ(n) , where ? is the set of characteristic vectors of ... spanning trees arborescences forests branchings connected edge sets for the complete (undirected/directed) graph on n nodes.
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
11 / 15
Motivation
Further Results
( EVENn :=
n
n X
n
i=1 n X
x ∈ {0, 1} : (
ODDn :=
x ∈ {0, 1} :
Basics
Lower bounds
Conclusion
) xi even ) xi odd
i=1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
12 / 15
Motivation
Further Results
( EVENn :=
n
n X
n
i=1 n X
x ∈ {0, 1} : (
ODDn :=
x ∈ {0, 1} :
Basics
Lower bounds
Conclusion
) xi even ) xi odd
i=1
Theorem (Jeroslow 1973) ODDn is a hiding set for EVENn .
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
12 / 15
Motivation
Further Results
( EVENn :=
n
n X
n
i=1 n X
x ∈ {0, 1} : (
ODDn :=
x ∈ {0, 1} :
Basics
Lower bounds
Conclusion
) xi even ) xi odd
i=1
Theorem (Jeroslow 1973) ODDn is a hiding set for EVENn .
Corollary rc(EVENn ) ≥ 2n−1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
12 / 15
Motivation
Basics
Lower bounds
Conclusion
Further Results (2) For T ⊆ V with n, |T | even: T -JOINSn := set of characteristic vectors of T -joins in (V , E )
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
13 / 15
Motivation
Basics
Lower bounds
Conclusion
Further Results (2) For T ⊆ V with n, |T | even: T -JOINSn := set of characteristic vectors of T -joins in (V , E )
Theorem n
rc(T -JOINSn ) ≥ 2 4 −1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
13 / 15
Motivation
Basics
Lower bounds
Conclusion
Further Results (2) For T ⊆ V with n, |T | even: T -JOINSn := set of characteristic vectors of T -joins in (V , E )
Theorem n
rc(T -JOINSn ) ≥ 2 4 −1
DIFFm,n := x ∈ {0, 1}m×n : x has pairwise distinct rows
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
13 / 15
Motivation
Basics
Lower bounds
Conclusion
Further Results (2) For T ⊆ V with n, |T | even: T -JOINSn := set of characteristic vectors of T -joins in (V , E )
Theorem n
rc(T -JOINSn ) ≥ 2 4 −1
DIFFm,n := x ∈ {0, 1}m×n : x has pairwise distinct rows
Theorem rc(DIFF2,n ) ≥ 2n
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
13 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: - Acyclicity
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: - Acyclicity - Connectivity
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: - Acyclicity - Connectivity - Parity
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: -
Stefan Weltge
Acyclicity Connectivity Parity Distinctness
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: -
Acyclicity Connectivity Parity Distinctness
Everything becomes easy with additional variables!
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
Motivation
Conclusion
Basics
Lower bounds
Conclusion
Hard to model without additional variables: -
Acyclicity Connectivity Parity Distinctness
Everything becomes easy with additional variables!
Projection is a powerful tool!
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
14 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Let ∆d := {O, e1 , . . . , ed }
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Let ∆d := {O, e1 , . . . , ed } Clearly: rc(∆d ) ≤ d + 1
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Let ∆d := {O, e1 , . . . , ed } Clearly: rc(∆d ) ≤ d + 1
Is there a polyhedron P with less than d + 1 facets such that P ∩ Zd = ∆d ?
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
My Favorite Open Question
Motivation
Basics
Lower bounds
Conclusion
What is the relaxation complexity of the standard simplex’ vertices?
Let ∆d := {O, e1 , . . . , ed } Clearly: rc(∆d ) ≤ d + 1
Is there a polyhedron P with less than d + 1 facets such that P ∩ Zd = ∆d ? (If yes, P must be unbounded and hence irrational!)
Stefan Weltge
Lower Bounds on the Sizes of Integer Programs w/o Additional Variables
Aussois, 2014
15 / 15
