Honors Combinatorics CMSC-27410 = Math-28410 CMSC-37200 Instructor: - - PowerPoint PPT Presentation

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Honors Combinatorics CMSC-27410 = Math-28410 CMSC-37200 Instructor: - - PowerPoint PPT Presentation

Oct 04, 2023 352 likes •1.02k views

Honors Combinatorics CMSC-27410 = Math-28410 CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 4, Tuesday, April 28, 2020 CMSC-27410=Math-28410 CMSC-3720 Honors Combinatorics Todays material Linear Program (LP)

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Honors Combinatorics

CMSC-27410 = Math-28410 ∼ CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 4, Tuesday, April 28, 2020

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Today’s material

Linear Program (LP) Primal/Dual LP , Duality Theorem Integer Linear Program (ILP), integrality gap Hypergraph cover and matching Hypergraph fractional cover and fractional matching Lovász’s greedy vs. fractional cover theorem

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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System of linear equations k linear equations in n unknowns a11x1 + a12x2 + . . . + a1nxn = b1 a21x1 + a22x2 + . . . + a2nxn = b2 . . . . . . . . . ak1x1 + ak2x2 + . . . + aknxn = bk

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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System of linear equations k linear equations in n unknowns a11x1 + a12x2 + . . . + a1nxn = b1 a21x1 + a22x2 + . . . + a2nxn = b2 . . . . . . . . . ak1x1 + ak2x2 + . . . + aknxn = bk

Ax = b

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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System of linear equations

Ax = b

A =               a11 a12 . . . a1n a21 a22 . . . a2n . . . . . . . . . ak1 ak2 . . . akn               x =               x1 x2 . . . xn               b =               b1 b2 . . . bk              

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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System of linear inequalities k linear inequatlities in n unknowns a11x1 + a12x2 + . . . + a1nxn ≤ b1 a21x1 + a22x2 + . . . + a2nxn ≤ b2 . . . . . . . . . ak1x1 + ak2x2 + . . . + aknxn ≤ bk

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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System of linear inequalities k linear inequatlities in n unknowns a11x1 + a12x2 + . . . + a1nxn ≤ b1 a21x1 + a22x2 + . . . + a2nxn ≤ b2 . . . . . . . . . ak1x1 + ak2x2 + . . . + aknxn ≤ bk

Ax ≤ b

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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System of linear inequalities partially ordering the vectors: coordinatewise               u1 u2 . . . uk               ≤               v1 v2 . . . vk               if (∀i)(ui ≤ vi) Same for matrices: A ≤ B if (∀i, j)(aij ≤ bij) DO: A ≤ B ∧ C ≥ 0 =⇒ AC ≤ BC

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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System of linear inequalities

Ax ≤ b feasible system: solution exists

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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System of linear inequalities

Ax ≤ b feasible system: solution exists

Example of infeasible system: x1 ≤ 2 x1 ≥ 3

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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System of linear inequalities

Ax ≤ b feasible system: solution exists

Example of infeasible system: x1 ≤ 2 x1 ≥ 3 equivalently x1 ≤ 2 −x1 ≤ −3

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Linear Program (LP)

Input: A ∈ Rk×n, b ∈ Rk, c ∈ Rn Constraints: Ax ≤ b, x ≥ 0 Objective: max ← c · x

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Linear Program (LP)

Input: A ∈ Rk×n, b ∈ Rk, c ∈ Rn Constraints: Ax ≤ b, x ≥ 0 Objective: max ← c · x max ← {c · x | Ax ≤ b, x ≥ 0} Goal: maximize objective function subject to constraints

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Feasible LP

max ← {c · x | Ax ≤ b, x ≥ 0} feasible solution: x ∈ Rn that satisfies the constraints: Ax ≤ b, x ≥ 0 feasible LP: ∃ feasible solution, i.e., (∃x ∈ Rn)(Ax ≤ b, x ≥ 0)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Linear Program (LP)

Input: A ∈ Rk×n, b ∈ Rk, c ∈ Rk Constraints: Ax ≤ b, x ≥ 0 Objective: max ← c · x

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Linear Program (LP)

Input: A ∈ Rk×n, b ∈ Rk, c ∈ Rk Constraints: Ax ≤ b, x ≥ 0 Objective: max ← c · x max ← {c · x | Ax ≤ b, x ≥ 0}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Linear Program (LP)

Input: A ∈ Rk×n, b ∈ Rk, c ∈ Rk Constraints: Ax ≤ b, x ≥ 0 Objective: max ← c · x max ← {c · x | Ax ≤ b, x ≥ 0} feasible LP: constraints feasible (∃x ∈ Rn)(Ax ≤ b, x ≥ 0)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Primal/Dual Linear Program

INPUT: A ∈ Rk×n, b ∈ Rk, c ∈ Rn UNKNOWNS: primal variables x ∈ Rn, dual variables y ∈ Rk PRIMAL LP: max ← {c · x | Ax ≤ b, x ≥ 0}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Primal/Dual Linear Program

INPUT: A ∈ Rk×n, b ∈ Rk, c ∈ Rn UNKNOWNS: primal variables x ∈ Rn, dual variables y ∈ Rk PRIMAL LP: max ← {c · x | Ax ≤ b, x ≥ 0} DUAL LP: min ← {b · y | ATy ≥ c, y ≥ 0}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Primal/Dual Linear Program INPUT: A ∈ Rk×n, b ∈ Rk, c ∈ Rn PRIMAL: max ← {c · x | Ax ≤ b, x ≥ 0} DUAL: min ← {b · y | ATy ≥ c, y ≥ 0}

  • Lemma. If x0 is a feasible solution to the Primal

and y0 a feasible solution to the Dual then c · x0 ≤ b · y0 Gives upper bound for MAX lower bound for MIN

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Primal/Dual Linear Program PRIMAL: max ← {c · x | Ax ≤ b, x ≥ 0} DUAL: min ← {b · y | ATy ≥ c, y ≥ 0}

  • Lemma. If x0 is a feasible solution to the Primal

and y0 a feasible solution to the Dual then c · x0 ≤ b · y0

  • Proof. Note: (a) a · b = aTb

(b) (AB)T = BTAT

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Primal/Dual Linear Program PRIMAL: max ← {c · x | Ax ≤ b, x ≥ 0} DUAL: min ← {b · y | ATy ≥ c, y ≥ 0}

  • Lemma. If x0 is a feasible solution to the Primal

and y0 a feasible solution to the Dual then c · x0 ≤ b · y0

  • Proof. Note: (a) a · b = aTb

(b) (AB)T = BTAT yT

0 (Ax0) ≤ yT 0 b = y0 · b = b · y0

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Primal/Dual Linear Program PRIMAL: max ← {c · x | Ax ≤ b, x ≥ 0} DUAL: min ← {b · y | ATy ≥ c, y ≥ 0}

  • Lemma. If x0 is a feasible solution to the Primal

and y0 a feasible solution to the Dual then c · x0 ≤ b · y0

  • Proof. Note: (a) a · b = aTb

(b) (AB)T = BTAT yT

0 (Ax0) ≤ yT 0 b = y0 · b = b · y0

(yT

0 A)x0 = (ATy0)Tx0 ≥ cTx0 = c · x0

QED

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Duality Theorem of Linear Programming PRIMAL: max ← {c · x | Ax ≤ b, x ≥ 0} DUAL: min ← {b · y | ATy ≥ c, y ≥ 0}

  • Lemma. If x0 is a feasible solution to the Primal

and y0 a feasible solution to the Dual then c · x0 ≤ b · y0 Max PRIMAL Min DUAL

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Duality Theorem of Linear Programming PRIMAL: max ← {c · x | Ax ≤ b, x ≥ 0} DUAL: min ← {b · y | ATy ≥ c, y ≥ 0}

  • Lemma. If x0 is a feasible solution to the Primal

and y0 a feasible solution to the Dual then c · x0 ≤ b · y0 Max PRIMAL ≤ Min DUAL

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Duality Theorem of Linear Programming PRIMAL: max ← {c · x | Ax ≤ b, x ≥ 0} DUAL: min ← {b · y | ATy ≥ c, y ≥ 0}

  • Lemma. If x0 is a feasible solution to the Primal

and y0 a feasible solution to the Dual then c · x0 ≤ b · y0 Max PRIMAL ≤ Min DUAL LP Duality Theorem If Primal feasible and range of obj functn bounded from above then Dual feasible and the two attain equal optima: (∃ feasible x ∈ Rn, y ∈ Rk)(c · x = b · y)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Duality Theorem of Linear Programming PRIMAL: max ← {c · x | Ax ≤ b, x ≥ 0} DUAL: min ← {b · y | ATy ≥ c, y ≥ 0}

  • Lemma. If x0 is a feasible solution to the Primal

and y0 a feasible solution to the Dual then c · x0 ≤ b · y0 Max PRIMAL ≤ Min DUAL LP Duality Theorem If Primal feasible and range of obj functn bounded from above then Dual feasible and the two attain equal optima: (∃ feasible x ∈ Rn, y ∈ Rk)(c · x = b · y) Max PRIMAL = Min DUAL

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Program checking Untrusted party wishes to sell us a cloud service to solve huge LPs Money-back Guarantee: gives optimal solution whenever one exists Can we catch them at cheating without running a trusted LP solver on our problems? We can check that the solution they bring is

  • feasible. Can we check it is optimal?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Program checking Untrusted party wishes to sell us a cloud service to solve huge LPs Money-back Guarantee: gives optimal solution whenever one exists Can we catch them at cheating without running a trusted LP solver on our problems? We can check that the solution they bring is

  • feasible. Can we check it is optimal?

Run it on the dual

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Program checking Untrusted party wishes to sell us a cloud service to solve huge LPs Money-back Guarantee: gives optimal solution whenever one exists Can we catch them at cheating without running a trusted LP solver on our problems? We can check that the solution they bring is

  • feasible. Can we check it is optimal?

Run it on the dual

  • ptimum proposed by computationally superior

but untrused party verifiable

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Integer Linear Program (ILP) LP: max ← {c · x | x ∈ Rn, Ax ≤ b, x ≥ 0}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Integer Linear Program (ILP) LP: max ← {c · x | x ∈ Rn, Ax ≤ b, x ≥ 0} ILP: max ← {c · x | x ∈ Zn, Ax ≤ b, x ≥ 0}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Integer Linear Program (ILP) LP: max ← {c · x | x ∈ Rn, Ax ≤ b, x ≥ 0} ILP: max ← {c · x | x ∈ Zn, Ax ≤ b, x ≥ 0} OPT(ILP) ≤ OPT(LP)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Integer Linear Program (ILP) LP: max ← {c · x | x ∈ Rn, Ax ≤ b, x ≥ 0} ILP: max ← {c · x | x ∈ Zn, Ax ≤ b, x ≥ 0} OPT(ILP) ≤ OPT(LP) Integrality gap: OPT(LP)/ OPT(ILP)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Integer Linear Program (ILP) LP: max ← {c · x | x ∈ Rn, Ax ≤ b, x ≥ 0} ILP: max ← {c · x | x ∈ Zn, Ax ≤ b, x ≥ 0} OPT(ILP) ≤ OPT(LP) Integrality gap: OPT(LP)/ OPT(ILP) If minimization problem: OPT(ILP)/ OPT(LP)

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Hypergraphs Hypergraph H = (V, E) E ⊆ P(V) V: vertices E: edges V = {v1, . . . , vn} E = {E1, . . . , Em} n × m incidence matrix M = (mij) mij =        1 if vi ∈ Ej if vi Ej Columns: incidence vectors of edges

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Matchings and covers Hypergraph H = (V, E) E ⊆ P(V) V: vertices E: edges matching: set of disjoint edges matching number ν(H): max # disjoint edges cover: set C ⊆ V that hits every edge (“hitting set”): (∀E ∈ E)(C ∩ E ∅) covering number τ(H): min cover size Both quantities NP-hard ∴ not believed to be computable in polynomial time

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Matchings and covers Observation ν ≤ τ

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Matchings and covers Observation ν ≤ τ max ≤ min

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Matchings and covers Observation ν ≤ τ max ≤ min Proof: E1, . . . , Eν max matching =⇒ C ∩ Ei ∅ disjoint sets =⇒ |C| ≥ ν QED

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Matchings and covers Observation ν ≤ τ max ≤ min Proof: E1, . . . , Eν max matching =⇒ C ∩ Ei ∅ disjoint sets =⇒ |C| ≥ ν QED Duality?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Matchings and covers Observation ν ≤ τ max ≤ min Proof: E1, . . . , Eν max matching =⇒ C ∩ Ei ∅ disjoint sets =⇒ |C| ≥ ν QED Duality? describe ν, τ by LPs (y1, . . . , yn) – incidence vector of cover yi ∈ {0, 1} must hit E ∈ E – describe this by lin inequality

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Matchings and covers Observation ν ≤ τ max ≤ min Proof: E1, . . . , Eν max matching =⇒ C ∩ Ei ∅ disjoint sets =⇒ |C| ≥ ν QED Duality? describe ν, τ by LPs (y1, . . . , yn) – incidence vector of cover yi ∈ {0, 1} must hit E ∈ E – describe this by lin inequality

  • i∈E

yi ≥ 1 Objective:

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Matchings and covers Observation ν ≤ τ max ≤ min Proof: E1, . . . , Eν max matching =⇒ C ∩ Ei ∅ disjoint sets =⇒ |C| ≥ ν QED Duality? describe ν, τ by LPs (y1, . . . , yn) – incidence vector of cover yi ∈ {0, 1} must hit E ∈ E – describe this by lin inequality

  • i∈E

yi ≥ 1 Objective: min ← n

i=1 yi

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover V = {v1, . . . , vn} E = {E1, . . . , Em} Constraints: yi ≥ 0 (i ∈ [n])

  • i:vi∈Ej

yi ≥ 1 (j ∈ [m]) Objective: min ← n

i=1 yi

What is the matrix of this LP?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover V = {v1, . . . , vn} E = {E1, . . . , Em} Constraints: yi ≥ 0 (i ∈ [n])

  • i:vi∈Ej

yi ≥ 1 (j ∈ [m]) Objective: min ← n

i=1 yi

What is the matrix of this LP?

  • i:vi∈Ej

yi = ej · y = eT

j y

where ej: incidence vector of Ej, i.e., j-th column

  • f incidence matrix M

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover Incidence matrix: M = (e1, . . . , em) (columns) Constrains: eT

j y ≥ 1

(j ∈ [m])

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover Incidence matrix: M = (e1, . . . , em) (columns) Constrains: eT

j y ≥ 1

(j ∈ [m])

MTy ≥ 1m all-ones vector

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover Incidence matrix: M = (e1, . . . , em) (columns) Constrains: eT

j y ≥ 1

(j ∈ [m])

MTy ≥ 1m all-ones vector So the matrix of the LP is

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover Incidence matrix: M = (e1, . . . , em) (columns) Constrains: eT

j y ≥ 1

(j ∈ [m])

MTy ≥ 1m all-ones vector So the matrix of the LP is MT Right-hand side b = 1m Objective vectors c =

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover Incidence matrix: M = (e1, . . . , em) (columns) Constrains: eT

j y ≥ 1

(j ∈ [m])

MTy ≥ 1m all-ones vector So the matrix of the LP is MT Right-hand side b = 1m Objective vectors c = 1n

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover

y = (y1, . . . , yn)T min ← {1T

n y | y ≥ 0, MTy ≥ 1m}

Integral optimum: y ∈ Zn

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover

y = (y1, . . . , yn)T min ← {1T

n y | y ≥ 0, MTy ≥ 1m}

Integral optimum: y ∈ Zn (DO) τ(H) = min{1T

n y | y ∈ Zn, y ≥ 0, MTy ≥ 1m}

fractional cover: y ∈ Rn such that y ≥ 0 and MTy ≥ 1 Optimal fractional cover: τ∗(H) = min{1T

n y | y ∈ Rn, y ≥ 0, MTy ≥ 1m}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover

τ(H) = min{1T

n y | y ∈ Zn, y ≥ 0, MTy ≥ 1m}

τ∗(H) = min{1T

n y | y ∈ Rn, y ≥ 0, MTy ≥ 1m}

Which is bigger?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover

τ(H) = min{1T

n y | y ∈ Zn, y ≥ 0, MTy ≥ 1m}

τ∗(H) = min{1T

n y | y ∈ Rn, y ≥ 0, MTy ≥ 1m}

Which is bigger? τ∗ ≤ τ DUAL: max{1T

mx | x ∈ Zm, x ≥ 0, Mx ≤ 1n}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover

τ(H) = min{1T

n y | y ∈ Zn, y ≥ 0, MTy ≥ 1m}

τ∗(H) = min{1T

n y | y ∈ Rn, y ≥ 0, MTy ≥ 1m}

Which is bigger? τ∗ ≤ τ DUAL: ν(H) = max{1T

mx | x ∈ Zm, x ≥ 0, Mx ≤ 1n}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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LP for min cover

τ(H) = min{1T

n y | y ∈ Zn, y ≥ 0, MTy ≥ 1m}

τ∗(H) = min{1T

n y | y ∈ Rn, y ≥ 0, MTy ≥ 1m}

Which is bigger? τ∗ ≤ τ DUAL: ν(H) = max{1T

mx | x ∈ Zm, x ≥ 0, Mx ≤ 1n}

ν∗(H) = max{1T

mx | x ∈ Zm, x ≥ 0, Mx ≤ 1n}

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Linear relaxation

τ∗

  • ptimal fractional cover

ν∗

  • ptimal fractional matching

ν∗ = τ∗ Why?

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Linear relaxation

τ∗

  • ptimal fractional cover

ν∗

  • ptimal fractional matching

ν∗ = τ∗ Why? LP Duality ν ≤ ν∗ = τ∗ ≤ τ

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Linear relaxation

τ∗

  • ptimal fractional cover

ν∗

  • ptimal fractional matching

ν ≤ ν∗ = τ∗ ≤ τ Theorem (Lovász) τ < (1 + ln m)τ∗ In fact, Greedy Cover Algorithm produces cover of size < (1 + ln m)τ∗

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Integrality gap, approximation ratio

Theorem (Lovász) τ < (1 + ln m)τ∗ In fact, Greedy Cover Algorithm produces cover of size < (1 + ln m)τ∗ τ∗ ≤ τ ≤ greedy cover lower and upper bounds via efficient algorithms

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Integrality gap, approximation ratio

τ∗ ≤ τ ≤ greedy cover Lovász: greedy cover < (1 + ln m)τ∗

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Integrality gap, approximation ratio

τ∗ ≤ τ ≤ greedy cover Lovász: greedy cover < (1 + ln m)τ∗ Integrality gap: τ/τ∗ < 1 + ln m

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Integrality gap, approximation ratio

τ∗ ≤ τ ≤ greedy cover Lovász: greedy cover < (1 + ln m)τ∗ Integrality gap: τ/τ∗ < 1 + ln m Approximation ratio: greedy cover/τ < 1 + ln m

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics

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Lovász’s Fractional Cover Theorem (1975)

Lovász’s actual upper bound is even stronger: Theorem (greedy vs. fractional cover) greedy cover < (1 + ln degmax)τ∗ where degmax = maxv∈V deg(v) maximum degree

replace ln m by ln degmax in results stated

CMSC-27410=Math-28410∼CMSC-3720 Honors Combinatorics