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

Nov 14, 2023 419 likes •753 views

Honors Combinatorics CMSC-27410 = Math-28410 CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 5, Thursday, May 7, 2020 CMSC-27410=Math-28410 CMSC-3720 Honors Combinatorics Fractional cover hypergraph H = ( V , E ) V = { v

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Honors Combinatorics

CMSC-27410 = Math-28410 ∼ CMSC-37200 Instructor: Laszlo Babai University of Chicago Week 5, Thursday, May 7, 2020

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

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Fractional cover hypergraph H = (V, E) V = {v1, . . . , vn} E = {E1, . . . , Em} fractional cover y = (y1, . . . , yn) ∈ Rn putting weight yi on vertex vi, s.t. (∀i ∈ [n])(yi ≥ 0) (: weights non-negative :) (∀j ∈ [m])(

i:vi∈Ej yi ≥ 1)

(: total weight on each edge ≥ 1 :) τ∗(H) = min{n

i=1 yi | constraints }

Integral optimum: τ(H) ≥ τ∗(H)

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

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Fractional matching hypergraph H = (V, E) V = {v1, . . . , vn} E = {E1, . . . , Em} fractional matching x = (x1, . . . , xm) ∈ Rm putting weight xj on edge Ej, s.t. (∀j ∈ [m])(xj ≥ 0) (: weights non-negative :) (∀i ∈ [n])(

j:vi∈Ej xj ≤ 1)

(: total weight on each vertex ≤ 1 :) ν∗(H) = max{m

j=1 xj | constraints }

Integral optimum: ν(H) ≤ ν∗(H) = τ∗(H)

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

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Greedy cover algorithm hypergraph H = (V, E) V = {v1, . . . , vn}, E = {E1, . . . , Em} T ← ∅ while E ∅ do v ← argmax{deg(v) | v ∈ V} (: max degree :) E ← E \ {edges incident with v} T ← T ∪ {x} end(while)

τgreedy := |T|

(: size of greedy cover :)

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

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Fractional vs. greedy cover τgreedy size of greedy cover ν ≤ ν∗ = τ∗ ≤ τ ≤ τgreedy

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

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Fractional vs. greedy cover τgreedy size of greedy cover ν ≤ ν∗ = τ∗ ≤ τ ≤ τgreedy

Theorem (Lovász 1975) τgreedy < (1 + ln degmax) · τ∗

Corollary: Integrality gap τ/τ∗ < 1 + ln degmax Corollary: Approximation ratio τgreedy/τ < 1 + ln degmax

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

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Fractional vs. greedy cover: proof k-matching: set of edges s.t. ∀ vertex covered ≤ k times νk max size of k-matching ν = ν1 i.e. subhypergraph with degmax ≤ k

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

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Fractional vs. greedy cover: proof k-matching: set of edges s.t. ∀ vertex covered ≤ k times νk max size of k-matching ν = ν1 i.e. subhypergraph with degmax ≤ k ν∗ ≥ νk/k

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

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Fractional vs. greedy cover: proof k-matching: set of edges s.t. ∀ vertex covered ≤ k times νk max size of k-matching ν = ν1 i.e. subhypergraph with degmax ≤ k ν∗ ≥ νk/k why?

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

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Fractional vs. greedy cover: proof k-matching: set of edges s.t. ∀ vertex covered ≤ k times νk max size of k-matching ν = ν1 i.e. subhypergraph with degmax ≤ k ν∗ ≥ νk/k why? put weight 1/k on each edge of k-matching total weight on ∀ vertex ≤ 1

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

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Fractional vs. greedy cover: proof k-matching: set of edges s.t. ∀ vertex covered ≤ k times νk max size of k-matching ν = ν1 i.e. subhypergraph with degmax ≤ k ν∗ ≥ νk/k why? put weight 1/k on each edge of k-matching total weight on ∀ vertex ≤ 1 Max degree starts at d := degmax then goes down ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds)

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Ei set of edges left over after td + td−1 + · · · + ti+1 rounds

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Ei set of edges left over after td + td−1 + · · · + ti+1 rounds Ei has degmax ≤ i for the first time starts sequence of ti rounds with degmax = i

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

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SLIDE 14

Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Ei set of edges left over after td + td−1 + · · · + ti+1 rounds Ei has degmax ≤ i for the first time starts sequence of ti rounds with degmax = i (ti = 0 possible; then Ei = Ei−1)

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Ei set of edges left over after td + td−1 + · · · + ti+1 rounds Ei has degmax ≤ i for the first time starts sequence of ti rounds with degmax = i (ti = 0 possible; then Ei = Ei−1) |Ei| = i · ti + (i − 1) · ti−1 + · · · + 1 · t1

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Ei set of edges left over after td + td−1 + · · · + ti+1 rounds

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Ei set of edges left over after td + td−1 + · · · + ti+1 rounds |Ei| = i · ti + (i − 1) · ti−1 + · · · + 1 · t1

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Ei set of edges left over after td + td−1 + · · · + ti+1 rounds |Ei| = i · ti + (i − 1) · ti−1 + · · · + 1 · t1 Ei i-matching: degmax ≤ i ∴ |Ei| ≤ νi ≤ i · ν∗

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Ei set of edges left over after td + td−1 + · · · + ti+1 rounds |Ei| = i · ti + (i − 1) · ti−1 + · · · + 1 · t1 Ei i-matching: degmax ≤ i ∴ |Ei| ≤ νi ≤ i · ν∗ i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Li := i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Li := i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

K := 1 1 · 2L1 + 1 2 · 3L2 + · · · + 1 (d − 1) · d Ld−1 + 1 d Ld

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

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Fractional vs. greedy cover: proof ti := # rounds while max degree = i τgreedy = td + td−1 + · · · + t1 (total # rounds) Li := i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

K := 1 1 · 2L1 + 1 2 · 3L2 + · · · + 1 (d − 1) · d Ld−1 + 1 d Ld coeff(ti) =

i

  • 1

i(i + 1) + 1 (i + 1)(i + 2) + · · · + 1 (d − 1)d + 1 d

  • CMSC-27410=Math-28410∼CMSC-3720

Honors Combinatorics

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A telescoping sum

1 i(i + 1) + 1 (i + 1)(i + 2) + · · · + 1 (d − 1)d + 1 d

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

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A telescoping sum

1 i(i + 1) + 1 (i + 1)(i + 2) + · · · + 1 (d − 1)d + 1 d = 1 i − 1 i + 1

  • +
  • 1

i + 1 − 1 i + 2

  • + · · · +
  • 1

d − 1 − 1 d

  • + 1

d

= 1 i

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

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Fractional vs. greedy cover: proof τgreedy = td + td−1 + · · · + t1 (total # rounds) Li := i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

K := 1 1 · 2L1 + 1 2 · 3L2 + · · · + 1 (d − 1) · d Ld−1 + 1 d Ld coeff(ti) =

i

  • 1

i(i + 1) + 1 (i + 1)(i + 2) + · · · + 1 (d − 1)d + 1 d

  • CMSC-27410=Math-28410∼CMSC-3720

Honors Combinatorics

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SLIDE 26

Fractional vs. greedy cover: proof τgreedy = td + td−1 + · · · + t1 (total # rounds) Li := i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

K := 1 1 · 2L1 + 1 2 · 3L2 + · · · + 1 (d − 1) · d Ld−1 + 1 d Ld coeff(ti) =

i

  • 1

i(i + 1) + 1 (i + 1)(i + 2) + · · · + 1 (d − 1)d + 1 d

  • =1

K = td + td−1 + · · · + d1 =

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

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SLIDE 27

Fractional vs. greedy cover: proof τgreedy = td + td−1 + · · · + t1 (total # rounds) Li := i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

K := 1 1 · 2L1 + 1 2 · 3L2 + · · · + 1 (d − 1) · d Ld−1 + 1 d Ld coeff(ti) =

i

  • 1

i(i + 1) + 1 (i + 1)(i + 2) + · · · + 1 (d − 1)d + 1 d

  • =1

K = td + td−1 + · · · + d1 = τgreedy

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

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Fractional vs. greedy cover: proof Li := i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

τgreedy = 1 1 · 2L1 + 1 2 · 3L2 + · · · + 1 (d − 1) · d Ld−1 + 1 d Ld

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

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Fractional vs. greedy cover: proof Li := i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

τgreedy = 1 1 · 2L1 + 1 2 · 3L2 + · · · + 1 (d − 1) · d Ld−1 + 1 d Ld ≤ ν∗ · 1 2 + 1 3 + · · · + 1 d + 1

  • < ν∗ · (1 + ln d) = τ∗ · (1 + ln d)

QED

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

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Fractional vs. greedy cover: proof Li := i · ti + (i − 1) · ti−1 + · · · + 1 · t1 ≤ i · ν∗

τgreedy = 1 1 · 2L1 + 1 2 · 3L2 + · · · + 1 (d − 1) · d Ld−1 + 1 d Ld ≤ ν∗ · 1 2 + 1 3 + · · · + 1 d + 1

  • < ν∗ · (1 + ln d) = τ∗ · (1 + ln d)

QED

Actually, τ∗ · (γ + ln n) where γ = 0.5772 . . . is the Euler–Mascheroni constant

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

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Fractional vs. greedy cover: applications Lovász gives applications to combinatorics, additive number theory, information theory, including Theorem (Erd˝

  • s 1964) ∃ r-uniform hypergraph

with m = O(r 22r) edges that is not 2-colorable. Erd˝

  • s’s proof: probabilistic method

Lovász: simulates coloring by covering

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

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Fractional vs. greedy cover: applications How to construct a non-2-colorable r-uniform hypergraph? Define hypergraph H = (V, E): Pick set A of size TBA (will be |A| = r 2) ❘ V := A

r

  • for B ⊆ A define edge E(B) = B

r

∪ A\B

r

⊆ V ❘ E = {E(B) | B ⊆ A}

❉

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

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Fractional vs. greedy cover: applications How to construct a non-2-colorable r-uniform hypergraph? Define hypergraph H = (V, E): Pick set A of size TBA (will be |A| = r 2) ❘ V := A

r

  • for B ⊆ A define edge E(B) = B

r

∪ A\B

r

⊆ V ❘ E = {E(B) | B ⊆ A} DO Let T ⊆ V. Then K = (A, T) is an r-uniform hypergraph on N vertices. K is not 2-colorable if and only if T is a cover of H.

Uniform weight gives fractional cover . . . ❉

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