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

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 1 , . . . , v n } E = { E 1 , . . . , E m } fractional cover y = ( y 1 , . . . , y n ) ∈ R n putting weight y i on vertex v i , s.t. ( ∀ i ∈ [ n ])( y i ≥ 0 ) (: weights non-negative :) ( ∀ j ∈ [ m ])( � i : v i ∈ E j y i ≥ 1 ) (: total weight on each edge ≥ 1 :) τ ∗ ( H ) = min { � n i = 1 y i | constraints } τ ( H ) ≥ τ ∗ ( H ) Integral optimum: CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional matching hypergraph H = ( V , E ) V = { v 1 , . . . , v n } E = { E 1 , . . . , E m } fractional matching x = ( x 1 , . . . , x m ) ∈ R m putting weight x j on edge E j , s.t. ( ∀ j ∈ [ m ])( x j ≥ 0 ) (: weights non-negative :) ( ∀ i ∈ [ n ])( � j : v i ∈ E j x j ≤ 1 ) (: total weight on each vertex ≤ 1 :) ν ∗ ( H ) = max { � m j = 1 x j | constraints } ν ( H ) ≤ ν ∗ ( H ) = τ ∗ ( H ) Integral optimum: CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Greedy cover algorithm hypergraph H = ( V , E ) V = { v 1 , . . . , v n } , E = { E 1 , . . . , E m } T ← ∅ while E � ∅ do v ← argmax { deg ( v ) | v ∈ V } (: max degree :) E ← E \ { edges incident with v } T ← T ∪ { x } end ( while ) τ greedy := | T | (: size of greedy cover :) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover size of greedy cover τ greedy ν ≤ ν ∗ = τ ∗ ≤ τ ≤ τ greedy CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover size of greedy cover τ greedy ν ≤ ν ∗ = τ ∗ ≤ τ ≤ τ greedy Theorem (Lovász 1975) τ greedy < ( 1 + ln deg max ) · τ ∗ Corollary: Integrality gap τ/τ ∗ < 1 + ln deg max Corollary: Approximation ratio τ greedy /τ < 1 + ln deg max CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof k -matching: set of edges s.t. ∀ vertex covered ≤ k times ν k max size of k -matching ν = ν 1 i.e. subhypergraph with deg max ≤ k CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof k -matching: set of edges s.t. ∀ vertex covered ≤ k times ν k max size of k -matching ν = ν 1 i.e. subhypergraph with deg max ≤ k ν ∗ ≥ ν k / k CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof k -matching: set of edges s.t. ∀ vertex covered ≤ k times ν k max size of k -matching ν = ν 1 i.e. subhypergraph with deg max ≤ k ν ∗ ≥ ν k / k why? CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof k -matching: set of edges s.t. ∀ vertex covered ≤ k times ν k max size of k -matching ν = ν 1 i.e. subhypergraph with deg max ≤ k ν ∗ ≥ ν k / 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

Fractional vs. greedy cover: proof k -matching: set of edges s.t. ∀ vertex covered ≤ k times ν k max size of k -matching ν = ν 1 i.e. subhypergraph with deg max ≤ k ν ∗ ≥ ν k / k why? put weight 1 / k on each edge of k -matching total weight on ∀ vertex ≤ 1 Max degree starts at d := deg max then goes down t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) E i set of edges left over after t d + t d − 1 + · · · + t i + 1 rounds CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) E i set of edges left over after t d + t d − 1 + · · · + t i + 1 rounds E i has deg max ≤ i for the first time starts sequence of t i rounds with deg max = i CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) E i set of edges left over after t d + t d − 1 + · · · + t i + 1 rounds E i has deg max ≤ i for the first time starts sequence of t i rounds with deg max = i ( t i = 0 possible; then E i = E i − 1 ) CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) E i set of edges left over after t d + t d − 1 + · · · + t i + 1 rounds E i has deg max ≤ i for the first time starts sequence of t i rounds with deg max = i ( t i = 0 possible; then E i = E i − 1 ) |E i | = i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) E i set of edges left over after t d + t d − 1 + · · · + t i + 1 rounds CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) E i set of edges left over after t d + t d − 1 + · · · + t i + 1 rounds |E i | = i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) E i set of edges left over after t d + t d − 1 + · · · + t i + 1 rounds |E i | = i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 E i i -matching: deg max ≤ i |E i | ≤ ν i ≤ i · ν ∗ ∴ CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) E i set of edges left over after t d + t d − 1 + · · · + t i + 1 rounds |E i | = i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 E i i -matching: deg max ≤ i |E i | ≤ ν i ≤ i · ν ∗ ∴ i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 ≤ i · ν ∗ CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) L i := i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 ≤ i · ν ∗ CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) L i := i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 ≤ i · ν ∗ 1 1 ( d − 1 ) · d L d − 1 + 1 1 K := 1 · 2 L 1 + 2 · 3 L 2 + · · · + d L d CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof t i := # rounds while max degree = i τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) L i := i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 ≤ i · ν ∗ 1 1 ( d − 1 ) · d L d − 1 + 1 1 K := 1 · 2 L 1 + 2 · 3 L 2 + · · · + d L d coeff ( t i ) = � � 1 1 ( d − 1 ) d + 1 1 i i ( i + 1 ) + ( i + 1 )( i + 2 ) + · · · + d CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

A telescoping sum 1 1 ( d − 1 ) d + 1 1 i ( i + 1 ) + ( i + 1 )( i + 2 ) + · · · + d CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

A telescoping sum 1 1 ( d − 1 ) d + 1 1 i ( i + 1 ) + ( i + 1 )( i + 2 ) + · · · + d � 1 1 1 1 d − 1 − 1 1 + 1 � � � � � = i − + i + 1 − + · · · + i + 1 i + 2 d d = 1 i CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) L i := i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 ≤ i · ν ∗ 1 1 ( d − 1 ) · d L d − 1 + 1 1 2 · 3 L 2 + · · · + K := 1 · 2 L 1 + d L d coeff ( t i ) = � � 1 1 ( d − 1 ) d + 1 1 i i ( i + 1 ) + ( i + 1 )( i + 2 ) + · · · + d CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) L i := i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 ≤ i · ν ∗ 1 1 ( d − 1 ) · d L d − 1 + 1 1 K := 1 · 2 L 1 + 2 · 3 L 2 + · · · + d L d coeff ( t i ) = � � 1 1 ( d − 1 ) d + 1 1 = 1 i i ( i + 1 ) + ( i + 1 )( i + 2 ) + · · · + d K = t d + t d − 1 + · · · + d 1 = CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics

Fractional vs. greedy cover: proof τ greedy = t d + t d − 1 + · · · + t 1 (total # rounds) L i := i · t i + ( i − 1 ) · t i − 1 + · · · + 1 · t 1 ≤ i · ν ∗ 1 1 ( d − 1 ) · d L d − 1 + 1 1 K := 1 · 2 L 1 + 2 · 3 L 2 + · · · + d L d coeff ( t i ) = � � 1 1 ( d − 1 ) d + 1 1 = 1 i i ( i + 1 ) + ( i + 1 )( i + 2 ) + · · · + d K = t d + t d − 1 + · · · + d 1 = τ greedy CMSC-27410=Math-28410 ∼ CMSC-3720 Honors Combinatorics