Thin Trees and Interlacing Families on Strongly Rayleigh - PowerPoint PPT Presentation

Thin Trees and Interlacing Families on Strongly Rayleigh Distributions Nima Anari / based on joint work with Shayan Oveis Gharan 1 / 25

Exponentially large set . There is always an such that There is always an such that Example 2 Brief Intro to Interlacing Families Example 1 Exponentially large set { a s } s ∈ { 0,1 } n . 2 / 25

Exponentially large set . There is always an such that Example 2 Brief Intro to Interlacing Families Example 1 Exponentially large set { a s } s ∈ { 0,1 } n . There is always an s such that a s ⩽ E [ a s ] 2 / 25

There is always an such that Brief Intro to Interlacing Families Example 1 Exponentially large set { a s } s ∈ { 0,1 } n . There is always an s such that a s ⩽ E [ a s ] Example 2 Exponentially large set { a s b s } s ∈ { 0,1 } n . 2 / 25

Brief Intro to Interlacing Families Example 1 Exponentially large set { a s } s ∈ { 0,1 } n . There is always an s such that a s ⩽ E [ a s ] Example 2 Exponentially large set { a s b s } s ∈ { 0,1 } n . There is always an s such that ⩽ E [ a s ] a s E [ b s ] b s 2 / 25

Brief Intro to Interlacing Families Example 1 E [ a s ] Exponentially large set { a s } s ∈ { 0,1 } n . E [ b s ] There is always an s such that a s ⩽ E [ a s ] E [ a s | s 1 = 0 ] E [ a s | s 1 = 1 ] E [ b s | s 1 = 0 ] E [ b s | s 1 = 1 ] Example 2 . . . . . . Exponentially large set { a s b s } s ∈ { 0,1 } n . E [ a s | s 1 = 0,s 2 = 0 ] E [ a s | s 1 = 0,s 2 = 1 ] There is always an s such that E [ b s | s 1 = 0,s 2 = 0 ] E [ b s | s 1 = 0,s 2 = 1 ] . . . . . . . . ⩽ E [ a s ] a s . . . . E [ b s ] b s 2 / 25

Instead of chasing fractions in the hierarchy, chase roots of polynomials. Interlacing families are the generalization of this idea to polynomials of higher degree [Marcus- Spielman-Srivastava’13] . Polynomials: Let p s ( x ) = b s x − a s . Then b s and root ( p s ) = a s root ( E [ p s ]) = E [ a s ] E [ b s ] . 3 / 25

Interlacing families are the generalization of this idea to polynomials of higher degree [Marcus- Spielman-Srivastava’13] . Polynomials: Let p s ( x ) = b s x − a s . Then b s and root ( p s ) = a s root ( E [ p s ]) = E [ a s ] E [ b s ] . Instead of chasing fractions in the hierarchy, chase roots of polynomials. 3 / 25

Polynomials: Let p s ( x ) = b s x − a s . Then root i ( E [ p s ]) b s and root ( p s ) = a s root ( E [ p s ]) = E [ a s ] E [ b s ] . Instead of chasing root i ( E [ p s | s 1 = 0 ]) root i ( E [ p s | s 1 = 1 ]) fractions in the hierarchy, chase roots . . . . of polynomials. . . Interlacing families are root i ( E [ p s | s 1 = 0, s 2 = 0 ]) root i ( E [ p s | s 1 = 0, s 2 = 1 ]) the generalization of this idea to . . . . . . . . . . . . polynomials of higher degree [Marcus- Spielman-Srivastava’13] . 3 / 25

Polynomials: Let p s ( x ) = b s x − a s . Then root i ( E [ p s ]) b s and root ( p s ) = a s root ( E [ p s ]) = E [ a s ] E [ b s ] . Instead of chasing root i ( E [ p s | s 1 = 0 ]) root i ( E [ p s | s 1 = 1 ]) fractions in the hierarchy, chase roots . . . . of polynomials. . . Interlacing families are root i ( E [ p s | s 1 = 0, s 2 = 0 ]) root i ( E [ p s | s 1 = 0, s 2 = 1 ]) the generalization of this idea to . . . . . . . . . . . . polynomials of higher degree [Marcus- Works as long as all nodes are real-rooted and so Spielman-Srivastava’13] . are all convex combinations of siblings. 3 / 25

Spectral Thinness is -spectrally thin w.r.t. ifg or in other words for every , Thin Tree and Spectrally Thin Tree Thinness S T is α -thin w.r.t. G ifg | T ( S, ¯ S ) | ⩽ α · | G ( S, ¯ S ) | , for every subset of vertices S . ¯ S 4 / 25

Thin Tree and Spectrally Thin Tree Thinness S T is α -thin w.r.t. G ifg | T ( S, ¯ S ) | ⩽ α · | G ( S, ¯ S ) | , for every subset of vertices S . ¯ Spectral Thinness S T is α -spectrally thin w.r.t. G ifg L T ⪯ α · L G , or in other words for every x ∈ R n , x ⊺ L T x ⩽ x ⊺ L G x. 4 / 25

Thin Tree and Spectrally Thin Tree Thinness S T is α -thin w.r.t. G ifg | T ( S, ¯ S ) | ⩽ α · | G ( S, ¯ S ) | , for every subset of vertices S . ¯ Spectral Thinness S T is α -spectrally thin w.r.t. G ifg α -spectrally thin ⇒ α -thin L T ⪯ α · L G , = [on board . . . ] or in other words for every x ∈ R n , x ⊺ L T x ⩽ x ⊺ L G x. 4 / 25

Structure of the Talk 1 Thin Trees Random Spanning Trees Statement Needed from Interlacing Families Well-Conditioning 2 Interlacing Families on Strongly Rayleigh Distributions Statement Needed from Interlacing Families Proof Sketch 5 / 25

Existence of -thin trees implies upper bound for integrality gap of LP relaxation for asymmetric traveling salesman problem. integrality gap already proved for ATSP [Svensson-Tarnawski-Végh’17] , but thin tree remains open. Weighted random spanning trees are -thin [Asadpour-Goemans-Madry-Oveis Gharan-Saberi’10] [on board ]. [A-Oveis Gharan’15] There is always a -thin tree. Thin Tree Conjecture Strong Form of [Goddyn] Every k -edge connected graph has O ( 1/k ) -thin spanning tree. 6 / 25

integrality gap already proved for ATSP [Svensson-Tarnawski-Végh’17] , but thin tree remains open. Weighted random spanning trees are -thin [Asadpour-Goemans-Madry-Oveis Gharan-Saberi’10] [on board ]. [A-Oveis Gharan’15] There is always a -thin tree. Thin Tree Conjecture Strong Form of [Goddyn] Every k -edge connected graph has O ( 1/k ) -thin spanning tree. Existence of f ( n ) /k -thin trees implies O ( f ( n )) upper bound for integrality gap of LP relaxation for asymmetric traveling salesman problem. 6 / 25

Weighted random spanning trees are -thin [Asadpour-Goemans-Madry-Oveis Gharan-Saberi’10] [on board ]. [A-Oveis Gharan’15] There is always a -thin tree. Thin Tree Conjecture Strong Form of [Goddyn] Every k -edge connected graph has O ( 1/k ) -thin spanning tree. Existence of f ( n ) /k -thin trees implies O ( f ( n )) upper bound for integrality gap of LP relaxation for asymmetric traveling salesman problem. O ( 1 ) integrality gap already proved for ATSP [Svensson-Tarnawski-Végh’17] , but thin tree remains open. 6 / 25

[A-Oveis Gharan’15] There is always a -thin tree. Thin Tree Conjecture Strong Form of [Goddyn] Every k -edge connected graph has O ( 1/k ) -thin spanning tree. Existence of f ( n ) /k -thin trees implies O ( f ( n )) upper bound for integrality gap of LP relaxation for asymmetric traveling salesman problem. O ( 1 ) integrality gap already proved for ATSP [Svensson-Tarnawski-Végh’17] , but thin tree remains open. Weighted random spanning trees are O ( log n/ log log n ) /k -thin [Asadpour-Goemans-Madry-Oveis Gharan-Saberi’10] [on board . . . ]. 6 / 25

Thin Tree Conjecture Strong Form of [Goddyn] Every k -edge connected graph has O ( 1/k ) -thin spanning tree. Existence of f ( n ) /k -thin trees implies O ( f ( n )) upper bound for integrality gap of LP relaxation for asymmetric traveling salesman problem. O ( 1 ) integrality gap already proved for ATSP [Svensson-Tarnawski-Végh’17] , but thin tree remains open. Weighted random spanning trees are O ( log n/ log log n ) /k -thin [Asadpour-Goemans-Madry-Oveis Gharan-Saberi’10] [on board . . . ]. [A-Oveis Gharan’15] There is always a log log O ( 1 ) ( n ) /k -thin tree. 6 / 25

Electrical Connectivity Spectrally Thin Tree Spectral Thinness Edge Connectivity | G ( S, ¯ S ) | ⩾ k [Harvey- Olver’14, Marcus- Goal Spielman- ⩾ k Srivastava’14] Thin Tree | T ( S, ¯ S ) | ⩽ α · | G ( S, ¯ S ) | 7 / 25

Spectral Thinness Electrical Connectivity Edge Connectivity Reff ( u, v ) ⩽ 1 | G ( S, ¯ k S ) | ⩾ k [Harvey- u Olver’14, Marcus- Goal Spielman- ⩾ k Srivastava’14] v Thin Tree Spectrally Thin Tree | T ( S, ¯ S ) | ⩽ α · | G ( S, ¯ x ⊺ L T x ⩽ α · x ⊺ L G x S ) | 7 / 25

Spectral Thinness Electrical Connectivity Edge Connectivity Reff ( u, v ) ⩽ 1 | G ( S, ¯ k S ) | ⩾ k [Harvey- u Olver’14, Marcus- Goal Spielman- ⩾ k Srivastava’14] v Thin Tree Spectrally Thin Tree | T ( S, ¯ S ) | ⩽ α · | G ( S, ¯ x ⊺ L T x ⩽ α · x ⊺ L G x S ) | 7 / 25

Problem: Electrical connectivity is needed for the existence of spectrally thin trees. For any : Obstacles Problem: Edge connectivity does not imply electrical connectivity. · · · · · · 8 / 25

Obstacles Problem: Edge connectivity does not imply electrical connectivity. · · · · · · Problem: Electrical connectivity is needed for the existence of spectrally thin trees. For any e = ( u, v ) ∈ T : T b e ⩾ 1 G b e = 1 1 ⩾ Reff T ( u, v ) = e ⊺ L − e L − α · b ⊺ α · Reff G ( u, v ) . 8 / 25

Key Idea : Well-condition the graph spectrally without changing cuts much. 9 / 25

If admits an -spectrally thin tree , then Goal: Find that brings down. Problem 1: How do we ensure does not use any newly added edges? Problem 2: How do we certify is -thin w.r.t. ? Well-Conditioning Scheme Add “graph” H to G ensuring | H ( S, ¯ S ) | ⩽ O ( 1 ) · | G ( S, ¯ S ) | . 10 / 25