Bipartite Vertex Cover Mika Gs University of Toronto & HIIT - PowerPoint PPT Presentation

Bipartite Vertex Cover Mika Göös University of Toronto & HIIT Jukka Suomela University of Helsinki & HIIT Göös and Suomela Bipartite Vertex Cover 18th October 2012 1 / 12

LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October 2012 2 / 12

LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October 2012 2 / 12

LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October 2012 2 / 12

LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October 2012 2 / 12

LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October 2012 2 / 12

LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October 2012 2 / 12

LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October 2012 2 / 12

LOCAL model �→ { 0, 1 } Göös and Suomela Bipartite Vertex Cover 18th October 2012 2 / 12

LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October 2012 2 / 12

LOCAL model Definition: A : { } → { 0, 1 } Run-time R = radius- R neighbourhood: 1 Nodes have unique IDs 2 Nodes get random strings as input Göös and Suomela Bipartite Vertex Cover 18th October 2012 3 / 12

❬❑▼❲ ❪ ❬➴❙ ❪ ❬P❘ ❪ Prior work on Min Vertex Cover (M IN -VC) Apx ratio Run-time � ❬❑▼❲ PODC’04 ❪ General graphs O ( 1 ) Ω ( log n ) Göös and Suomela Bipartite Vertex Cover 18th October 2012 4 / 12

Prior work on Min Vertex Cover (M IN -VC) Apx ratio Run-time � ❬❑▼❲ PODC’04 ❪ General graphs O ( 1 ) Ω ( log n ) O ( 1 ) Bounded degree 0 ❬❑▼❲ SODA’06 ❪ 2 + ǫ O ǫ ( 1 ) ❬➴❙ SPAA’10 ❪ 2 O ( 1 ) ❬P❘ ’07 ❪ 2 − ǫ Ω ( log n ) Göös and Suomela Bipartite Vertex Cover 18th October 2012 4 / 12

Prior work on Min Vertex Cover (M IN -VC) Apx ratio Run-time � ❬❑▼❲ PODC’04 ❪ General graphs O ( 1 ) Ω ( log n ) O ( 1 ) Bounded degree 0 ❬❑▼❲ SODA’06 ❪ 2 + ǫ O ǫ ( 1 ) ❬➴❙ SPAA’10 ❪ 2 O ( 1 ) ❬P❘ ’07 ❪ 2 − ǫ Ω ( log n ) Note: M IN -VC is solvable on bipartite graphs using sequential polynomial-time algorithms! Göös and Suomela Bipartite Vertex Cover 18th October 2012 4 / 12

The bipartite case Question: Can we approximate M IN -VC fast on bipartite graphs? Göös and Suomela Bipartite Vertex Cover 18th October 2012 5 / 12

The bipartite case Question: Can we approximate M IN -VC fast on bipartite graphs? ( 1 + ǫ ) -approximation scheme? Göös and Suomela Bipartite Vertex Cover 18th October 2012 5 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case − Bipartite 2-coloured graph Setting: − Bounded degree ∆ = O ( 1 ) − Compute ( 1 + ǫ ) -approximation Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case − Bipartite 2-coloured graph Setting: − Bounded degree ∆ = O ( 1 ) − Compute ( 1 + ǫ ) -approximation Covering Min Integer VC Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case − Bipartite 2-coloured graph Setting: − Bounded degree ∆ = O ( 1 ) − Compute ( 1 + ǫ ) -approximation Covering Min Integer VC Min LP Frac. VC Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case − Bipartite 2-coloured graph Setting: − Bounded degree ∆ = O ( 1 ) − Compute ( 1 + ǫ ) -approximation Covering Packing Min Max Integer VC Matching Min Max LP Frac. VC Frac. Matching Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case − Bipartite 2-coloured graph Setting: − Bounded degree ∆ = O ( 1 ) − Compute ( 1 + ǫ ) -approximation Covering Packing Min Max Integer VC Matching = Min Max LP Frac. VC Frac. Matching = LP duality Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case − Bipartite 2-coloured graph Setting: − Bounded degree ∆ = O ( 1 ) − Compute ( 1 + ǫ ) -approximation Covering Packing Min Max Integer VC Matching = = = Min Max LP Frac. VC Frac. Matching = LP duality = Total unimodularity Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case − Bipartite 2-coloured graph Setting: − Bounded degree ∆ = O ( 1 ) − Compute ( 1 + ǫ ) -approximation Covering Packing = Min Max Integer VC Matching = = = Min Max LP Frac. VC Frac. Matching = LP duality = Total unimodularity = König’s theorem Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case − Bipartite 2-coloured graph Setting: − Bounded degree ∆ = O ( 1 ) − Compute ( 1 + ǫ ) -approximation Covering Packing Min Max Integer VC Matching O ǫ ( 1 ) O ǫ ( 1 ) LP ❬❑▼❲ SODA’06 ❪ Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

The bipartite case − Bipartite 2-coloured graph Setting: − Bounded degree ∆ = O ( 1 ) − Compute ( 1 + ǫ ) -approximation Covering Packing Min O ǫ ( 1 ) Integer VC O ǫ ( 1 ) O ǫ ( 1 ) LP ❬❑▼❲ SODA’06 ❪ ❬◆❖ FOCS’08 ❪ , ❬➴P❘❙❯ ’10 ❪ Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case Covering Packing ??? O ǫ ( 1 ) Integer O ǫ ( 1 ) O ǫ ( 1 ) LP Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case Covering Packing Ω ( log n ) O ǫ ( 1 ) Integer O ǫ ( 1 ) O ǫ ( 1 ) LP Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪ The bipartite case Surprise: No Sublogarithmic-Time Approximation Scheme for Bipartite Vertex Cover! Covering Packing Ω ( log n ) O ǫ ( 1 ) Integer O ǫ ( 1 ) O ǫ ( 1 ) LP Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

❬▲❙ ❪ Our result Main Theorem ∃ δ > 0 : No o ( log n ) -time algorithm to ( 1 + δ ) -approximate M IN -VC on 2-coloured graphs of max degree ∆ = 3 Göös and Suomela Bipartite Vertex Cover 18th October 2012 7 / 12

Our result Main Theorem ∃ δ > 0 : No o ( log n ) -time algorithm to ( 1 + δ ) -approximate M IN -VC on 2-coloured graphs of max degree ∆ = 3 Lower bound is tight 1 There is O ǫ ( log n ) -time approx. scheme ❬▲❙ ’93 ❪ 2 If ∆ = 2 there is O ǫ ( 1 ) -time approx. scheme Göös and Suomela Bipartite Vertex Cover 18th October 2012 7 / 12

Why is M IN -VC difficult for distributed graph algorithms? Short answer: Solving M IN -VC requires solving a hard cut minimisation problem Göös and Suomela Bipartite Vertex Cover 18th October 2012 8 / 12

Why is M IN -VC difficult for distributed graph algorithms? Short answer: Solving M IN -VC requires solving a hard cut minimisation problem Strategy: 1. Reduce cut problem to M IN -VC 2. Prove that cut problem is hard Göös and Suomela Bipartite Vertex Cover 18th October 2012 8 / 12

♦✉t ♦✉t Reduction formalised R ECUT problem Input: Labelled graph ( G , ℓ ✐♥ ) where ℓ ✐♥ : V → { r❡❞ , ❜❧✉❡ } Output: Labelling ℓ ♦✉t : V → { r❡❞ , ❜❧✉❡ } that minimises the size of the cut | ℓ ♦✉t | subject to − If ℓ ✐♥ is all- r❡❞ then ℓ ♦✉t is all- r❡❞ − If ℓ ✐♥ is all- ❜❧✉❡ then ℓ ♦✉t is all- ❜❧✉❡ ℓ ✐♥ Göös and Suomela Bipartite Vertex Cover 18th October 2012 9 / 12

♦✉t Reduction formalised R ECUT problem Input: Labelled graph ( G , ℓ ✐♥ ) where ℓ ✐♥ : V → { r❡❞ , ❜❧✉❡ } Output: Labelling ℓ ♦✉t : V → { r❡❞ , ❜❧✉❡ } that minimises the size of the cut | ℓ ♦✉t | subject to − If ℓ ✐♥ is all- r❡❞ then ℓ ♦✉t is all- r❡❞ − If ℓ ✐♥ is all- ❜❧✉❡ then ℓ ♦✉t is all- ❜❧✉❡ ℓ ✐♥ Global optimum �− → Göös and Suomela Bipartite Vertex Cover 18th October 2012 9 / 12

♦✉t Reduction formalised R ECUT problem Input: Labelled graph ( G , ℓ ✐♥ ) where ℓ ✐♥ : V → { r❡❞ , ❜❧✉❡ } Output: Labelling ℓ ♦✉t : V → { r❡❞ , ❜❧✉❡ } that minimises the size of the cut | ℓ ♦✉t | subject to − If ℓ ✐♥ is all- r❡❞ then ℓ ♦✉t is all- r❡❞ − If ℓ ✐♥ is all- ❜❧✉❡ then ℓ ♦✉t is all- ❜❧✉❡ ℓ ✐♥ ℓ ♦✉t �− → Göös and Suomela Bipartite Vertex Cover 18th October 2012 9 / 12