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

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Bipartite Vertex Cover

Mika Göös University of Toronto & HIIT Jukka Suomela University of Helsinki & HIIT

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LOCAL model

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LOCAL model

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LOCAL model

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LOCAL model

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LOCAL model

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LOCAL model

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LOCAL model

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LOCAL model

→ {0, 1}

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LOCAL model

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LOCAL model

Definition:

A : {

} → {0, 1}

Run-time R = radius-R neighbourhood:

1 Nodes have unique IDs 2 Nodes get random strings as input

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Prior work on Min Vertex Cover (MIN-VC)

Apx ratio Run-time

General graphs O(1) Ω(

  • log n)

❬❑▼❲ PODC’04❪ ❬❑▼❲ ❪ ❬➴❙ ❪ ❬P❘ ❪

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Prior work on Min Vertex Cover (MIN-VC)

Apx ratio Run-time

General graphs O(1) Ω(

  • log n)

❬❑▼❲ PODC’04❪

Bounded degree O(1) 2 + ǫ Oǫ(1)

❬❑▼❲ SODA’06❪

2 O(1)

❬➴❙ SPAA’10❪

2 − ǫ Ω(log n)

❬P❘ ’07❪

Göös and Suomela Bipartite Vertex Cover 18th October 2012 4 / 12

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Prior work on Min Vertex Cover (MIN-VC)

Apx ratio Run-time

General graphs O(1) Ω(

  • log n)

❬❑▼❲ PODC’04❪

Bounded degree O(1) 2 + ǫ Oǫ(1)

❬❑▼❲ SODA’06❪

2 O(1)

❬➴❙ SPAA’10❪

2 − ǫ Ω(log n)

❬P❘ ’07❪ Note: MIN-VC is solvable on bipartite graphs using sequential polynomial-time algorithms!

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The bipartite case Question: Can we approximate MIN-VC fast on bipartite graphs?

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The bipartite case Question: Can we approximate MIN-VC fast on bipartite graphs?

(1 + ǫ)-approximation scheme?

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The bipartite case

Setting:

− Bipartite 2-coloured graph − Bounded degree ∆ = O(1) − Compute (1 + ǫ)-approximation

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

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The bipartite case

Setting:

− Bipartite 2-coloured graph − Bounded degree ∆ = O(1) − Compute (1 + ǫ)-approximation

Covering Integer Min VC

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

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The bipartite case

Setting:

− Bipartite 2-coloured graph − Bounded degree ∆ = O(1) − Compute (1 + ǫ)-approximation

Covering Integer Min VC LP Min

  • Frac. VC

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

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The bipartite case

Setting:

− Bipartite 2-coloured graph − Bounded degree ∆ = O(1) − Compute (1 + ǫ)-approximation

Covering Packing Integer Min VC Max Matching LP Min

  • Frac. VC

Max

  • Frac. Matching

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

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The bipartite case

Setting:

− Bipartite 2-coloured graph − Bounded degree ∆ = O(1) − Compute (1 + ǫ)-approximation

Covering Packing Integer Min VC Max Matching LP Min

  • Frac. VC

Max

  • Frac. Matching

= LP duality

=

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

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The bipartite case

Setting:

− Bipartite 2-coloured graph − Bounded degree ∆ = O(1) − Compute (1 + ǫ)-approximation

Covering Packing Integer Min VC Max Matching LP Min

  • Frac. VC

Max

  • Frac. Matching

= LP duality = Total unimodularity

=

= =

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

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The bipartite case

Setting:

− Bipartite 2-coloured graph − Bounded degree ∆ = O(1) − Compute (1 + ǫ)-approximation

Covering Packing Integer Min VC Max Matching LP Min

  • Frac. VC

Max

  • Frac. Matching

= LP duality = Total unimodularity = König’s theorem

= =

= =

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

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The bipartite case

Setting:

− Bipartite 2-coloured graph − Bounded degree ∆ = O(1) − Compute (1 + ǫ)-approximation

Covering Packing Integer Min VC Max Matching LP

Oǫ(1) Oǫ(1)

❬❑▼❲ SODA’06❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12

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The bipartite case

Setting:

− Bipartite 2-coloured graph − Bounded degree ∆ = O(1) − Compute (1 + ǫ)-approximation

Covering Packing Integer Min VC

Oǫ(1)

LP

Oǫ(1) Oǫ(1)

❬❑▼❲ SODA’06❪ ❬◆❖ FOCS’08❪, ❬➴P❘❙❯ ’10❪

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The bipartite case

Covering Packing Integer

???

Oǫ(1)

LP

Oǫ(1) Oǫ(1)

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

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The bipartite case

Covering Packing Integer

Ω(log n) Oǫ(1)

LP

Oǫ(1) Oǫ(1)

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

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The bipartite case

Surprise: No Sublogarithmic-Time Approximation Scheme for Bipartite Vertex Cover!

Covering Packing Integer

Ω(log n) Oǫ(1)

LP

Oǫ(1) Oǫ(1)

❬❑▼❲ ❪ ❬◆❖ ❪ ❬➴P❘❙❯ ❪

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Our result Main Theorem

∃δ > 0: No o(log n)-time algorithm to (1 + δ)-approximate MIN-VC on 2-coloured graphs of max degree ∆ = 3

❬▲❙ ❪

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Our result Main Theorem

∃δ > 0: No o(log n)-time algorithm to (1 + δ)-approximate MIN-VC on 2-coloured graphs of max degree ∆ = 3

Lower bound is tight

1 There is Oǫ(log n)-time approx. scheme ❬▲❙ ’93❪ 2 If ∆ = 2 there is Oǫ(1)-time approx. scheme

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Why is MIN-VC difficult for distributed graph algorithms?

Short answer: Solving MIN-VC requires solving a hard cut minimisation problem

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Why is MIN-VC difficult for distributed graph algorithms?

Short answer: Solving MIN-VC requires solving a hard cut minimisation problem Strategy:

  • 1. Reduce cut problem to MIN-VC
  • 2. Prove that cut problem is hard

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Reduction formalised

ℓ✐♥

♦✉t

RECUT 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-❜❧✉❡

♦✉t

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Reduction formalised

ℓ✐♥

Global optimum

− →

RECUT 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-❜❧✉❡

♦✉t

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Reduction formalised

ℓ✐♥ ℓ♦✉t

− →

RECUT 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-❜❧✉❡

♦✉t

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Reduction formalised

✐♥ ♦✉t

RECUT 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-❜❧✉❡

RECUT ≤ MIN-VC

If: MIN-VC can be (1 + ǫ)-approximated in time R Then: We can compute in time R a RECUT of density

|ℓ♦✉t| |E| ≤ ǫ

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Reduction in pictures

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Reduction in pictures

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Reduction in pictures

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Reduction in pictures

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Reduction in pictures

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Reduction in pictures

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Reduction in pictures

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On RECUT

✐♥ ♦✉t

Theorem: RECUT ≤ MIN-VC r❡❞ r❡❞ r❡❞ ❜❧✉❡

♦✉t r❡❞ ♦✉t ❜❧✉❡

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On RECUT

ℓ✐♥ ℓ♦✉t

− →

Theorem: RECUT ≤ MIN-VC Sometimes RECUT Is Easy: The algorithm “Output r❡❞ iff you see any r❡❞ nodes” computes a small RECUT on grid-like graphs r❡❞ ❜❧✉❡

♦✉t r❡❞ ♦✉t ❜❧✉❡

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On RECUT

✐♥ ♦✉t

Theorem: RECUT ≤ MIN-VC Sometimes RECUT Is Easy: The algorithm “Output r❡❞ iff you see any r❡❞ nodes” computes a small RECUT on grid-like graphs Therefore: We consider expander graphs that satisfy |ℓ| ≥ δ · min(|ℓ−1(r❡❞)|, |ℓ−1(❜❧✉❡)|)

♦✉t r❡❞ ♦✉t ❜❧✉❡

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On RECUT

✐♥ ♦✉t

Theorem: RECUT ≤ MIN-VC Sometimes RECUT Is Easy: The algorithm “Output r❡❞ iff you see any r❡❞ nodes” computes a small RECUT on grid-like graphs Therefore: We consider expander graphs that satisfy |ℓ| ≥ δ · min(|ℓ−1(r❡❞)|, |ℓ−1(❜❧✉❡)|)

Main Technical Lemma: We fool a fast algorithm into producing a balanced RECUT |ℓ−1

♦✉t(r❡❞)| ≈ |ℓ−1 ♦✉t(❜❧✉❡)| ≈ n/2

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Conclusions Our result:

No o(log n)-time approximation scheme for MIN-VC on 2-coloured graphs with ∆ = 3

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Conclusions Our result:

No o(log n)-time approximation scheme for MIN-VC on 2-coloured graphs with ∆ = 3

Open problems:

Approximation ratios for O(1)-time algorithms? Derandomising Linial–Saks requires designing deterministic algorithms for RECUT

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Conclusions Our result:

No o(log n)-time approximation scheme for MIN-VC on 2-coloured graphs with ∆ = 3

Open problems:

Approximation ratios for O(1)-time algorithms? Derandomising Linial–Saks requires designing deterministic algorithms for RECUT

Cheers!

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