Bipartite Vertex Cover Mika Göös Jukka Suomela University of Toronto & HIIT 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
Prior work on Min Vertex Cover (MIN-VC) Apx ratio Run-time General graphs O(1) Ω( log n) [KMW PODC 04] Göös and Suomela Bipartite Vertex Cover 18th October 2012 4 / 12
Prior work on Min Vertex Cover (MIN-VC) Apx ratio Run-time General graphs O(1) Ω( log n) [KMW PODC 04] Bounded degree O(1) 0 2 + ɛ O ɛ (1) [KMW SODA 06] 2 O(1) [ÅS SPAA 10] 2 ɛ Ω(log n) [PR 07] Göös and Suomela Bipartite Vertex Cover 18th October 2012 4 / 12
Prior work on Min Vertex Cover (MIN-VC) Apx ratio Run-time General graphs O(1) Ω( log n) [KMW PODC 04] Bounded degree O(1) 0 2 + ɛ O ɛ (1) [KMW SODA 06] 2 O(1) [ÅS SPAA 10] 2 ɛ Ω(log n) [PR 07] Note: MIN-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 MIN-VC fast on bipartite graphs? Göös and Suomela Bipartite Vertex Cover 18th October 2012 5 / 12
The bipartite case Question: Can we approximate MIN-VC fast on bipartite graphs? (1 + ɛ)-approximation scheme? Göös and Suomela Bipartite Vertex Cover 18th October 2012 5 / 12
The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Covering Integer Min VC Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Covering Integer LP Min VC Min Frac. VC Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Integer LP Covering Min VC Min Frac. VC Packing Max Matching Max Frac. Matching Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Covering Packing Integer Min VC Max Matching LP Min Frac. VC = Max Frac. Matching = LP duality Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Covering Packing Integer Min VC Max Matching LP = Min Frac. VC = = Max Frac. Matching = LP duality = Total unimodularity Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Covering Packing Integer Min VC = Max Matching LP = Min Frac. VC = = Max Frac. Matching = LP duality = Total unimodularity = König s theorem Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Integer Covering Min VC Packing Max Matching LP O ɛ (1) O ɛ (1) [KMW SODA 06] Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Covering Packing Integer Min VC O ɛ (1) LP O ɛ (1) O ɛ (1) [KMW SODA 06] [NO FOCS 08], [ÅPRSU 10] Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Covering Packing Integer??? O ɛ (1) LP O ɛ (1) O ɛ (1) Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Covering Packing Integer Ω(log n) O ɛ (1) LP O ɛ (1) O ɛ (1) Göös and Suomela Bipartite Vertex Cover 18th October 2012 6 / 12
The bipartite case Surprise: No Sublogarithmic-Time Approximation Scheme for Bipartite Vertex Cover! Covering Packing Integer Ω(log n) O ɛ (1) LP O ɛ (1) O ɛ (1) 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 MIN-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 MIN-VC on 2-coloured graphs of max degree = 3 Lower bound is tight 1 There is O ɛ (log n)-time approx. scheme [LS 93] 2 If = 2 there is O ɛ (1)-time approx. scheme Göös and Suomela Bipartite Vertex Cover 18th October 2012 7 / 12
Why is MIN-VC difficult for distributed graph algorithms? Short answer: Solving MIN-VC requires solving a hard cut minimisation problem Göös and Suomela Bipartite Vertex Cover 18th October 2012 8 / 12
Why is MIN-VC difficult for distributed graph algorithms? Short answer: Strategy: Solving MIN-VC requires solving a hard cut minimisation problem 1. Reduce cut problem to MIN-VC 2. Prove that cut problem is hard Göös and Suomela Bipartite Vertex Cover 18th October 2012 8 / 12
Reduction formalised RECUT problem Input: Labelled graph (G, l in ) where l in : V {red, blue} Output: Labelling l out : V {red, blue} that minimises the size of the cut l out subject to If l in is all-red then l out is all-red If l in is all-blue then l out is all-blue l in Göös and Suomela Bipartite Vertex Cover 18th October 2012 9 / 12
Reduction formalised RECUT problem Input: Labelled graph (G, l in ) where l in : V {red, blue} Output: Labelling l out : V {red, blue} that minimises the size of the cut l out subject to If l in is all-red then l out is all-red If l in is all-blue then l out is all-blue l in Global optimum Göös and Suomela Bipartite Vertex Cover 18th October 2012 9 / 12
Reduction formalised RECUT problem Input: Labelled graph (G, l in ) where l in : V {red, blue} Output: Labelling l out : V {red, blue} that minimises the size of the cut l out subject to If l in is all-red then l out is all-red If l in is all-blue then l out is all-blue l in l out Göös and Suomela Bipartite Vertex Cover 18th October 2012 9 / 12
Reduction formalised RECUT problem Input: Labelled graph (G, l in ) where l in : V {red, blue} Output: Labelling l out : V {red, blue} that minimises the size of the cut l out subject to If l in is all-red then l out is all-red If l in is all-blue then l out is all-blue RECUT MIN-VC If: MIN-VC can be (1 + ɛ)-approximated in time R Then: We can compute in time R a RECUT of density l out E ɛ Göös and Suomela Bipartite Vertex Cover 18th October 2012 9 / 12
Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October 2012 10 / 12
Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October 2012 10 / 12
Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October 2012 10 / 12
Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October 2012 10 / 12
Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October 2012 10 / 12
Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October 2012 10 / 12
Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October 2012 10 / 12
On RECUT Theorem: RECUT MIN-VC Göös and Suomela Bipartite Vertex Cover 18th October 2012 11 / 12
On RECUT Theorem: RECUT MIN-VC Sometimes RECUT Is Easy: The algorithm Output red iff you see any red nodes computes a small RECUT on grid-like graphs l in l out Göös and Suomela Bipartite Vertex Cover 18th October 2012 11 / 12
On RECUT Theorem: RECUT MIN-VC Sometimes RECUT Is Easy: The algorithm Output red iff you see any red nodes computes a small RECUT on grid-like graphs Therefore: We consider expander graphs that satisfy l δ min( l 1 (red), l 1 (blue) ) Göös and Suomela Bipartite Vertex Cover 18th October 2012 11 / 12
On RECUT Theorem: RECUT MIN-VC Sometimes RECUT Is Easy: The algorithm Output red iff you see any red nodes computes a small RECUT on grid-like graphs Therefore: We consider expander graphs that satisfy l δ min( l 1 (red), l 1 (blue) ) Main Technical Lemma: We fool a fast algorithm into producing a balanced RECUT l 1 out (red) l 1 (blue) n/2 out Göös and Suomela Bipartite Vertex Cover 18th October 2012 11 / 12
Conclusions Our result: No o(log n)-time approximation scheme for MIN-VC on 2-coloured graphs with = 3 Göös and Suomela Bipartite Vertex Cover 18th October 2012 12 / 12
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 Göös and Suomela Bipartite Vertex Cover 18th October 2012 12 / 12
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! Göös and Suomela Bipartite Vertex Cover 18th October 2012 12 / 12