Bipartite Vertex Cover
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1 Bipartite Vertex Cover Mika Göös Jukka Suomela University of Toronto & HIIT University of Helsinki & HIIT Göös and Suomela Bipartite Vertex Cover 18th October / 12
2 LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October / 12
3 LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October / 12
4 LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October / 12
5 LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October / 12
6 LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October / 12
7 LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October / 12
8 LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October / 12
9 LOCAL model {0, 1} Göös and Suomela Bipartite Vertex Cover 18th October / 12
10 LOCAL model Göös and Suomela Bipartite Vertex Cover 18th October / 12
11 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 / 12
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 / 12
13 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) 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 / 12
14 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) 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 / 12
15 The bipartite case Question: Can we approximate MIN-VC fast on bipartite graphs? Göös and Suomela Bipartite Vertex Cover 18th October / 12
16 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 / 12
17 The bipartite case Setting: Bipartite 2-coloured graph Bounded degree = O(1) Compute (1 + ɛ)-approximation Göös and Suomela Bipartite Vertex Cover 18th October / 12
18 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 / 12
19 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 / 12
20 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 / 12
21 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 / 12
22 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 / 12
23 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 / 12
24 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 / 12
25 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 / 12
26 The bipartite case Covering Packing Integer??? O ɛ (1) LP O ɛ (1) O ɛ (1) Göös and Suomela Bipartite Vertex Cover 18th October / 12
27 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 / 12
28 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 / 12
29 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 / 12
30 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 / 12
31 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 / 12
32 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 / 12
33 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 / 12
34 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 / 12
35 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 / 12
36 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 / 12
37 Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October / 12
38 Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October / 12
39 Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October / 12
40 Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October / 12
41 Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October / 12
42 Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October / 12
43 Reduction in pictures Göös and Suomela Bipartite Vertex Cover 18th October / 12
44 On RECUT Theorem: RECUT MIN-VC Göös and Suomela Bipartite Vertex Cover 18th October / 12
45 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 / 12
46 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 / 12
47 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 / 12
48 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 / 12
49 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 / 12
50 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 / 12
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