Parameters of Two-Prover-One-Round Game and The Hardness of - - PowerPoint PPT Presentation
Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems Bundit Laekhanukit McGill University 1 This Talk Investigate the connection between 2-Prover-1-Round Game (2P1R) and Hardness of Approximating
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– 2-Prover-1-Round Game (2P1R) and – Hardness of Approximating Connectivity Problems
– 2P1R with small alphabets, degree reduction, random sampling
– Rooted k-Connectivity – Vertex Connectivity Survivable Network Design (VC-SNDP) – Vertex Connectivity k-Route Cut (VC k-Route Cut)
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(complexity)
– View 2P1R as
– Know:
– Obsecure:
(algorithm)
– View 2P1R as
– Know:
– Obsecure:
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No Cooperation
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No Cooperation
Ask 1 question Ask 1 question
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No Cooperation
Give an answer Give an answer
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No Cooperation
Give an answer Give an answer Accept / Reject
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set of colors L e
admissible color pairs (constraint) on each edge. admissible color pairs on e u w
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set of colors L e
admissible color pairs (constraint) on each edge. The coloring satisfies a constraint on an edge e since Red-Green is admissible. admissible color pairs on e u w
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set of colors L e
admissible color pairs (constraint) on each edge. We may need more than one colors on each vertex. admissible color pairs on e u w
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Total cost = 12
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– Maximum Degree : D – Alphabet Size (# of available colors) : L
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Cheriyan, L, Naves, Vetta SODA 2012 Kortsarz, Krauthgamer, Lee SICOMP 2003 Chkraborty, Chuzhoy, Khanna STOC 2008 Chuzhoy, Makarychev, Vijayaraghavan, Zhou SODA 2012
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– A graph G=(V,E) with costs on edges – A root vertex r – A set of terminals T ⊆ V
– Find a min-cost subgraph H ⊆ G : H has k-vertex
r T
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– A graph G=(V,E) with costs on edges – A root vertex r – A set of terminals T ⊆ V
– Find a min-cost subgraph H ⊆ G : H has k-vertex
r T
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– A graph G=(V,E) with costs on edges – A requirement k(s,t) for each pair s,t ∈ V
– Find a min-cost subgraph H ⊆ G : H has k(s,t)
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– A graph G=(V,E) with costs on edges – A requirement k(s,t) for each pair s,t ∈ V
– Find a min-cost subgraph H ⊆ G : H has k(s,t)
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– A graph G=(V,E) with costs on edges – Source-sink pairs (s1,t1), ..., (sq, tq)
– Find a min-cost subset of edges F ⊆ E : G – F
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– A graph G=(V,E) with costs on edges – Source-sink pairs (s1,t1), ..., (sq, tq)
– Find a min-cost subset of edges F ⊆ E : G – F
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Cheriyan et al., 2012 Chkraborty et al., 2008 Chuzhoy et al., 2012
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O(ℓ) and |L| = 2 O(ℓ)
γℓ
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O(ℓ) and |L| = 2 O(ℓ)
γℓ
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– Moshkovitz-Raz, J.ACM 2010 / FOCS 2008
– Dinur-Harsha, FOCS 2009
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– Moshkovitz-Raz, J.ACM 2010 / FOCS 2008
– Dinur-Harsha, FOCS 2009
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– Khot-Safra, FOCS 2011:
– Chan, STOC 2013:
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– Khot-Safra, FOCS 2011:
– Chan, STOC 2013:
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Each step must preserves Hardness Factor
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