Advice Complexity of Adaptive Priority Algorithms: Part 2 – Lower Bounds Joan Boyar 1 ∗ , Kim S. Larsen 1 , Denis Pankratov 2 1 University of Southern Denmark 2 Concordia University OLAWA 2020 Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 1 / 23
Overview 1 Old lower bound for priority algorithms without advice 2 Gadget pairs 3 Lower bound for optimality — Vertex Cover 4 Lower bounds for approximation Template for proving hardness results 1 Example results 2 Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 2 / 23
Section 1 Old Lower Bound Result for Priority Algorithms without Advice Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 3 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) For Vertex Cover, no adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 . Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 4 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) For Vertex Cover, no adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 . Pf. 1 4 7 2 6 1 3 6 3 7 5 4 2 5 Graph G 1 Graph G 2 Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 4 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) For Vertex Cover, no adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 . Pf. cont. 1 2 6 3 7 5 4 In G 1 , rejecting vertex 1, Graph G 1 gives a vertex cover of size 4. Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 5 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) For Vertex Cover, no adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 . Pf. cont. r 2 6 3 7 5 4 In G 1 , rejecting vertex 1, Graph G 1 gives a vertex cover of size 4. Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 5 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) For Vertex Cover, no adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 . Pf. cont. r a a 2 6 3 7 5 4 In G 1 , rejecting vertex 1, Graph G 1 gives a vertex cover of size 4. Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 5 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) For Vertex Cover, no adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 . Pf. cont. r a a 2 6 3 7 5 4 In G 1 , rejecting vertex 1, Graph G 1 gives a vertex cover of size 4. Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 5 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) No adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 for the Vertex Cover problem. Pf. cont. (Adversary argument) Input items are (Vertex name, Names of neighbors). Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 6 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) No adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 for the Vertex Cover problem. Pf. cont. (Adversary argument) Input items are (Vertex name, Names of neighbors). Input will be an isomorphic copy of G 1 or G 2 . The input universe contains all possible input items consistent with that. Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 6 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) No adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 for the Vertex Cover problem. Pf. cont. (Adversary argument) Input items are (Vertex name, Names of neighbors). Input will be an isomorphic copy of G 1 or G 2 . The input universe contains all possible input items consistent with that. If Alg ’s first priority function selects some vertex v , the adversary, Adv , can make it be any vertex of the same degree, in either G 1 or G 2 . Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 6 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) No adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 for the Vertex Cover problem. Pf. cont. (Adversary argument) Input items are (Vertex name, Names of neighbors). Input will be an isomorphic copy of G 1 or G 2 . The input universe contains all possible input items consistent with that. If Alg ’s first priority function selects some vertex v , the adversary, Adv , can make it be any vertex of the same degree, in either G 1 or G 2 . If v has degree 2 and Alg accepts, Adv chooses vertex 2 in G 1 . If v has degree 2 and Alg rejects, Adv chooses vertex 1 in G 1 . Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 6 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem (Borodin,B.,Larsen,Mirmohammadi, 2010) No adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 for the Vertex Cover problem. Pf. cont. (Adversary argument) Input items are (Vertex name, Names of neighbors). Input will be an isomorphic copy of G 1 or G 2 . The input universe contains all possible input items consistent with that. If Alg ’s first priority function selects some vertex v , the adversary, Adv , can make it be any vertex of the same degree, in either G 1 or G 2 . If v has degree 2 and Alg accepts, Adv chooses vertex 2 in G 1 . If v has degree 2 and Alg rejects, Adv chooses vertex 1 in G 1 . If v has degree 3, and Alg accepts, Adv chooses vertex 1 in G 2 . If v has degree 3 and Alg rejects, Adv chooses vertex 3 in G 1 . Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 6 / 23
1 4 7 2 6 1 3 6 3 7 5 4 2 5 Graph G 1 Graph G 2 Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 7 / 23
Adaptive Priority Algorithms for Vertex Cover Theorem ([Borodin,B.,Larsen,Mirmohammadi, 2010) For Vertex Cover, no adaptive priority algorithm can achieve an approximation ratio better than 4 / 3 . Pf. cont. In all cases, Alg accepts ≥ 4 vertices, but 3 is optimal. � Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 8 / 23
Section 2 Gadget Pairs Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 9 / 23
Gadget pairs 2 1 1 3 2 6 6 7 5 3 7 5 4 4 Graph G ′ Graph G 1 1 G 1 and G ′ 1 are a gadget pair (degree-2 chosen first – call v – vertex 1). Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 10 / 23
Gadget pairs 2 1 1 3 2 6 6 7 5 3 7 5 4 4 Graph G ′ Graph G 1 1 G 1 and G ′ 1 are a gadget pair (degree-2 chosen first – call v – vertex 1). Properties: Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 10 / 23
Gadget pairs 2 1 1 3 2 6 6 7 5 3 7 5 4 4 Graph G ′ Graph G 1 1 G 1 and G ′ 1 are a gadget pair (degree-2 chosen first – call v – vertex 1). Properties: First item condition: Input item for v gives no info as to which of G 1 / G ′ 1 . Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 10 / 23
Gadget pairs 2 1 1 3 2 6 6 7 5 3 7 5 4 4 Graph G ′ Graph G 1 1 G 1 and G ′ 1 are a gadget pair (degree-2 chosen first – call v – vertex 1). Properties: First item condition: Input item for v gives no info as to which of G 1 / G ′ 1 . Distinguishing decision condition: Decisions for v giving optimum for G 1 and G ′ 1 are opposite. Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 10 / 23
Gadget pairs 7 4 7 2 6 1 3 6 1 3 5 4 2 5 Graph G 1 Graph G 2 G 1 and G 2 are a gadget pair (degree-3 chosen first – call v – vertex 1). Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 11 / 23
Gadget pairs 7 4 7 2 6 1 3 6 1 3 5 4 2 5 Graph G 1 Graph G 2 G 1 and G 2 are a gadget pair (degree-3 chosen first – call v – vertex 1). Properties: Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 11 / 23
Gadget pairs 7 4 7 2 6 1 3 6 1 3 5 4 2 5 Graph G 1 Graph G 2 G 1 and G 2 are a gadget pair (degree-3 chosen first – call v – vertex 1). Properties: First item condition: Input item for v gives no info as to which of G 1 / G 2 . Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 11 / 23
Gadget pairs 7 4 7 2 6 1 3 6 1 3 5 4 2 5 Graph G 1 Graph G 2 G 1 and G 2 are a gadget pair (degree-3 chosen first – call v – vertex 1). Properties: First item condition: Input item for v gives no info as to which of G 1 / G 2 . Distinguishing decision condition: Decisions for v giving optimum for G 1 and G 2 are opposite. Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 11 / 23
Section 3 Lower Bound for Optimality – Vertex Cover Boyar, Larsen, Pankratov Priority Algorithms – Lower Bounds (1) OLAWA 2020 12 / 23
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