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Testing Consumer Rationality using Perfect Graphs and Oriented Discs Shant Boodaghians and Adrian Vetta McGill University WINE 2015 Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and

  1. Degree of Rationality [Afriat, ‘67] The agent is rational ⇐ ⇒ preference graph is acyclic. We ask: how rational is agent? Definition. An agent’s degree of rationality is the least number of data pairs to ignore for the agent to seem rational. Rational iff DoR = 0. Note 1. This was defined by Houtman and Maks in ‘88. Note 2. Equivalent to Min. Feedback Vertex Set on the Pref. Graph. Example. In the following preference graph, the degree of rationality is 1. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  2. Degree of Rationality [Afriat, ‘67] The agent is rational ⇐ ⇒ preference graph is acyclic. We ask: how rational is agent? Definition. An agent’s degree of rationality is the least number of data pairs to ignore for the agent to seem rational. Rational iff DoR = 0. Note 1. This was defined by Houtman and Maks in ‘88. Note 2. Equivalent to Min. Feedback Vertex Set on the Pref. Graph. Example. In the following preference graph, the degree of rationality is 1. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  3. Degree of Rationality [Afriat, ‘67] The agent is rational ⇐ ⇒ preference graph is acyclic. We ask: how rational is agent? Definition. An agent’s degree of rationality is the least number of data pairs to ignore for the agent to seem rational. Rational iff DoR = 0. Note 1. This was defined by Houtman and Maks in ‘88. Note 2. Equivalent to Min. Feedback Vertex Set on the Pref. Graph. Example. In the following preference graph, the degree of rationality is 1. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  4. Degree of Rationality, Complexity What is the complexity of MFVS in preference graphs? ◮ If market size ( n ) large, then any graph is a preference graph  1 if i = j �  1 if i = j  p i x i j = j = 0 if ( i , j ) ∈ D 0 if i � = j  2 if ( i , j ) / ∈ D  Our Results ◮ Theorem 1. If n = 2, Degree of Rationality is in P. ◮ Theorem 2. If n ≥ 3, DoR decision problem is NP-complete. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  5. Degree of Rationality, Complexity What is the complexity of MFVS in preference graphs? ◮ If market size ( n ) large, then any graph is a preference graph  1 if i = j �  1 if i = j  p i x i j = j = 0 if ( i , j ) ∈ D 0 if i � = j  2 if ( i , j ) / ∈ D  Our Results ◮ Theorem 1. If n = 2, Degree of Rationality is in P. ◮ Theorem 2. If n ≥ 3, DoR decision problem is NP-complete. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  6. Degree of Rationality, Complexity What is the complexity of MFVS in preference graphs? ◮ If market size ( n ) large, then any graph is a preference graph  1 if i = j �  1 if i = j  p i x i j = j = 0 if ( i , j ) ∈ D 0 if i � = j  2 if ( i , j ) / ∈ D  Our Results ◮ Theorem 1. If n = 2, Degree of Rationality is in P. ◮ Theorem 2. If n ≥ 3, DoR decision problem is NP-complete. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  7. Degree of Rationality, Complexity What is the complexity of MFVS in preference graphs? ◮ If market size ( n ) large, then any graph is a preference graph So complexity and approximation complexity same as that of MFVS. ◮ What if n small? Set of attainable graphs is strictly smaller. Approximation complexity likely improves. Our Results ◮ Theorem 1. If n = 2, Degree of Rationality is in P. ◮ Theorem 2. If n ≥ 3, DoR decision problem is NP-complete. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  8. Degree of Rationality, Complexity What is the complexity of MFVS in preference graphs? ◮ If market size ( n ) large, then any graph is a preference graph So complexity and approximation complexity same as that of MFVS. ◮ What if n small? Set of attainable graphs is strictly smaller. Approximation complexity likely improves. Our Results ◮ Theorem 1. If n = 2, Degree of Rationality is in P. ◮ Theorem 2. If n ≥ 3, DoR decision problem is NP-complete. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  9. Degree of Rationality, Complexity What is the complexity of MFVS in preference graphs? ◮ If market size ( n ) large, then any graph is a preference graph So complexity and approximation complexity same as that of MFVS. ◮ What if n small? Set of attainable graphs is strictly smaller. Approximation complexity likely improves. Our Results ◮ Theorem 1. If n = 2, Degree of Rationality is in P. ◮ Theorem 2. If n ≥ 3, DoR decision problem is NP-complete. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  10. Degree of Rationality, Complexity What is the complexity of MFVS in preference graphs? ◮ If market size ( n ) large, then any graph is a preference graph So complexity and approximation complexity same as that of MFVS. ◮ What if n small? Set of attainable graphs is strictly smaller. Approximation complexity likely improves. Our Results ◮ Theorem 1. If n = 2, Degree of Rationality is in P. ◮ Theorem 2. If n ≥ 3, DoR decision problem is NP-complete. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  11. 2-Commodity Markets Theorem 1. Degree of rationality is poly-time for n = 2. Proof Sketch. 1. [Rose, ‘58] : If n = 2, every cycle contains a digon (2-cycle) 2. Corollary. MFVS is Min. Vertex Cover on the set of digons 3. Lemma. Digon graphs of pref. graphs are perfect for n = 2. (New class of perfect graphs?) 4. [GLS, ‘84] : Min Vertex Cover on a Perfect Graph is polytime. So Theorem 1. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  12. 2-Commodity Markets Theorem 1. Degree of rationality is poly-time for n = 2. Proof Sketch. 1. [Rose, ‘58] : If n = 2, every cycle contains a digon (2-cycle) 2. Corollary. MFVS is Min. Vertex Cover on the set of digons 3. Lemma. Digon graphs of pref. graphs are perfect for n = 2. (New class of perfect graphs?) 4. [GLS, ‘84] : Min Vertex Cover on a Perfect Graph is polytime. So Theorem 1. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  13. 2-Commodity Markets Theorem 1. Degree of rationality is poly-time for n = 2. Proof Sketch. 1. [Rose, ‘58] : If n = 2, every cycle contains a digon (2-cycle) 2. Corollary. MFVS is Min. Vertex Cover on the set of digons 3. Lemma. Digon graphs of pref. graphs are perfect for n = 2. (New class of perfect graphs?) 4. [GLS, ‘84] : Min Vertex Cover on a Perfect Graph is polytime. So Theorem 1. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  14. 2-Commodity Markets Theorem 1. Degree of rationality is poly-time for n = 2. Proof Sketch. 1. [Rose, ‘58] : If n = 2, every cycle contains a digon (2-cycle) 2. Corollary. MFVS is Min. Vertex Cover on the set of digons 3. Lemma. Digon graphs of pref. graphs are perfect for n = 2. (New class of perfect graphs?) 4. [GLS, ‘84] : Min Vertex Cover on a Perfect Graph is polytime. So Theorem 1. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  15. 2-Commodity Markets Theorem 1. Degree of rationality is poly-time for n = 2. Proof Sketch. 1. [Rose, ‘58] : If n = 2, every cycle contains a digon (2-cycle) 2. Corollary. MFVS is Min. Vertex Cover on the set of digons 3. Lemma. Digon graphs of pref. graphs are perfect for n = 2. (New class of perfect graphs?) 4. [GLS, ‘84] : Min Vertex Cover on a Perfect Graph is polytime. So Theorem 1. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  16. 2-Commodity Markets Theorem 1. Degree of rationality is poly-time for n = 2. Proof Sketch. 1. [Rose, ‘58] : If n = 2, every cycle contains a digon (2-cycle) 2. Corollary. MFVS is Min. Vertex Cover on the set of digons 3. Lemma. Digon graphs of pref. graphs are perfect for n = 2. (New class of perfect graphs?) 4. [GLS, ‘84] : Min Vertex Cover on a Perfect Graph is polytime. So Theorem 1. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  17. 2-Commodity Markets Theorem 1. Degree of rationality is poly-time for n = 2. Proof Sketch. 1. [Rose, ‘58] : If n = 2, every cycle contains a digon (2-cycle) 2. Corollary. MFVS is Min. Vertex Cover on the set of digons 3. Lemma. Digon graphs of pref. graphs are perfect for n = 2. (New class of perfect graphs?) 4. [GLS, ‘84] : Min Vertex Cover on a Perfect Graph is polytime. So Theorem 1. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  18. The digon graph [Rose, ‘58] : If n = 2, every cycle contains the vertex set of a digon. Therefore, hitting every digon hits every cycle. Definition. The digon graph of a directed graph D = ( V , A ) is G = ( V , E ), where E is exactly the set of digons in D . Example. Corollary. Degree of Rationality reduces to Min. Vertex Cover on G . Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  19. The digon graph [Rose, ‘58] : If n = 2, every cycle contains the vertex set of a digon. Therefore, hitting every digon hits every cycle. Definition. The digon graph of a directed graph D = ( V , A ) is G = ( V , E ), where E is exactly the set of digons in D . Example. Corollary. Degree of Rationality reduces to Min. Vertex Cover on G . Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  20. The digon graph [Rose, ‘58] : If n = 2, every cycle contains the vertex set of a digon. Therefore, hitting every digon hits every cycle. Definition. The digon graph of a directed graph D = ( V , A ) is G = ( V , E ), where E is exactly the set of digons in D . Example. Corollary. Degree of Rationality reduces to Min. Vertex Cover on G . Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  21. The digon graph [Rose, ‘58] : If n = 2, every cycle contains the vertex set of a digon. Therefore, hitting every digon hits every cycle. Definition. The digon graph of a directed graph D = ( V , A ) is G = ( V , E ), where E is exactly the set of digons in D . Example. Corollary. Degree of Rationality reduces to Min. Vertex Cover on G . Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  22. The digon graph [Rose, ‘58] : If n = 2, every cycle contains the vertex set of a digon. Therefore, hitting every digon hits every cycle. Definition. The digon graph of a directed graph D = ( V , A ) is G = ( V , E ), where E is exactly the set of digons in D . Example. Corollary. Degree of Rationality reduces to Min. Vertex Cover on G . Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  23. The digon graph [Rose, ‘58] : If n = 2, every cycle contains the vertex set of a digon. Therefore, hitting every digon hits every cycle. Definition. The digon graph of a directed graph D = ( V , A ) is G = ( V , E ), where E is exactly the set of digons in D . Example. Corollary. Degree of Rationality reduces to Min. Vertex Cover on G . Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  24. Examples of Digon Graphs Which graphs can be constructed as digon graphs? Some examples Complete r -part. Complete Bip. Graphs, Complete Graphs, Graphs, ◮ Complements of complete r -partite graphs, ◮ Many small graphs, ◮ etc. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  25. Examples of Digon Graphs Which graphs can be constructed as digon graphs? Some examples Complete r -part. Complete Bip. Graphs, Complete Graphs, Graphs, ◮ Complements of complete r -partite graphs, ◮ Many small graphs, ◮ etc. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  26. Examples of Digon Graphs Which graphs can be constructed as digon graphs? Some examples Complete r -part. Complete Bip. Graphs, Complete Graphs, Graphs, ◮ Complements of complete r -partite graphs, ◮ Many small graphs, ◮ etc. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  27. Examples of Digon Graphs Which graphs can be constructed as digon graphs? Some examples Complete r -part. Complete Bip. Graphs, Complete Graphs, Graphs, ◮ Complements of complete r -partite graphs, ◮ Many small graphs, ◮ etc. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  28. Examples of Digon Graphs Which graphs can be constructed as digon graphs? Some examples Complete r -part. Complete Bip. Graphs, Complete Graphs, Graphs, ◮ Complements of complete r -partite graphs, ◮ Many small graphs, ◮ etc. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  29. Digon Graphs of Pref. Graphs are Perfect when n = 2 Sub-Lemma. If x ≶ y , x ≶ z and y � ≶ z , then either x y z y or not y x z z x Proof : Suppose not, y y x x z z Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  30. Digon Graphs of Pref. Graphs are Perfect when n = 2 Sub-Lemma. If x ≶ y , x ≶ z and y � ≶ z , then either x y z y or not y x z z x Proof : Suppose not, y y x x z z Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  31. Digon Graphs of Pref. Graphs are Perfect when n = 2 Sub-Lemma. If x ≶ y , x ≶ z and y � ≶ z , then either x y z y or not y x z z x Proof : Suppose not, y y x x z z Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  32. Digon Graphs of Pref. Graphs are Perfect when n = 2 Sub-Lemma. If x ≶ y , x ≶ z and y � ≶ z , then either x y z y or not y x z z x Proof : Suppose not, y y x x z z Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  33. Digon Graphs of Pref. Graphs are Perfect when n = 2 Sub-Lemma. If x ≶ y , x ≶ z and y � ≶ z , then either x y z y or not y x z z x Proof : Suppose not, y y x x z z Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  34. Corollary. Long paths “alternate” in the digon graph Corollary. 1. The digon graph contains no odd holes on ≥ 5 vertices. 2. The digon graph contains no antiholes on ≥ 5 vertices. [CRST, ‘06] Strong Perfect Graph Theorem A graph is perfect ⇐ ⇒ contains no odd holes and no odd antiholes. Corollary. Digon Graphs of Pref. Graphs are perfect when n = 2. “So Theorem 1. follows as a corollary.” Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  35. Corollary. Long paths “alternate” in the digon graph Corollary. 1. The digon graph contains no odd holes on ≥ 5 vertices. 2. The digon graph contains no antiholes on ≥ 5 vertices. [CRST, ‘06] Strong Perfect Graph Theorem A graph is perfect ⇐ ⇒ contains no odd holes and no odd antiholes. Corollary. Digon Graphs of Pref. Graphs are perfect when n = 2. “So Theorem 1. follows as a corollary.” Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  36. Corollary. Long paths “alternate” in the digon graph Corollary. 1. The digon graph contains no odd holes on ≥ 5 vertices. 2. The digon graph contains no antiholes on ≥ 5 vertices. [CRST, ‘06] Strong Perfect Graph Theorem A graph is perfect ⇐ ⇒ contains no odd holes and no odd antiholes. Corollary. Digon Graphs of Pref. Graphs are perfect when n = 2. “So Theorem 1. follows as a corollary.” Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  37. Corollary. Long paths “alternate” in the digon graph Corollary. 1. The digon graph contains no odd holes on ≥ 5 vertices. 2. The digon graph contains no antiholes on ≥ 5 vertices. [CRST, ‘06] Strong Perfect Graph Theorem A graph is perfect ⇐ ⇒ contains no odd holes and no odd antiholes. Corollary. Digon Graphs of Pref. Graphs are perfect when n = 2. “So Theorem 1. follows as a corollary.” Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  38. Corollary. Long paths “alternate” in the digon graph Corollary. 1. The digon graph contains no odd holes on ≥ 5 vertices. 2. The digon graph contains no antiholes on ≥ 5 vertices. [CRST, ‘06] Strong Perfect Graph Theorem A graph is perfect ⇐ ⇒ contains no odd holes and no odd antiholes. Corollary. Digon Graphs of Pref. Graphs are perfect when n = 2. “So Theorem 1. follows as a corollary.” Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  39. Digon Graphs as a Class of Perfect Graphs Open problem: Characterize the class of digon graphs of pref. graphs for n = 2. We know: ◮ DG � ≶ Complements of bipartite graphs ◮ DG � ≶ Complements of line gr. of bipartite graphs ◮ DG �⊂ Bipartite graphs ◮ DG �⊂ Line gr. of bipartite graphs And maybe DG � Comparability graphs Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  40. 3-Commodity Markets Theorem 2. The decision problem is NP-complete for n ≥ 3. Proof Sketch. 1. [WK, ‘88] : Reduce planar 3-SAT to vert. cover on “gadget graphs” 2. Remark. Can reduce vert. cover to MFVS on a graph of digons. 3. Lemma. This graph of digons is an “oriented-disk graph” 4. Lemma. Any ODG is a valid preference graph for n = 3. So Theorem 2. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  41. 3-Commodity Markets Theorem 2. The decision problem is NP-complete for n ≥ 3. Proof Sketch. 1. [WK, ‘88] : Reduce planar 3-SAT to vert. cover on “gadget graphs” 2. Remark. Can reduce vert. cover to MFVS on a graph of digons. 3. Lemma. This graph of digons is an “oriented-disk graph” 4. Lemma. Any ODG is a valid preference graph for n = 3. So Theorem 2. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  42. 3-Commodity Markets Theorem 2. The decision problem is NP-complete for n ≥ 3. Proof Sketch. 1. [WK, ‘88] : Reduce planar 3-SAT to vert. cover on “gadget graphs” 2. Remark. Can reduce vert. cover to MFVS on a graph of digons. 3. Lemma. This graph of digons is an “oriented-disk graph” 4. Lemma. Any ODG is a valid preference graph for n = 3. So Theorem 2. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  43. 3-Commodity Markets Theorem 2. The decision problem is NP-complete for n ≥ 3. Proof Sketch. 1. [WK, ‘88] : Reduce planar 3-SAT to vert. cover on “gadget graphs” 2. Remark. Can reduce vert. cover to MFVS on a graph of digons. 3. Lemma. This graph of digons is an “oriented-disk graph” 4. Lemma. Any ODG is a valid preference graph for n = 3. So Theorem 2. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  44. 3-Commodity Markets Theorem 2. The decision problem is NP-complete for n ≥ 3. Proof Sketch. 1. [WK, ‘88] : Reduce planar 3-SAT to vert. cover on “gadget graphs” 2. Remark. Can reduce vert. cover to MFVS on a graph of digons. 3. Lemma. This graph of digons is an “oriented-disk graph” 4. Lemma. Any ODG is a valid preference graph for n = 3. So Theorem 2. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  45. 3-Commodity Markets Theorem 2. The decision problem is NP-complete for n ≥ 3. Proof Sketch. 1. [WK, ‘88] : Reduce planar 3-SAT to vert. cover on “gadget graphs” 2. Remark. Can reduce vert. cover to MFVS on a graph of digons. 3. Lemma. This graph of digons is an “oriented-disk graph” 4. Lemma. Any ODG is a valid preference graph for n = 3. So Theorem 2. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  46. [WK, ‘88] : Reduce planar 3-SAT to vertex cover on “gadget graphs”: ◮ Add a triangle per clause, labelled with variable type ◮ Add a cycle per variable, alternating labels ◮ Connect each clause to the cycles, opposite label. Example. Graph for ϕ = (¯ u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ ¯ u 2 ∨ u 3 ) u 1 u 1 ¯ . . . u 1 gadget ¯ u 1 u 1 . . . C 1 gadget C 2 gadget ¯ . . . u 1 u 1 u 2 ¯ ¯ u 2 u 2 u 2 u 2 gadget . . . u 3 u 3 . . . u 3 u 3 gadget u 3 . . . ¯ u 3 Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  47. [WK, ‘88] : Reduce planar 3-SAT to vertex cover on “gadget graphs”: ◮ Add a triangle per clause, labelled with variable type ◮ Add a cycle per variable, alternating labels ◮ Connect each clause to the cycles, opposite label. Example. Graph for ϕ = (¯ u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ ¯ u 2 ∨ u 3 ) u 1 u 1 ¯ . . . u 1 gadget ¯ u 1 u 1 . . . C 1 gadget C 2 gadget ¯ . . . u 1 u 1 u 2 ¯ ¯ u 2 u 2 u 2 u 2 gadget . . . u 3 u 3 . . . u 3 u 3 gadget u 3 . . . ¯ u 3 Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  48. [WK, ‘88] : Reduce planar 3-SAT to vertex cover on “gadget graphs”: ◮ Add a triangle per clause, labelled with variable type ◮ Add a cycle per variable, alternating labels ◮ Connect each clause to the cycles, opposite label. Example. Graph for ϕ = (¯ u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ ¯ u 2 ∨ u 3 ) u 1 u 1 ¯ . . . u 1 gadget ¯ u 1 u 1 . . . C 1 gadget C 2 gadget ¯ . . . u 1 u 1 u 2 ¯ ¯ u 2 u 2 u 2 u 2 gadget . . . u 3 u 3 . . . u 3 u 3 gadget u 3 . . . ¯ u 3 Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  49. [WK, ‘88] : Reduce planar 3-SAT to vertex cover on “gadget graphs”: ◮ Add a triangle per clause, labelled with variable type ◮ Add a cycle per variable, alternating labels ◮ Connect each clause to the cycles, opposite label. Example. Graph for ϕ = (¯ u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ ¯ u 2 ∨ u 3 ) u 1 u 1 ¯ . . . u 1 gadget ¯ u 1 u 1 . . . C 1 gadget C 2 gadget ¯ . . . u 1 u 1 u 2 ¯ ¯ u 2 u 2 u 2 u 2 gadget . . . u 3 u 3 . . . u 3 u 3 gadget u 3 . . . ¯ u 3 Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  50. [WK, ‘88] : Reduce planar 3-SAT to vertex cover on “gadget graphs”: ◮ Add a triangle per clause, labelled with variable type ◮ Add a cycle per variable, alternating labels ◮ Connect each clause to the cycles, opposite label. Example. Graph for ϕ = (¯ u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ ¯ u 2 ∨ u 3 ) u 1 u 1 ¯ . . . u 1 gadget ¯ u 1 u 1 . . . C 1 gadget C 2 gadget ¯ . . . u 1 u 1 u 2 ¯ ¯ u 2 u 2 u 2 u 2 gadget . . . u 3 u 3 . . . u 3 u 3 gadget u 3 . . . ¯ u 3 Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  51. Reduction To Vertex Cover ... ... ... ... ... ... Note. Have a lower bound on vertex cover: ◮ Every clause needs at least 2 vertices to cover ◮ Every cycle needs at least half [WK,‘88] This bound is attainable if and only if the expression is satisfiable. = ⇒ 3-SAT reduces to Vertex Cover on these graphs. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  52. Reduction To Vertex Cover ... ... ... ... ... ... Note. Have a lower bound on vertex cover: ◮ Every clause needs at least 2 vertices to cover ◮ Every cycle needs at least half [WK,‘88] This bound is attainable if and only if the expression is satisfiable. = ⇒ 3-SAT reduces to Vertex Cover on these graphs. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  53. 3-Commodity Markets Theorem 2. The decision problem is NP-complete for n ≥ 3. Proof Sketch. 1. [WK, ‘88] : Reduce planar 3-SAT to vert. cover on “gadget graphs” 2. Remark. Can reduce vert. cover to MFVS on a graph of digons. 3. Lemma. This graph of digons is an “oriented-disk graph” 4. Lemma. Any ODG is a valid preference graph for n = 3. So Theorem 2. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  54. Oriented Disk Graphs Definition. Let x 1 , . . . , x n be points on the plane, and D 1 , . . . D n be disks such that x i is on D i ’s boundary. Add an arc from x i to x j if x j ∈ D i . Any such graph is an oriented-disk graph. Example. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  55. Oriented Disk Graphs Definition. Let x 1 , . . . , x n be points on the plane, and D 1 , . . . D n be disks such that x i is on D i ’s boundary. Add an arc from x i to x j if x j ∈ D i . Any such graph is an oriented-disk graph. Example. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  56. Oriented Disk Graphs Definition. Let x 1 , . . . , x n be points on the plane, and D 1 , . . . D n be disks such that x i is on D i ’s boundary. Add an arc from x i to x j if x j ∈ D i . Any such graph is an oriented-disk graph. Example. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  57. Oriented Disk Graphs Definition. Let x 1 , . . . , x n be points on the plane, and D 1 , . . . D n be disks such that x i is on D i ’s boundary. Add an arc from x i to x j if x j ∈ D i . Any such graph is an oriented-disk graph. Example. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  58. Oriented Disk Graphs Definition. Let x 1 , . . . , x n be points on the plane, and D 1 , . . . D n be disks such that x i is on D i ’s boundary. Add an arc from x i to x j if x j ∈ D i . Any such graph is an oriented-disk graph. Example. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  59. Vertex Cover Reduces to FVS on a Digon Graph Remark. ◮ Take any undirected graph, and replace each edge with a digon. ◮ Add an arbitrary acyclic set of arcs. ◮ Min. Vertex Cover on the underlying graph is equivalent to MFVS on this digraph. (Any new cycle passes through a digon) Claim. There exists an oriented-disk graph which is a graph of digons for the gadget graph, plus an acyclic set of edges. This would give a reduction from planar 3-SAT to MFVS on ODG’s. Clause Gadgets, Long Paths and Cycles Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  60. Vertex Cover Reduces to FVS on a Digon Graph Remark. ◮ Take any undirected graph, and replace each edge with a digon. ◮ Add an arbitrary acyclic set of arcs. ◮ Min. Vertex Cover on the underlying graph is equivalent to MFVS on this digraph. (Any new cycle passes through a digon) Claim. There exists an oriented-disk graph which is a graph of digons for the gadget graph, plus an acyclic set of edges. This would give a reduction from planar 3-SAT to MFVS on ODG’s. Clause Gadgets, Long Paths and Cycles Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  61. Vertex Cover Reduces to FVS on a Digon Graph Remark. ◮ Take any undirected graph, and replace each edge with a digon. ◮ Add an arbitrary acyclic set of arcs. ◮ Min. Vertex Cover on the underlying graph is equivalent to MFVS on this digraph. (Any new cycle passes through a digon) Claim. There exists an oriented-disk graph which is a graph of digons for the gadget graph, plus an acyclic set of edges. This would give a reduction from planar 3-SAT to MFVS on ODG’s. Clause Gadgets, Long Paths and Cycles Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  62. Vertex Cover Reduces to FVS on a Digon Graph Remark. ◮ Take any undirected graph, and replace each edge with a digon. ◮ Add an arbitrary acyclic set of arcs. ◮ Min. Vertex Cover on the underlying graph is equivalent to MFVS on this digraph. (Any new cycle passes through a digon) Claim. There exists an oriented-disk graph which is a graph of digons for the gadget graph, plus an acyclic set of edges. This would give a reduction from planar 3-SAT to MFVS on ODG’s. Clause Gadgets, Long Paths and Cycles Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  63. Vertex Cover Reduces to FVS on a Digon Graph Remark. ◮ Take any undirected graph, and replace each edge with a digon. ◮ Add an arbitrary acyclic set of arcs. ◮ Min. Vertex Cover on the underlying graph is equivalent to MFVS on this digraph. (Any new cycle passes through a digon) Claim. There exists an oriented-disk graph which is a graph of digons for the gadget graph, plus an acyclic set of edges. This would give a reduction from planar 3-SAT to MFVS on ODG’s. Clause Gadgets, Long Paths and Cycles Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  64. The Gadget Graph is Planar What does this oriented-disk drawing look like? 1. Take a planar drawing of the variable-clause graph. 2. Replace the clause vertices with clause gadgets. ∗ 3. Trace around the edges incident to each variable with its cycle. 4. ∗ How do we connect the cycles to the clause gadgets? C 1 u 3 u 1 u 2 u 4 C 2 C 3 E.g. ϕ = ( u 1 ∨ u 2 ∨ u 4 ) ∧ ( u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ u 3 ∨ u 4 ) Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  65. The Gadget Graph is Planar What does this oriented-disk drawing look like? 1. Take a planar drawing of the variable-clause graph. 2. Replace the clause vertices with clause gadgets. ∗ 3. Trace around the edges incident to each variable with its cycle. 4. ∗ How do we connect the cycles to the clause gadgets? C 1 u 3 u 1 u 2 u 4 C 2 C 3 E.g. ϕ = ( u 1 ∨ u 2 ∨ u 4 ) ∧ ( u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ u 3 ∨ u 4 ) Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  66. The Gadget Graph is Planar What does this oriented-disk drawing look like? 1. Take a planar drawing of the variable-clause graph. 2. Replace the clause vertices with clause gadgets. ∗ 3. Trace around the edges incident to each variable with its cycle. 4. ∗ How do we connect the cycles to the clause gadgets? C 1 u 3 u 1 u 2 u 4 C 3 C 2 E.g. ϕ = ( u 1 ∨ u 2 ∨ u 4 ) ∧ ( u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ u 3 ∨ u 4 ) Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  67. The Gadget Graph is Planar What does this oriented-disk drawing look like? 1. Take a planar drawing of the variable-clause graph. 2. Replace the clause vertices with clause gadgets. ∗ 3. Trace around the edges incident to each variable with its cycle. 4. ∗ How do we connect the cycles to the clause gadgets? C 1 u 3 u 1 u 2 u 4 C 3 C 2 E.g. ϕ = ( u 1 ∨ u 2 ∨ u 4 ) ∧ ( u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ u 3 ∨ u 4 ) Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  68. The Gadget Graph is Planar What does this oriented-disk drawing look like? 1. Take a planar drawing of the variable-clause graph. 2. Replace the clause vertices with clause gadgets. ∗ 3. Trace around the edges incident to each variable with its cycle. 4. ∗ How do we connect the cycles to the clause gadgets? C 1 u 3 u 1 u 2 u 4 C 3 C 2 E.g. ϕ = ( u 1 ∨ u 2 ∨ u 4 ) ∧ ( u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ u 3 ∨ u 4 ) Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  69. The Gadget Graph is Planar What does this oriented-disk drawing look like? 1. Take a planar drawing of the variable-clause graph. 2. Replace the clause vertices with clause gadgets. ∗ 3. Trace around the edges incident to each variable with its cycle. 4. ∗ How do we connect the cycles to the clause gadgets? C 1 u 3 u 1 u 2 u 4 C 3 C 2 E.g. ϕ = ( u 1 ∨ u 2 ∨ u 4 ) ∧ ( u 1 ∨ u 2 ∨ u 3 ) ∧ ( u 1 ∨ u 3 ∨ u 4 ) Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  70. Clause Gadget to Cycle Connection Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  71. Clause Gadget to Cycle Connection Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  72. 3-Commodity Markets Theorem 2. The decision problem is NP-complete for n ≥ 3. Proof Sketch. 1. [WK, ‘88] : Reduce planar 3-SAT to vert. cover on “gadget graphs” 2. Remark. Can reduce vert. cover to MFVS on a graph of digons. 3. Lemma. This graph of digons is an “oriented-disk graph” 4. Lemma. Any ODG is a valid preference graph for n = 3. So Theorem 2. follows as a corollary. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  73. Lemma. Oriented-disk graphs are valid preference graphs for n = 3. Can map any planar drawing onto a small sphere section Map points onto sphere, cut through sphere with plane to get disks (Price vector is plane normal) Vertex revealed preferred to any point inside disk So any ODG is a preference graph, and “ Theorem 2. follows as a corollary.” Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  74. Lemma. Oriented-disk graphs are valid preference graphs for n = 3. Can map any planar drawing onto a small sphere section Map points onto sphere, cut through sphere with plane to get disks (Price vector is plane normal) Vertex revealed preferred to any point inside disk So any ODG is a preference graph, and “ Theorem 2. follows as a corollary.” Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  75. Lemma. Oriented-disk graphs are valid preference graphs for n = 3. Can map any planar drawing onto a small sphere section Map points onto sphere, cut through sphere with plane to get disks (Price vector is plane normal) Vertex revealed preferred to any point inside disk So any ODG is a preference graph, and “ Theorem 2. follows as a corollary.” Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  76. Lemma. Oriented-disk graphs are valid preference graphs for n = 3. Can map any planar drawing onto a small sphere section Map points onto sphere, cut through sphere with plane to get disks (Price vector is plane normal) Vertex revealed preferred to any point inside disk So any ODG is a preference graph, and “ Theorem 2. follows as a corollary.” Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  77. Further Research We have shown that n = 2 is the threshold for poly-time solvability ◮ What about approximation complexity? ◮ Can we provide combinatorial algorithm for n = 2? ◮ Can we characterise preference graphs for fixed n ? ◮ Does bounding DoR affect welfare in a combinatorial auction? (RP constraints are used to impose truthfulness.) Thank you. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  78. Further Research We have shown that n = 2 is the threshold for poly-time solvability ◮ What about approximation complexity? ◮ Can we provide combinatorial algorithm for n = 2? ◮ Can we characterise preference graphs for fixed n ? ◮ Does bounding DoR affect welfare in a combinatorial auction? (RP constraints are used to impose truthfulness.) Thank you. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  79. Further Research We have shown that n = 2 is the threshold for poly-time solvability ◮ What about approximation complexity? ◮ Can we provide combinatorial algorithm for n = 2? ◮ Can we characterise preference graphs for fixed n ? ◮ Does bounding DoR affect welfare in a combinatorial auction? (RP constraints are used to impose truthfulness.) Thank you. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

  80. Further Research We have shown that n = 2 is the threshold for poly-time solvability ◮ What about approximation complexity? ◮ Can we provide combinatorial algorithm for n = 2? ◮ Can we characterise preference graphs for fixed n ? ◮ Does bounding DoR affect welfare in a combinatorial auction? (RP constraints are used to impose truthfulness.) Thank you. Shant Boodaghians and Adrian Vetta McGill University Testing Consumer Rationality using Perfect Graphs and Oriented Discs

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