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1. 1. On the complexity of turning a graph into the analogue of a clique Julien Bensmail 1 Romaric Duvignau 1 Sergey Kirgizov 2 1Combinatoire et Algorithmique ⊂ LaBRI ⊂ (Université de Bordeaux × CNRS) 2Complex Networks ⊂ LIP6 ⊂ (UPMC × CNRS) 30 June – 4 July 2014
2. 2. G orientation −−−−−−−→ −→ G
3. 3. Sometimes −→ G is o-clique
4. 4. Does such orientation exist for a given graph?
5. 5. How diﬃcult is it to ﬁnd the answer?
6. 6. Outline 1 Some deﬁnitions 2 Can a given graph be turned into o-clique? 3 Family of related problems 4 Open question 3
7. 7. Deﬁnitions: orientations, o-cliques, oriented distances and diameters, Klostermeyer-MacGillivray lemma
8. 8. Orientation G orientation −−−−−−→ −→ G is an assignment of a direction to each edge from undirected graph. •• • • •• • • •• • • o1 o2 4
9. 9. Oriented chromatic number χo( −→ G ) is the minimal number of colors such that: 1 Colors of adjacent vertices are diﬀerent • • 2 All arcs between two colors have the same direction • • • • • • 5
10. 10. O-clique oriented analogue of usual clique Undirected G is clique ⇐⇒ χ(G) = n • • •• Oriented −→ G is o-clique ⇐⇒ χo( −→ G ) = n • • •• • • • 6
11. 11. Oriented distances and diameters Strong dists = max dist(u, v) dist(v, u) diams = max u,v∈ −→ G dists(u, v) Weak distw = min dist(u, v) dist(v, u) diamw = max u,v∈ −→ G distw (u, v) 7
12. 12. O-cliques & weak diameter −→ G is o-clique χo( −→ G ) = n diamw ( −→ G ) ≤ 2 [Klostermeyer, MacGillivray, 2004] 8
13. 13. Decision problem: Does G admit a 2-weak orientation?
14. 14. Does G admit a 2-weak orientation? An orientation G → −→ G is called 2-weak when diamw ( −→ G ) ≤ 2 • • • ? 9
15. 15. Does G admit a 2-weak orientation? An orientation G → −→ G is called 2-weak when diamw ( −→ G ) ≤ 2 • • • • • • 9
16. 16. Does G admit a 2-weak orientation? An orientation G → −→ G is called 2-weak when diamw ( −→ G ) ≤ 2 • • • •• • • • • • ? 9
17. 17. Does G admit a 2-weak orientation? An orientation G → −→ G is called 2-weak when diamw ( −→ G ) ≤ 2 • • • •• • • • • • •• • • •• • • •• • • •• • • 9
18. 18. Does G admit a 2-weak orientation? An orientation G → −→ G is called 2-weak when diamw ( −→ G ) ≤ 2 • • • •• • • • • • •• • • •• • • •• • • •• • • diamw = ∞ for any orientation of a claw 9
19. 19. 2-weak orientation is NP-complete 2-weak orientation is in NP We just run BFS from all vertices to check whether diamw ( −→ G ) ≤ 2 or not. 2-weak orientation is NP-hard We prove this by reduction from the monotone version of Not-All-Equal 3SAT 10
20. 20. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Monotone Not-All-Equal 3SAT F is 3CNF formula without negations. Example: (x1 ∨ x2 ∨ x3) ∧ (x3 ∨ x3 ∨ x4) ∧ . . . Is F satisﬁable in such way that no clause have all literals set to same value? 11
21. 21. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Monotone Not-All-Equal 3SAT F is 3CNF formula without negations. Example: (x1 ∨ x2 ∨ x3) ∧ (x3 ∨ x3 ∨ x4) ∧ . . . Is F satisﬁable in such way that no clause have all literals set to same value? NP-complete even when in every clause all variables are diﬀerent 11
22. 22. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Reduction overview: F G 2 this diagram commutes F = nae-3-formulae G = graphs 2 = {good, bad} 12
23. 23. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G F → G F is nae-satisﬁable ⇐⇒ ∃ −→ G : diamw ( −→ G ) ≤ 2 (•, • ) is a representative pair from G (•, • ) is a non-representative pair from G ∃ −→ G : distw (•, • ) ≤ 2 ⇐⇒ F is nae-satisﬁable ∃ −→ G : distw (•, • ) ≤ 2 by construction of G 13
24. 24. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Variable gadget: xi −−→ • vi • vi xi = TRUE −−→ • vi • vi xi = FALSE −−→ • vi • vi 14
25. 25. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
26. 26. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
27. 27. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
28. 28. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
29. 29. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
30. 30. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
31. 31. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
32. 32. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
33. 33. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
34. 34. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
35. 35. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Gadget for clause: Ct = xi ∨ xj ∨ xk −−→ • vi • vi • vi,t • vj • vj • vj,t • vk • vk • vk,t • wt • wt Ct is nae-satisﬁable ⇐⇒ ∃ −→ G : distw (•, • ) ≤ 2 Using wt we can make distw (•, • ) ≤ 2 when it is needed 15
36. 36. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G GF is a union of clause gadgets. G = GF + smth. For any pair of vertices (u, v) from GF we add the following: • u • • • • • v distw (u, •) ≤ 2 distw (v, •) ≤ 2 Here distw (u, v) = 3, so we don’t break our reduction by adding blue vertices. 16
37. 37. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Also, for any non-representative pair of vertices (u,v) from GF we add the following: • u • • • • • v • distw (u, •) = 1 distw (v, •) = 1 Here distw (u, v) = 2, so we ensure that the distance between non-representative vertices is ≤ 2 17
38. 38. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Next, in order to ensure that ∀y ∈ GF : distw (y, •) ≤ 2 we add the following: • u • • • • • v • • • y — any other vertex from GF • distw (y, •) = 2 18
39. 39. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G Finally, turn all brown and blue vertices into one big clique, by adding all possible edges between them. 19
40. 40. 2-weak orientation is NP-complete (xi ∨ xj ∨ xk) ∧ . . . ∧ (xi ∨ xk ∨ xl ) → G ∃ −→ G : distw (•, • ) ≤ 2 ⇐⇒ F is nae-satisﬁable ∃ −→ G : distw (•, • ) ≤ 2 ∃ −→ G : distw (•, • ) ≤ 2 ∃ −→ G : distw (•, • ) ≤ 2 ∃ −→ G : distw (•, • ) ≤ 2 ∃ −→ G : distw (•, • ) ≤ 2 ∃ −→ G : distw (•, • ) ≤ 2    by construction of G F is nae-satisﬁable ⇐⇒ ∃ −→ G : diamw ( −→ G ) ≤ 2 20
41. 41. Family of related problems: Does G admit an orientation G → −→ G such that prop( −→ G )?
42. 42. Family of o-problems Does G admit an orientation G → −→ G such that prop( −→ G )? Strong orientation: diams( −→ G ) < ∞ ⇐⇒ G is bridgeless [Robbins, 1939] Linear time using Tarjan’s bridge-ﬁnding algorithm [1974] k-strong orientation: diams( −→ G ) ≤ k NP-complete for k = 2 [Chvátal, Thomassen, 1978] 21
43. 43. Family of o-problems Does G admit an orientation G → −→ G such that prop( −→ G )? Weak orientation: diamw ( −→ G ) < ∞ ⇐⇒ B-contraction(G) is claw-free. B-contraction means “replace all bridgeless subgraphs by vertices” Linear time [Bensmail, Duvignau, Kirgizov 2013] 22
44. 44. Family of o-problems Does G admit an orientation G → −→ G such that prop( −→ G )? k-weak orientation: diamw ( −→ G ) ≤ k NP-complete for k ≥ 2 [Bensmail, Duvignau, Kirgizov, 2013] Gk is a k-edge-coloured graph (e.g. 2-edge coloured graphs are just signiﬁed graphs) ∃ −→ G : diamw ( −→ G ) ≤ k ⇐⇒ ∃Gk : χk(Gk) = n 23
45. 45. Open question: Complexity of k-strong orientation when k > 2 Thank you for your attention kirgizov.complexnetworks.fr/some-problems/orientation.html