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# Kernelization algorithms for graph and other structure modiﬁcation problems

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Thesis defense on November 14th, 2011, in Montpellier.

Jury:
Stéphane Bessy, Bruno Durand, Frédéric Havet, Rolf Niedermeier, Christophe Paul & Ioan Todinca.

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### Kernelization algorithms for graph and other structure modiﬁcation problems

1. 1. Kernelization algorithms for graph and other structure modiﬁcation problems Anthony P EREZ ´ Supervisors: Stephane B ESSY and Christophe PAUL (AlGCo Team) November 14
2. 2. I NTRODUCTION (Graph) Modiﬁcation problemsInput: A graph (or another structure) and a (graph) property G.Output: A minimum number of modiﬁcation of the graph in order tosatisfy G.modiﬁcation: adding edges, deleting edges, deleting vertices, ... 2 / 42
3. 3. I NTRODUCTION (Graph) Modiﬁcation problemsInput: A graph (or another structure) and a (graph) property G.Output: A minimum number of modiﬁcation of the graph in order tosatisfy G.modiﬁcation: adding edges, deleting edges, deleting vertices, ... 2 / 42
4. 4. I NTRODUCTION (Graph) Modiﬁcation problems C LUSTER E DITING Input: A graph G = (V , E). Output: A set F ⊆ (V × V ) of minimum size such that the graph H = (V , E F ) is a disjoint union of cliques. 3 / 42
5. 5. I NTRODUCTION (Graph) Modiﬁcation problems C LUSTER E DITING Input: A graph G = (V , E). Output: A set F ⊆ (V × V ) of minimum size such that the graph H = (V , E F ) is a disjoint union of cliques. 3 / 42
6. 6. I NTRODUCTION (Graph) Modiﬁcation problemsCover a broad range of NP-Hard problems: VERTEX COVER FEEDBACK VERTEX SET More general: F - MINOR DELETION EDGE - MULTICUT 4 / 42
7. 7. I NTRODUCTION (Graph) Modiﬁcation problemsFind applications in various domains: bioinformatics machine learning relational databases image processing 4 / 42
8. 8. I NTRODUCTION Different approachesMost modiﬁcation problems are NP-hard.How to solve them efﬁciently? Approximation algorithms Exact exponential algorithms Preprocessing steps (heuristics) 5 / 42
9. 9. I NTRODUCTION Different approachesMost modiﬁcation problems are NP-hard.How to solve them efﬁciently? Approximation algorithms Exact exponential algorithms Preprocessing steps (heuristics) 5 / 42
10. 10. I NTRODUCTION Different approachesMost modiﬁcation problems are NP-hard.How to solve them efﬁciently? Approximation algorithms Exact exponential algorithms Preprocessing steps (heuristics)How to measure the efﬁciency of heuristics? 5 / 42
11. 11. I NTRODUCTION Different approachesMost modiﬁcation problems are NP-hard.How to solve them efﬁciently? Approximation algorithms Exact exponential algorithms Preprocessing steps (heuristics)Exploit the fact that the number of modiﬁcations needed should besmall compared to the instance size n. 5 / 42
12. 12. Outline of the talk1 Parameterized complexity Part I. Graph Modiﬁcation Problems2 Branches and generic reduction rules3 P ROPER I NTERVAL C OMPLETION Part II. Different modiﬁcation problems4 Considered problems5 F EEDBACK A RC S ET IN TOURNAMENTS
13. 13. PARAMETERIZED COMPLEXITY Parameterized problem G-M ODIFICATION Input: A graph G = (V , E), k ∈ N. Parameter: k . Output: A set F ⊆ (V × V ) of size at most k such that the graph H = (V , E F ) belongs to G.Idea. Measure the complexity of a problem with respect tosome parameter k. 7 / 42
14. 14. PARAMETERIZED COMPLEXITY Parameterized problem G-M ODIFICATION Input: A graph G = (V , E), k ∈ N. Parameter: k . Output: A set F ⊆ (V × V ) of size at most k such that the graph H = (V , E F ) belongs to G. Parameterized algorithmA problem parameterized by some k ∈ N admits a parameterizedalgorithm if it can be solved in time f (k ) · nO(1) . 7 / 42
15. 15. PARAMETERIZED COMPLEXITY KernelsGiven an instance (I, k ) of a parameterized problem L,a kernelization algorithm: runs in time Poly (|I| + k)and outputs an instance (I , k ) such that: (i) (I, k ) ∈ YES ⇔ (I , k ) ∈ YES (ii) |I | h(k ) and k k (I , k ) (I , k ) Poly (|I | + k ) |I | h(k ) k k 8 / 42
16. 16. PARAMETERIZED COMPLEXITY KernelsGiven an instance (I, k ) of a parameterized problem L,a kernelization algorithm: runs in time Poly (|I| + k)and outputs an instance (I , k ) such that: (i) (I, k ) ∈ YES ⇔ (I , k ) ∈ YES (ii) |I | h(k ) and k k Theorem (Folklore) Parameterized algorithm ⇔ Kernelization algorithm 8 / 42
17. 17. PARAMETERIZED COMPLEXITY KernelsGiven an instance (I, k ) of a parameterized problem L,a kernelization algorithm: runs in time Poly (|I| + k)and outputs an instance (I , k ) such that: (i) (I, k ) ∈ YES ⇔ (I , k ) ∈ YES (ii) |I | h(k ) and k kSize: super-polynomial 8 / 42
18. 18. PARAMETERIZED COMPLEXITY KernelsGiven an instance (I, k ) of a parameterized problem L,a kernelization algorithm: runs in time Poly (|I| + k)and outputs an instance (I , k ) such that: (i) (I, k ) ∈ YES ⇔ (I , k ) ∈ YES (ii) |I | h(k ) and k kSize: super-polynomialDo all parameterized problems admit polynomial kernels? 8 / 42
19. 19. PARAMETERIZED COMPLEXITY Lower bounds for kernelsThere exist some parameterized problems that do not admit polynomialkernels. (under a complexity-theoretic assumption) (i) Or-composition [Bodlaender et al., 2008 - Fortnow and Santhanam, 2008] (ii) Polynomial time and parameter transformations [Bodlaender et al., 2009] (iii) Cross-composition [Bodlaender et al., 2011] 9 / 42
20. 20. Graph modiﬁcation problems2 Branches and generic reduction rules3 P ROPER I NTERVAL C OMPLETION G-M ODIFICATION Input: A graph G = (V , E), k ∈ N. Parameter: k. Output: A set F ⊆ (V × V ) of size at most k s.t. the graph H = (V , E F ) belongs to G.
21. 21. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesConnected component.If G is hereditary and closed under disjoint union, remove anyconnected component C that belongs to G. 11 / 42
22. 22. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesSunﬂower. Consider a ﬁnite forbidden induced subgraph of G (obstruction). For any pair e ⊆ (V × V ) that belongs to a set of k + 1 obstructions pairwise intersecting exactly in e, transform G into (V , E {e}). 12 / 42
23. 23. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesSunﬂower. Consider a ﬁnite forbidden induced subgraph of G (obstruction). For any pair e ⊆ (V × V ) that belongs to a set of k + 1 obstructions pairwise intersecting exactly in e, transform G into (V , E {e}). 12 / 42
24. 24. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesCritical clique. Assume G is hereditary and closed under true twin addition. For any critical clique T with |T | > k + 1, remove |T | − (k + 1) arbitrary vertices from T . u v 13 / 42
25. 25. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesCritical clique. Assume G is hereditary and closed under true twin addition. For any critical clique T with |T | > k + 1, remove |T | − (k + 1) arbitrary vertices from T . u v 13 / 42
26. 26. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesCritical clique. Assume G is hereditary and closed under true twin addition. For any critical clique T with |T | > k + 1, remove |T | − (k + 1) arbitrary vertices from T . k =1Lemma [Bessy, Paul and P., 2010]There always exists an optimal editionthat preserves the critical cliques. k =1 13 / 42
27. 27. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches: a natural idea Reduce set of vertices that induce a graph belonging to G. The Connected Component rule is a Branch reduction rule. 14 / 42
28. 28. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches: a natural idea Reduce set of vertices that induce a graph belonging to G. The Connected Component rule is a Branch reduction rule. 14 / 42
29. 29. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches: a natural idea Reduce set of vertices that induce a graph belonging to G. The Connected Component rule is a Branch reduction rule.Context: can be used on problems where G admits a so-calledadjacency decomposition. Branch: set of vertices B ⊆ V such that: (i) G[B] ∈ G and, (ii) B is connected properly to the rest of the graph. 14 / 42
30. 30. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches: a natural idea Reduce set of vertices that induce a graph belonging to G. The Connected Component rule is a Branch reduction rule.Context: can be used on problems where G admits a so-calledadjacency decomposition. G [B ] ∈ G Branch: set of vertices B ⊆ V B such that: (i) G[B] ∈ G and, (ii) B is connected properly to the rest of the graph. GB 14 / 42
31. 31. Outline2 Branches and generic reduction rules Generic reduction rules Branches3 P ROPER I NTERVAL C OMPLETION Deﬁnition and known results Branches Reducing the branches
32. 32. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Deﬁnition P ROPER I NTERVAL C OMPLETION Input: A graph G = (V , E), k ∈ N. Parameter: k . Output: A set F ⊆ (V × V ) E of size at most k such that H = (V , E ∪ F ) is a proper interval graph. 16 / 42
33. 33. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Deﬁnition P ROPER I NTERVAL C OMPLETION Input: A graph G = (V , E), k ∈ N. Parameter: k . Output: A set F ⊆ (V × V ) E of size at most k such that H = (V , E ∪ F ) is a proper interval graph. 16 / 42
34. 34. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Deﬁnition P ROPER I NTERVAL C OMPLETION Input: A graph G = (V , E), k ∈ N. Parameter: k . Output: A set F ⊆ (V × V ) E of size at most k such that H = (V , E ∪ F ) is a proper interval graph. 16 / 42
35. 35. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Deﬁnition P ROPER I NTERVAL C OMPLETION Input: A graph G = (V , E), k ∈ N. Parameter: k . Output: A set F ⊆ (V × V ) E of size at most k such that H = (V , E ∪ F ) is a proper interval graph. 16 / 42
36. 36. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Deﬁnition P ROPER I NTERVAL C OMPLETION Input: A graph G = (V , E), k ∈ N. Parameter: k . Output: A set F ⊆ (V × V ) E of size at most k such that H = (V , E ∪ F ) is a proper interval graph. NP-Complete [Golumbic et al., 1994] FPT : O(24k m) (motivated by applications in genomic research) [Kaplan, Shamir and Tarjan, 1994] Polynomial kernel? 16 / 42
37. 37. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Deﬁnition P ROPER I NTERVAL C OMPLETION Input: A graph G = (V , E), k ∈ N. Parameter: k . Output: A set F ⊆ (V × V ) E of size at most k such that H = (V , E ∪ F ) is a proper interval graph. Theorem [Bessy and P., 2011]The P ROPER I NTERVAL C OMPLETION problem admits a kernel withO(k 4 ) vertices. 16 / 42
38. 38. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Some useful resultsA graph is a proper interval graph if and only if: it does not contain any of the following graphs as an induced subgraph. claw 3-sun net p-cycle (p ≥ 4)[Wegner, 1967] 17 / 42
39. 39. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Some useful resultsA graph is a proper interval graph if and only if: its vertices admit an ordering v1 . . . vn such that: vi vj ∈ E i < j ⇒ vi vl , vl vj ∈ E, i < l < j[Looges and Olartu, 1993] 17 / 42
40. 40. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesRemarks. Proper interval graphs are hereditary and: (i) closed under disjoint union: the Connected Component rule can be applied. (ii) do not admit any claw or C4 as an induced subgraph: the Sunﬂower rule can be applied. (iii) closed under true twin addition: the Critical Clique rule can be applied. 18 / 42
41. 41. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesRemarks. Proper interval graphs are hereditary and: (i) closed under disjoint union: the Connected Component rule can be applied. (ii) do not admit any claw or C4 as an induced subgraph: the Sunﬂower rule can be applied. (iii) closed under true twin addition: the Critical Clique rule can be applied. 18 / 42
42. 42. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesRemarks. Proper interval graphs are hereditary and: (i) closed under disjoint union: the Connected Component rule can be applied. (ii) do not admit any claw or C4 as an induced subgraph: the Sunﬂower rule can be applied. (iii) closed under true twin addition: the Critical Clique rule can be applied. 18 / 42
43. 43. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Generic reduction rulesRemarks. Proper interval graphs are hereditary and: (i) closed under disjoint union: the Connected Component rule can be applied. (ii) do not admit any claw or C4 as an induced subgraph: the Sunﬂower rule can be applied. (iii) closed under true twin addition: the Critical Clique rule can be applied.What about branches? 18 / 42
44. 44. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Adjacency decomposition 1 3 6 8(a) 2 4 5 7 9 4 6 7(b) 3 8 2 1 2 3 4 5 6 7 8 9 5 9 1 19 / 42
45. 45. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Adjacency decomposition 1 3 6 8(a) 2 4 5 7 9 4 6 7(b) 3 8 2 1 2 3 4 5 6 7 8 9 5 9 1 19 / 42
46. 46. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Adjacency decomposition 1 3 6 8(a) 2 4 5 7 9 4 6 7(b) 3 8 2 1 2 3 4 5 6 7 8 9 5 9 1 Branches can be used on P ROPER I NTERVAL C OMPLETION. 19 / 42
47. 47. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION How to deﬁne a branch? Consider the structure of a solution. Look at unaffected vertices. 20 / 42
48. 48. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches B B1 BR B2 b1 bl L R CA subset B of V is a branch if: (i) G[B] is a connected PIG with umbrella ordering σB = b1 , . . . , b|B| , (ii) The vertex set V B can be partitioned into sets L, R and C with: no edges between B and C every vertex in L (resp. R) has a neighbor in B NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl } NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| } NL (bi+1 ) ⊆ NL (bi ) for every 1 ≤ i < l and NR (bi ) ⊆ NR (bi+1 ) for every l ≤ i < |B| 21 / 42
49. 49. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches B B1 BR B2 b1 b|B| L R CA subset B of V is a branch if: (i) G[B] is a connected PIG with umbrella ordering σB = b1 , . . . , b|B| , (ii) The vertex set V B can be partitioned into sets L, R and C with: no edges between B and C every vertex in L (resp. R) has a neighbor in B NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl } NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| } NL (bi+1 ) ⊆ NL (bi ) for every 1 ≤ i < l and NR (bi ) ⊆ NR (bi+1 ) for every l ≤ i < |B| 21 / 42
50. 50. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches B B1 BR B2 b1 b|B| L R CA subset B of V is a branch if: (i) G[B] is a connected PIG with umbrella ordering σB = b1 , . . . , b|B| , (ii) The vertex set V B can be partitioned into sets L, R and C with: no edges between B and C every vertex in L (resp. R) has a neighbor in B NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl } NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| } NL (bi+1 ) ⊆ NL (bi ) for every 1 ≤ i < l and NR (bi ) ⊆ NR (bi+1 ) for every l ≤ i < |B| 21 / 42
51. 51. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches B B1 BR B2 b1 bl bl b|B| L R CA subset B of V is a branch if: (i) G[B] is a connected PIG with umbrella ordering σB = b1 , . . . , b|B| , (ii) The vertex set V B can be partitioned into sets L, R and C with: no edges between B and C every vertex in L (resp. R) has a neighbor in B NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl } NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| } NL (bi+1 ) ⊆ NL (bi ) for every 1 ≤ i < l and NR (bi ) ⊆ NR (bi+1 ) for every l ≤ i < |B| 21 / 42
52. 52. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches B B1 BR B2 b1 bl bl b|B| L R CA subset B of V is a branch if: (i) G[B] is a connected PIG with umbrella ordering σB = b1 , . . . , b|B| , (ii) The vertex set V B can be partitioned into sets L, R and C with: no edges between B and C every vertex in L (resp. R) has a neighbor in B NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl } NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| } NL (bi+1 ) ⊆ NL (bi ) for every 1 ≤ i < l and NR (bi ) ⊆ NR (bi+1 ) for every l ≤ i < |B| 21 / 42
53. 53. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches B B1 BR B2 b1 bl bl b|B| L R CA subset B of V is a branch if: (i) G[B] is a connected PIG with umbrella ordering σB = b1 , . . . , b|B| , (ii) The vertex set V B can be partitioned into sets L, R and C with: no edges between B and C every vertex in L (resp. R) has a neighbor in B NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl } NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| } NL (bi+1 ) ⊆ NL (bi ) for every 1 ≤ i < l and NR (bi ) ⊆ NR (bi+1 ) for every l ≤ i < |B| 21 / 42
54. 54. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches B B1 BR B2 b1 bl bl b|B| L R CA subset B of V is a branch if: (i) G[B] is a connected PIG with umbrella ordering σB = b1 , . . . , b|B| , (ii) The vertex set V B can be partitioned into sets L, R and C with: no edges between B and C every vertex in L (resp. R) has a neighbor in B NB (L) ⊂ NB [b1 ] = {b1 , . . . , bl } NB (R) ⊂ NB [b|B| ] = {bl , . . . , b|B| } NL (bi+1 ) ⊆ NL (bi ) for every 1 ≤ i < l and NR (bi ) ⊆ NR (bi+1 ) for every l ≤ i < |B| 21 / 42
55. 55. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches B B1 BR B2 b1 bl L R C If L = ∅ (or R = ∅), B is a 1-branch, otherwise B is a 2-branch If B is a clique, we call B a K-join 22 / 42
56. 56. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Branches B B1 BR B2 b1 bl L R C If L = ∅ (or R = ∅), B is a 1-branch, otherwise B is a 2-branch If B is a clique, we call B a K-join 22 / 42
57. 57. Outline2 Branches and generic reduction rules Generic reduction rules Branches3 P ROPER I NTERVAL C OMPLETION Deﬁnition and known results Branches Reducing the branches
58. 58. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Reducing the K -joinsCannot be done directly. x y z t 24 / 42
59. 59. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Reducing the K -joinsCannot be done directly.A clean K -join does not intersect any claw or C4 . 24 / 42
60. 60. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Reducing the K -joinsCannot be done directly.A clean K -join does not intersect any claw or C4 .Assuming the graph is reduced by the generic rules, we can removeO(k 3 ) vertices from any K -join to obtain a clean K -join. 24 / 42
61. 61. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Reducing the clean K -joinsLet B be a clean K -join of size at least 2k + 2. Let Bf be the k + 1 ﬁrstvertices of B, Bl be its k + 1 last vertices and M = B (Bf ∪ Bl ).Remove the set of vertices M from G. Bf (k + 1 vertices) M Bl (k + 1 vertices) 25 / 42
62. 62. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Reducing the clean K -joinsLet B be a clean K -join of size at least 2k + 2. Let Bf be the k + 1 ﬁrstvertices of B, Bl be its k + 1 last vertices and M = B (Bf ∪ Bl ).Remove the set of vertices M from G.Can be carried out in polynomial time! 25 / 42
63. 63. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Reducing the branchesIn polynomial time, the 1- and 2-branches can be reduced to O(k 3 )vertices. Remove 2k + 1 vertices BR B1 R G (B ∪ R ) B 26 / 42
64. 64. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Reducing the branchesIn polynomial time, the 1- and 2-branches can be reduced to O(k 3 )vertices. Remove 2k + 1 vertices BR B1 R G (B ∪ R ) B 2k + 1 vertices Remove 2k + 1 vertices B1 B1 BR B2 B2 L R B 26 / 42
65. 65. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Main result Theorem [Bessy and P., 2011]The P ROPER I NTERVAL C OMPLETION problem admits a kernel withO(k 4 ) vertices. 1-branch K -join K -join 2-branch K -join 1-branch 27 / 42
66. 66. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Main result Theorem [Bessy and P., 2011]The P ROPER I NTERVAL C OMPLETION problem admits a kernel withO(k 4 ) vertices. 1-branch K -join K -join 2-branch K -join 1-branch O (k 3 ) O (k 3 ) O (k 3 ) O (k 3 ) O (k 3 ) O (k 3 ) 27 / 42
67. 67. B RANCHES AND GENERIC REDUCTION RULES P ROPER I NTERVAL C OMPLETION Main result Theorem [Bessy and P., 2011]The P ROPER I NTERVAL C OMPLETION problem admits a kernel withO(k 4 ) vertices. Related result [Bessy, Paul and P., 2010]The C LOSEST 3-L EAF P OWER problem admits a kernel with O(k 3 )vertices. 27 / 42
68. 68. Different modiﬁcation problems4 Considered problems5 F EEDBACK A RC S ET IN TOURNAMENTS Π-E DITION Input: A dense set R of p-sized relations deﬁned over an universe V , an integer k ∈ N. Parameter: k. Output: A set F ⊆ R of size at most k whose modiﬁcation satisﬁes Π.
69. 69. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS F EEDBACK A RC S ET IN TOURNAMENTS (FAST)Input: A tournament T = (V , A) and an integer k ∈ N.Parameter: k .Output: A set at most k arcs whose reversal results in an acyclictournament. 1 4 3 1 4 2 2 3 29 / 42
70. 70. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS F EEDBACK A RC S ET IN TOURNAMENTS (FAST)Input: A tournament T = (V , A) and an integer k ∈ N.Parameter: k .Output: A set at most k arcs whose reversal results in an acyclictournament. NP-Complete [Charbit et al., 2007] Admits constant-factor approximation algorithms [Kenyon-Mathieu and Schudy, 2007] 29 / 42
71. 71. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS D ENSE R OOTED T RIPLET I NCONSISTENCY (RTI)Input: A set of leaves V and a dense collection R of rooted binarytrees over three leaves of V .Parameter: k .Output: A set of at most k triplets whose modiﬁcation leads to acollection admitting a consistent rooted binary tree deﬁned over V . t1 t2 t3 t4 a b c c d b a b d a c d R := {t1 , t2 , t3 , t4 } R := {ab|c, cd|b, ab|d, ac|d} 30 / 42
72. 72. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS D ENSE R OOTED T RIPLET I NCONSISTENCY (RTI)Input: A set of leaves V and a dense collection R of rooted binarytrees over three leaves of V .Parameter: k .Output: A set of at most k triplets whose modiﬁcation leads to acollection admitting a consistent rooted binary tree deﬁned over V . t1 t2 t3 t4 a b c c d b a b d a c d R := {t1 , t2 , t3 , t4 } R := {ab|c, cd|b, ab|d, ac|d} T is not consistent with R a b c d 30 / 42
73. 73. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS D ENSE R OOTED T RIPLET I NCONSISTENCY (RTI)Input: A set of leaves V and a dense collection R of rooted binarytrees over three leaves of V .Parameter: k .Output: A set of at most k triplets whose modiﬁcation leads to acollection admitting a consistent rooted binary tree deﬁned over V . t1 t2 t3 t4 a b c c d b a b d c d a R := {t1 , t2 , t3 , t4 } R := {ab|c, cd|b, ab|d, cd|a} T is consistent with R a b c d 30 / 42
74. 74. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS D ENSE R OOTED T RIPLET I NCONSISTENCY (RTI)Input: A set of leaves V and a dense collection R of rooted binarytrees over three leaves of V .Parameter: k .Output: A set of at most k triplets whose modiﬁcation leads to acollection admitting a consistent rooted binary tree deﬁned over V . NP-Complete [Barky et al., 2010] Does not admit a constant-factor approximation algorithm yet 30 / 42
75. 75. Outline4 Considered problems F EEDBACK A RC S ET IN TOURNAMENTS D ENSE R OOTED T RIPLET I NCONSISTENCY Conﬂict Packing5 F EEDBACK A RC S ET IN TOURNAMENTS Reduction rules Conﬂict Packing
76. 76. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS ConsistencyFAST (folklore)The following properties are equivalent: (i) T is acyclic (ii) T does not contain any directed triangle 32 / 42
77. 77. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS ConsistencyRTI [Guillemot and Mnich, 2010]The following properties are equivalent: (i) R is consistent (ii) R does not contain any conﬂict on four leavesConﬂict. Set of vertices C ⊆ V that does not admit a consistent rooted binary tree. 32 / 42
78. 78. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Parameterized complexity √ FAST RTI 1/3 FPT O ∗ (2 k log k ) a FPT O ∗ (2k log k ) b Kernel with O(k 2 ) verticesa Kernel with O(k 2 ) verticesb Linear vertex-kernel c No such result known before. a [Alon et al., 2009] b [Guillemot and Mnich, 2010] c [Bessy et al., 2009] 33 / 42
79. 79. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Parameterized complexity √ FAST RTI 1/3 FPT O ∗ (2 k log k ) a FPT O ∗ (2k log k ) b Kernel with O(k 2 ) verticesa Kernel with O(k 2 ) verticesb Linear vertex-kernel c No such result known before. a [Alon et al., 2009] b [Guillemot and Mnich, 2010] c [Bessy et al., 2009] 33 / 42
80. 80. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Parameterized complexity √ FAST RTI 1/3 FPT O ∗ (2 k log k ) a FPT O ∗ (2k log k ) b Kernel with O(k 2 ) verticesa Kernel with O(k 2 ) verticesb Linear vertex-kernel c No such result known before. a [Alon et al., 2009] b [Guillemot and Mnich, 2010] c [Bessy et al., 2009] 33 / 42
81. 81. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Parameterized complexity √ FAST RTI 1/3 FPT O ∗ (2 k log k ) a FPT O ∗ (2k log k ) b Kernel with O(k 2 ) verticesa Kernel with O(k 2 ) verticesb Linear vertex-kernel c No such result known before. a [Alon et al., 2009] b [Guillemot and Mnich, 2010] c [Bessy et al., 2009] The linear vertex-kernel for FAST described by [Bessy et al., 2009] uses a constant-factor approximation algorithm. Their reduction rules can be adapted to RTI. But no constant-factor approximation! 33 / 42
82. 82. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Parameterized complexity √ FAST RTI 1/3 FPT O ∗ (2 k log k ) a FPT O ∗ (2k log k ) b Kernel with O(k 2 ) verticesa Kernel with O(k 2 ) verticesb Linear vertex-kernel c No such result known before. a [Alon et al., 2009] b [Guillemot and Mnich, 2010] c [Bessy et al., 2009] The linear vertex-kernel for FAST described by [Bessy et al., 2009] uses a constant-factor approximation algorithm. Their reduction rules can be adapted to RTI. But no constant-factor approximation! 33 / 42
83. 83. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Conﬂict Packing ´[Paul, P. and Thomasse, 2011] works on problems characterized by some ﬁnite conﬂicts. maximal collection of p-uplets disjoint conﬂits C. provides a lower bound on the number of modiﬁcation required. implies that the instance induced by V V (C) is consistent. 34 / 42
84. 84. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Reduction rulesRemove any vertex that is not part of any directed triangle. a . a can be carried out in polynomial time. 35 / 42
85. 85. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Reduction rulesSafe partition Assume V (T ) is ordered under some ordering σ, and let P be a partition of σ into intervals. V1 V2 Vl AI := {uv ∈ A | ∃ i u , v ∈ Vi } AO := A AI B is the set of backward arcs of AO (arcs vi vj with i > j). 35 / 42
86. 86. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Reduction rulesSafe partition Assume V (T ) is ordered under some ordering σ, and let P be a partition of σ into intervals. V1 V2 Vl AI := {uv ∈ A | ∃ i u , v ∈ Vi } AO := A AI B is the set of backward arcs of AO (arcs vi vj with i > j). 35 / 42
87. 87. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Reduction rulesSafe partition P is safe if there exist |B| arc-disjoint conﬂicts using arcs of AO only. 35 / 42
88. 88. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Safe Partition Reduction Rule[Bessy et al., 2009]Let P be a safe partition of an ordered tournament T = (V , A, σ).Reverse every arc of B and decrease k accordingly. Use constant-factor approximation algorithm. Use Conﬂict Packing. 36 / 42
89. 89. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Safe Partition Reduction Rule[Bessy et al., 2009]Let P be a safe partition of an ordered tournament T = (V , A, σ).Reverse every arc of B and decrease k accordingly. Use constant-factor approximation algorithm. Use Conﬂict Packing. 36 / 42
90. 90. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Safe Partition Reduction Rule[Bessy et al., 2009]Let P be a safe partition of an ordered tournament T = (V , A, σ).Reverse every arc of B and decrease k accordingly.Main questionHow to compute a safe partition in polynomial time? Use constant-factor approximation algorithm. Use Conﬂict Packing. 36 / 42
91. 91. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Safe Partition Reduction Rule[Bessy et al., 2009]Let P be a safe partition of an ordered tournament T = (V , A, σ).Reverse every arc of B and decrease k accordingly.Main questionHow to compute a safe partition in polynomial time? Use constant-factor approximation algorithm. Use Conﬂict Packing. 36 / 42
92. 92. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Safe Partition Reduction Rule[Bessy et al., 2009]Let P be a safe partition of an ordered tournament T = (V , A, σ).Reverse every arc of B and decrease k accordingly.Main questionHow to compute a safe partition in polynomial time? Use constant-factor approximation algorithm. Use Conﬂict Packing. 36 / 42
93. 93. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Conﬂict PackingA conﬂict packing of a tournament is a maximal collection ofarc-disjoint directed triangles. 37 / 42
94. 94. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Conﬂict PackingA conﬂict packing of a tournament is a maximal collection ofarc-disjoint directed triangles. Can be computed greedily (i.e. in polynomial time). Let C be a conﬂict packing. If T = (V , A) is a positive instance then |V (C)| 3k. 37 / 42
95. 95. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Conﬂict PackingA conﬂict packing of a tournament is a maximal collection ofarc-disjoint directed triangles.Conﬂict Packing Lemma [Paul, P. and Thomasse, 2011] ´Let T = (V , A) be a tournament. There exists an ordering of T whosebackward arcs uv are such that u, v ∈ V (C). 37 / 42
96. 96. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Conﬂict PackingA conﬂict packing of a tournament is a maximal collection ofarc-disjoint directed triangles.Lemma [Paul, P. and Thomasse, 2011] ´Let T = (V , A) be a tournament such that |V | > 4k. There exists a safepartition that can be computed in polynomial time. proof 37 / 42
97. 97. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Conﬂict PackingA conﬂict packing of a tournament is a maximal collection ofarc-disjoint directed triangles.Corollary [Paul, P. and Thomasse, 2011] ´F EEDBACK A RC S ET IN TOURNAMENTS admits a kernel with at most 4kvertices. 37 / 42
98. 98. C ONSIDERED PROBLEMS F EEDBACK A RC S ET IN TOURNAMENTS Application to the RTI problem Remove vertices that do not belong to any conﬂict Safe Partition reduction rule Conﬂict Packing allows to ﬁnd a Safe PartitionTheorem [Paul, P. and Thomasse, 2011] ´D ENSE R OOTED T RIPLET I NCONSISTENCY admits a kernel with at most5k vertices. 38 / 42
99. 99. Conclusion6 Our results7 Open problems
100. 100. O UR RESULTS O PEN PROBLEMS Main resultsPolynomial kernels First polynomial kernels: (i) C LOSEST 3-L EAF P OWER (ii) P ROPER I NTERVAL C OMPLETION (iii) C OGRAPH E DGE -E DITION Improved polynomial kernels: (i) F EEDBACK A RC S ET IN TOURNAMENTS (ii) D ENSE R OOTED T RIPLET I NCONSISTENCY (iii) D ENSE B ETWEENNESS and D ENSE C IRCULAR O RDERINGjoint works with: S. Bessy, F. Fomin, S. Gaspers, S. Guillemot, F. Havet, C. Paul,S. Saurabh and S. Thomasse. ´ 40 / 42
101. 101. O UR RESULTS O PEN PROBLEMS Main resultsLower bounds on kernelization: (i) For any l 7, the Pl -F REE E DGE -D ELETION problem does not admit a polynomial kernel. (ii) For any l 4, the Cl -F REE E DGE -D ELETION problem does not admit a polynomial kernel.joint work with: S. Guillemot, F. Havet and C. Paul. 40 / 42
102. 102. O UR RESULTS O PEN PROBLEMS Open problems Do the F EEDBACK V ERTEX S ET IN TOURNAMENTS and C LUSTER V ERTEX D ELETION problems admit linear vertex-kernels? Characterize lower bounds for modiﬁcation problems. details Can we use branches on other problems? (e.g. C HORDAL D ELETION) Can we use Conﬂict Packing on other problems? (e.g. (weakly)-fragile constraint modiﬁcation problems) 41 / 42
103. 103. O UR RESULTS O PEN PROBLEMS Open problems Do the F EEDBACK V ERTEX S ET IN TOURNAMENTS and C LUSTER V ERTEX D ELETION problems admit linear vertex-kernels? Characterize lower bounds for modiﬁcation problems. details Can we use branches on other problems? (e.g. C HORDAL D ELETION) Can we use Conﬂict Packing on other problems? (e.g. (weakly)-fragile constraint modiﬁcation problems) 41 / 42
104. 104. O UR RESULTS O PEN PROBLEMS Open problems Do the F EEDBACK V ERTEX S ET IN TOURNAMENTS and C LUSTER V ERTEX D ELETION problems admit linear vertex-kernels? Characterize lower bounds for modiﬁcation problems. details Can we use branches on other problems? (e.g. C HORDAL D ELETION) Can we use Conﬂict Packing on other problems? (e.g. (weakly)-fragile constraint modiﬁcation problems) 41 / 42
105. 105. Merci de votre attention !