Lower bounds for polynomial kernelization

Transcription

1 Lower bounds for polynomial kernelization Part 2 Micha l Pilipczuk Institutt for Informatikk, Universitetet i Bergen August 20 th, 2014 Micha l Pilipczuk Kernelization lower bounds, part 2 1/33

2 Outline Goal: how to prove that for some problems polynomial kernels do not exist? Part 1: Introduction of the (cross)-composition framework. Basic example. Part 2: PPT reductions. Case study of several cross-compositions. Weak compositions. Micha l Pilipczuk Kernelization lower bounds, part 2 2/33

3 In the previous episode... Composition: an algorithm composing many instances into one instance simulating their OR. Micha l Pilipczuk Kernelization lower bounds, part 2 3/33

4 In the previous episode... Composition: an algorithm composing many instances into one instance simulating their OR. The new instance has parameter bounded polynomially in the maximum size of an input instance. Micha l Pilipczuk Kernelization lower bounds, part 2 3/33

5 In the previous episode... Composition: an algorithm composing many instances into one instance simulating their OR. The new instance has parameter bounded polynomially in the maximum size of an input instance. Composition+Compression gives OR-Distillation Micha l Pilipczuk Kernelization lower bounds, part 2 3/33

6 In the previous episode... Composition: an algorithm composing many instances into one instance simulating their OR. The new instance has parameter bounded polynomially in the maximum size of an input instance. Composition+Compression gives OR-Distillation OR-Distillation of an NP-hard language contradicts conp NP/poly. Micha l Pilipczuk Kernelization lower bounds, part 2 3/33

7 In the previous episode... Composition: an algorithm composing many instances into one instance simulating their OR. The new instance has parameter bounded polynomially in the maximum size of an input instance. Composition+Compression gives OR-Distillation OR-Distillation of an NP-hard language contradicts conp NP/poly. Corollary: To show no-poly-kernel it suffices to construct a composition algorithm. Micha l Pilipczuk Kernelization lower bounds, part 2 3/33

8 In the previous episode... Cross-composition An unparameterized problem Q cross-composes into a parameterized problem L, if there exists a polynomial equivalence relation R and an algorithm that, given R-equivalent strings x 1, x 2,..., x t, in time poly ( t + t i=1 x i ) produces one instance (y, k ) such that (y, k ) L iff x i Q for at least one i = 1, 2,..., t, k = poly (log t + max t i=1 x i ). Cross-composition theorem Bodlaender et al.; STACS 2011, SIDMA 2014 If some NP-hard problem Q cross-composes into L, then L does not admit a polynomial compression into any language R, unless NP conp/poly. Micha l Pilipczuk Kernelization lower bounds, part 2 4/33

9 PPTs Idea: Hardness of kernelization can be transferred via reductions, similarly to NP-hardness. Micha l Pilipczuk Kernelization lower bounds, part 2 5/33

10 PPTs Idea: Hardness of kernelization can be transferred via reductions, similarly to NP-hardness. Polynomial parameter transformation (PPT) A polynomial parameter transformation from a parameterized problem P to a parameterized problem Q is a polynomial-time algorithm that transforms a given instance (x, k) of P into an equivalent instance (y, k ) of Q such that k = poly(k). Micha l Pilipczuk Kernelization lower bounds, part 2 5/33

11 PPTs: properties Observation If problem P PPT-reduces to Q, and P does not admit a polynomial compression algorithm (into any language R), then neither does Q. Micha l Pilipczuk Kernelization lower bounds, part 2 6/33

12 PPTs: properties Observation If problem P PPT-reduces to Q, and P does not admit a polynomial compression algorithm (into any language R), then neither does Q. Proof: Compose the PPT-reduction with the assumed compression for Q. Micha l Pilipczuk Kernelization lower bounds, part 2 6/33

13 Application 2: Steiner Tree Steiner Tree Input: Parameter: k + T Question: Graph G with designated terminals T V (G), and an integer k Is there a set X V (G) \ T, such that X k and G[T X ] is connected? Micha l Pilipczuk Kernelization lower bounds, part 2 7/33

14 Application 2: Steiner Tree Steiner Tree Input: Parameter: k + T Question: Graph G with designated terminals T V (G), and an integer k Is there a set X V (G) \ T, such that X k and G[T X ] is connected? Follows from a PPT from Set Cover par. by U. Micha l Pilipczuk Kernelization lower bounds, part 2 7/33

15 Application 2: Steiner Tree Steiner Tree Input: Parameter: k + T Question: Graph G with designated terminals T V (G), and an integer k Is there a set X V (G) \ T, such that X k and G[T X ] is connected? Follows from a PPT from Set Cover par. by U. But we will present an alternative approach. Micha l Pilipczuk Kernelization lower bounds, part 2 7/33

16 The pivot problem technique Introduce an simpler problem P, which is almost trivially compositional. Micha l Pilipczuk Kernelization lower bounds, part 2 8/33

17 The pivot problem technique Introduce an simpler problem P, which is almost trivially compositional. Then design a PPT from P to the target problem. Micha l Pilipczuk Kernelization lower bounds, part 2 8/33

18 The pivot problem technique Introduce an simpler problem P, which is almost trivially compositional. Then design a PPT from P to the target problem. Move the weight of the proof to the transformation and the actual definition of P. Micha l Pilipczuk Kernelization lower bounds, part 2 8/33

19 The pivot problem technique Introduce an simpler problem P, which is almost trivially compositional. Then design a PPT from P to the target problem. Move the weight of the proof to the transformation and the actual definition of P. Idea: Extract the essence of the problem. Micha l Pilipczuk Kernelization lower bounds, part 2 8/33

20 Colourful Graph Motif Colourful Graph Motif Input: Parameter: Question: Graph G and a colouring function C : V (G) {1, 2,..., k} k Does there exists a connected subgraph H of G containing exactly one vertex of each colour? Micha l Pilipczuk Kernelization lower bounds, part 2 9/33

21 Colourful Graph Motif example Micha l Pilipczuk Kernelization lower bounds, part 2 10/33

22 Colourful Graph Motif example Micha l Pilipczuk Kernelization lower bounds, part 2 10/33

23 About CGM Introduced by Fellows et al. (ICALP 2007, JCSS 2011). Micha l Pilipczuk Kernelization lower bounds, part 2 11/33

24 About CGM Introduced by Fellows et al. (ICALP 2007, JCSS 2011). NP-hard even on trees. Micha l Pilipczuk Kernelization lower bounds, part 2 11/33

25 About CGM Introduced by Fellows et al. (ICALP 2007, JCSS 2011). NP-hard even on trees. Interesting FPT algorithms for various variants using the algebraic approach. Micha l Pilipczuk Kernelization lower bounds, part 2 11/33

26 About CGM Introduced by Fellows et al. (ICALP 2007, JCSS 2011). NP-hard even on trees. Interesting FPT algorithms for various variants using the algebraic approach. Trivial composition algorithm: take the disjoint union of instances, reuse colors. Micha l Pilipczuk Kernelization lower bounds, part 2 11/33

27 About CGM Introduced by Fellows et al. (ICALP 2007, JCSS 2011). NP-hard even on trees. Interesting FPT algorithms for various variants using the algebraic approach. Trivial composition algorithm: take the disjoint union of instances, reuse colors. Hence, CGM does not have a polykernel unless conp NP/poly. Micha l Pilipczuk Kernelization lower bounds, part 2 11/33

28 About CGM Introduced by Fellows et al. (ICALP 2007, JCSS 2011). NP-hard even on trees. Interesting FPT algorithms for various variants using the algebraic approach. Trivial composition algorithm: take the disjoint union of instances, reuse colors. Hence, CGM does not have a polykernel unless conp NP/poly. Now: PPT-reduction from CGM to ST. Micha l Pilipczuk Kernelization lower bounds, part 2 11/33

29 From CGM to ST Micha l Pilipczuk Kernelization lower bounds, part 2 12/33

30 From CGM to ST Attach a terminal to every colour class Give budget k for Steiner nodes Micha l Pilipczuk Kernelization lower bounds, part 2 12/33

31 From CGM to ST Attach a terminal to every colour class Give budget k for Steiner nodes Micha l Pilipczuk Kernelization lower bounds, part 2 12/33

32 Wrapping up CGM does not admit a polynomial kernel, unless conp NP/poly. Micha l Pilipczuk Kernelization lower bounds, part 2 13/33

33 Wrapping up CGM does not admit a polynomial kernel, unless conp NP/poly. CGM PPT-reduces to Steiner Tree par. by k + T. Micha l Pilipczuk Kernelization lower bounds, part 2 13/33

34 Wrapping up CGM does not admit a polynomial kernel, unless conp NP/poly. CGM PPT-reduces to Steiner Tree par. by k + T. Hence Steiner Tree par. by k + T does not admit a polynomial kernel, unless conp NP/poly. Micha l Pilipczuk Kernelization lower bounds, part 2 13/33

35 Application 3: Set Cover par. by U Set Cover Input: Universe U, a family of subsets F 2 U, integer k Parameter: U Question: Is there a subfamily G F, G k, such that G = U? Micha l Pilipczuk Kernelization lower bounds, part 2 14/33

36 Application 3: Set Cover par. by U Set Cover Input: Universe U, a family of subsets F 2 U, integer k Parameter: U Question: Is there a subfamily G F, G k, such that G = U? Convention: We view it as a bipartite graph with one side (blue) trying to dominate the other one (red). Micha l Pilipczuk Kernelization lower bounds, part 2 14/33

37 Application 3: Set Cover par. by U Set Cover Input: Universe U, a family of subsets F 2 U, integer k Parameter: U Question: Is there a subfamily G F, G k, such that G = U? Convention: We view it as a bipartite graph with one side (blue) trying to dominate the other one (red). W.l.o.g. k U. Micha l Pilipczuk Kernelization lower bounds, part 2 14/33

38 Colourful Set Cover Colourful Set Cover Input: Parameter: Question: Universe U and families F 1, F 2,..., F k 2 U U + k Is there a family G containing exactly one set from each family F i, such that G = U? Micha l Pilipczuk Kernelization lower bounds, part 2 15/33

39 Equivalence of the problems SC PPT CSC: Micha l Pilipczuk Kernelization lower bounds, part 2 16/33

40 Equivalence of the problems SC PPT CSC: Put F i = F for every i. Micha l Pilipczuk Kernelization lower bounds, part 2 16/33

41 Equivalence of the problems SC PPT CSC: Put F i = F for every i. CSC PPT SC: Micha l Pilipczuk Kernelization lower bounds, part 2 16/33

42 Equivalence of the problems SC PPT CSC: Put F i = F for every i. CSC PPT SC: Add k elements e 1, e 2,..., e k ; include e i in every set from F i. Micha l Pilipczuk Kernelization lower bounds, part 2 16/33

43 Equivalence of the problems SC PPT CSC: Put F i = F for every i. CSC PPT SC: Add k elements e 1, e 2,..., e k ; include e i in every set from F i. Then take F = F i. Micha l Pilipczuk Kernelization lower bounds, part 2 16/33

44 Equivalence of the problems SC PPT CSC: Put F i = F for every i. CSC PPT SC: Add k elements e 1, e 2,..., e k ; include e i in every set from F i. Then take F = F i. We will cross-compose Colourful Set Cover into itself. Micha l Pilipczuk Kernelization lower bounds, part 2 16/33

45 Equivalence of the problems SC PPT CSC: Put F i = F for every i. CSC PPT SC: Add k elements e 1, e 2,..., e k ; include e i in every set from F i. Then take F = F i. We will cross-compose Colourful Set Cover into itself. Assumption: the same universe U, the same k, and t being a power of 2. Micha l Pilipczuk Kernelization lower bounds, part 2 16/33

46 Cross-composing into Colourful Set Cover Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

47 Cross-composing into Colourful Set Cover U Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

48 Cross-composing into Colourful Set Cover k F 1 1 F 2 1 F 1 2 F 2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ U Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

49 Cross-composing into Colourful Set Cover k F 1 1 F 2 1 F 1 2 F 2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ U Problem: Ensure consistent instance choice (choices from the same row) Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

50 Cross-composing into Colourful Set Cover k F 1 1 F 2 1 F 1 2 F 2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ U Problem: Ensure consistent instance choice (choices from the same row) Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

51 Cross-composing into Colourful Set Cover k F 1 1 F 2 1 F 1 2 F 2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ Problem: Ensure consistent instance choice (choices from the same row) Solution: Equality gadgets Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

52 Cross-composing into Colourful Set Cover k F 1 1 F 2 1 F 1 2 F 2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ U 1 j 1 < j 2 k make a gadget Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

53 Cross-composing into Colourful Set Cover k F1 1 F1 2 F2 1 F2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ (3, 4) U 1 j 1 < j 2 k make a gadget log t pairs Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

54 Cross-composing into Colourful Set Cover k F1 1 F1 2 F2 1 F2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ (3, 4) U 1 j 1 < j 2 k make a gadget log t pairs i add bin(i) to sets from F i 3 Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

55 Cross-composing into Colourful Set Cover k F1 1 F1 2 F2 1 F2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ (3, 4) U 1 j 1 < j 2 k make a gadget log t pairs i add bin(i) to sets from F i 3 Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) i add bin(i) to sets from F i 4 Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

56 Cross-composing into Colourful Set Cover k F1 1 F1 2 F2 1 F2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ (3, 4) U Eq. gadgets are covered Instance choices are equal log t pairs i add bin(i) to sets from F i 3 Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) i add bin(i) to sets from F i 4 Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

57 Cross-composing into Colourful Set Cover k F1 1 F1 2 F2 1 F2 2 F 1 3 F 2 3 F 1 4 F 2 4 F 1 5 F F 1 k F 2 k t instances {}}{ (3, 4) U New parameter: U + O(k 2 log t) log t pairs i add bin(i) to sets from F i 3 Input: Instances (U, (F i j ) 1 j k ) Output: Instance (U, (F j ) 1 j k ) i add bin(i) to sets from F i 4 Micha l Pilipczuk Kernelization lower bounds, part 2 17/33

58 Wrapping up SC and CSC are equivalent wrt. PPTs. Micha l Pilipczuk Kernelization lower bounds, part 2 18/33

59 Wrapping up SC and CSC are equivalent wrt. PPTs. CSC does not admit a polynomial compression, unless NP conp/poly. Micha l Pilipczuk Kernelization lower bounds, part 2 18/33

60 Wrapping up SC and CSC are equivalent wrt. PPTs. CSC does not admit a polynomial compression, unless NP conp/poly. So neither does SC. Micha l Pilipczuk Kernelization lower bounds, part 2 18/33

61 Wrapping up SC and CSC are equivalent wrt. PPTs. CSC does not admit a polynomial compression, unless NP conp/poly. So neither does SC. Note: parameterization of Set Cover by F also does not admit a polynomial compression. Micha l Pilipczuk Kernelization lower bounds, part 2 18/33

62 Wrapping up SC and CSC are equivalent wrt. PPTs. CSC does not admit a polynomial compression, unless NP conp/poly. So neither does SC. Note: parameterization of Set Cover by F also does not admit a polynomial compression. The composition is quite different. Micha l Pilipczuk Kernelization lower bounds, part 2 18/33

63 Structural parameters Idea: Parameterize the problem by a quantitative measure of structure of the graph, rather than intended solution size. Micha l Pilipczuk Kernelization lower bounds, part 2 19/33

64 Structural parameters Idea: Parameterize the problem by a quantitative measure of structure of the graph, rather than intended solution size. Example: treewidth parameterizations Micha l Pilipczuk Kernelization lower bounds, part 2 19/33

65 Structural parameters Idea: Parameterize the problem by a quantitative measure of structure of the graph, rather than intended solution size. Example: treewidth parameterizations From kernelization point of view: work of Bodlaender, Jansen, and Kratsch. Micha l Pilipczuk Kernelization lower bounds, part 2 19/33

66 Structural parameters Idea: Parameterize the problem by a quantitative measure of structure of the graph, rather than intended solution size. Example: treewidth parameterizations From kernelization point of view: work of Bodlaender, Jansen, and Kratsch. Original motivation of cross-composition. Micha l Pilipczuk Kernelization lower bounds, part 2 19/33

67 Application 4: Clique par. by vertex cover Clique/VC Input: Graph G, a vertex cover X of G, integer k Parameter: X Question: Is there a clique of size k in G? Micha l Pilipczuk Kernelization lower bounds, part 2 20/33

68 Application 4: Clique par. by vertex cover Clique/VC Input: Graph G, a vertex cover X of G, integer k Parameter: X Question: Is there a clique of size k in G? W.l.o.g. k X + 1. Micha l Pilipczuk Kernelization lower bounds, part 2 20/33

69 Application 4: Clique par. by vertex cover Clique/VC Input: Graph G, a vertex cover X of G, integer k Parameter: X Question: Is there a clique of size k in G? W.l.o.g. k X + 1. Trivially FPT. Micha l Pilipczuk Kernelization lower bounds, part 2 20/33

70 Application 4: Clique par. by vertex cover Clique/VC Input: Graph G, a vertex cover X of G, integer k Parameter: X Question: Is there a clique of size k in G? W.l.o.g. k X + 1. Trivially FPT. We make a cross-composition from the standard Clique problem. Micha l Pilipczuk Kernelization lower bounds, part 2 20/33

71 Application 4: Clique par. by vertex cover Clique/VC Input: Graph G, a vertex cover X of G, integer k Parameter: X Question: Is there a clique of size k in G? W.l.o.g. k X + 1. Trivially FPT. We make a cross-composition from the standard Clique problem. Assume the same number of vertices n and the same target size of the clique k. Micha l Pilipczuk Kernelization lower bounds, part 2 20/33

72 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

73 Cross-composing into Clique/VC I t vertices Input: Instances (G i, k) Output: Instance (G, X, k ) Size constraints force us to take one vertex from I. X Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

74 Cross-composing into Clique/VC I t vertices Input: Instances (G i, k) Output: Instance (G, X, k ) Size constraints force us to take one vertex from I. Neighbourhood of i-th vertex from I acts as instance (G i, k). X Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

75 Cross-composing into Clique/VC I t vertices Input: Instances (G i, k) Output: Instance (G, X, k ) Size constraints force us to take one vertex from I. Neighbourhood of i-th vertex from I acts as instance (G i, k). X Problem: Design a universal modulator X. Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

76 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) 1... All connections are present except ones in the same row/column k n Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

77 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

78 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) all (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

79 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) all a (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

80 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) all a b (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

81 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) all a b (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

82 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) all a b (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

83 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) all a b (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

84 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) (a, b) E(G i ) all a b (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

85 Cross-composing into Clique/VC Input: Instances (G i, k) Output: Instance (G, X, k ) (a, b) / E(G i ) all a b (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

86 Cross-composing into Clique/VC Requested size of the clique: ( k = k + n 2) + 1 all a b (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

87 Cross-composing into Clique/VC i-th vertex is chosen from I (a, b) / E(G i ), columns a and b cannot be chosen simultaneously all a b (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

88 Cross-composing into Clique/VC New parameter: ( X = kn + 3 n 2) all a b (a, b) k n ( n 2) triples Micha l Pilipczuk Kernelization lower bounds, part 2 21/33

89 Conclusion of the case study Making compositions is highly non-trivial. Micha l Pilipczuk Kernelization lower bounds, part 2 22/33

90 Conclusion of the case study Making compositions is highly non-trivial. Requires good understanding of the problem. Micha l Pilipczuk Kernelization lower bounds, part 2 22/33

91 Conclusion of the case study Making compositions is highly non-trivial. Requires good understanding of the problem. Intuitively, what we exploit is choice. We need to identify it, and build a technical construction on top of it. Micha l Pilipczuk Kernelization lower bounds, part 2 22/33

92 Conclusion of the case study Making compositions is highly non-trivial. Requires good understanding of the problem. Intuitively, what we exploit is choice. We need to identify it, and build a technical construction on top of it. Often it requires a lot of gadgeteering... Micha l Pilipczuk Kernelization lower bounds, part 2 22/33

93 Conclusion of the case study Making compositions is highly non-trivial. Requires good understanding of the problem. Intuitively, what we exploit is choice. We need to identify it, and build a technical construction on top of it. Often it requires a lot of gadgeteering experience... Micha l Pilipczuk Kernelization lower bounds, part 2 22/33

94 Conclusion of the case study Making compositions is highly non-trivial. Requires good understanding of the problem. Intuitively, what we exploit is choice. We need to identify it, and build a technical construction on top of it. Often it requires a lot of gadgeteering experience tricks that I did not mention... Micha l Pilipczuk Kernelization lower bounds, part 2 22/33

95 Conclusion of the case study Making compositions is highly non-trivial. Requires good understanding of the problem. Intuitively, what we exploit is choice. We need to identify it, and build a technical construction on top of it. Often it requires a lot of gadgeteering experience tricks that I did not mention... or clever ideas. Micha l Pilipczuk Kernelization lower bounds, part 2 22/33

96 Weak compositions We have seen an O(k 2 ) kernel for Feedback Vertex Set (bitsize O(k 2 log k)). Micha l Pilipczuk Kernelization lower bounds, part 2 23/33

97 Weak compositions We have seen an O(k 2 ) kernel for Feedback Vertex Set (bitsize O(k 2 log k)). Can we prove that a subquadratic kernel is unlikely? Micha l Pilipczuk Kernelization lower bounds, part 2 23/33

98 Weak compositions We have seen an O(k 2 ) kernel for Feedback Vertex Set (bitsize O(k 2 log k)). Can we prove that a subquadratic kernel is unlikely? YES Micha l Pilipczuk Kernelization lower bounds, part 2 23/33

99 Weak compositions We have seen an O(k 2 ) kernel for Feedback Vertex Set (bitsize O(k 2 log k)). Can we prove that a subquadratic kernel is unlikely? YES Weak compositions: proving lower bounds on kernelization complexity for problems that do have polynomial kernels. Micha l Pilipczuk Kernelization lower bounds, part 2 23/33

100 Weak compositions We have seen an O(k 2 ) kernel for Feedback Vertex Set (bitsize O(k 2 log k)). Can we prove that a subquadratic kernel is unlikely? YES Weak compositions: proving lower bounds on kernelization complexity for problems that do have polynomial kernels. First results by Dell and van Melkebeek (STOC 2010), the framework here by (Dell, Marx; Hermelin, Wu; SODA 2012). Micha l Pilipczuk Kernelization lower bounds, part 2 23/33

101 Back to Fortnow and Santhanam OR-distillation of t = k 1000 instances of size k into one instance with bitsize k 7 is unlikely. Micha l Pilipczuk Kernelization lower bounds, part 2 24/33

102 Back to Fortnow and Santhanam OR-distillation of t = k 1000 instances of size k into one instance with bitsize k 7 is unlikely. Well, it should be unlikely even if we required any bitsize sublinear in t. Micha l Pilipczuk Kernelization lower bounds, part 2 24/33

103 Back to Fortnow and Santhanam OR-distillation of t = k 1000 instances of size k into one instance with bitsize k 7 is unlikely. Well, it should be unlikely even if we required any bitsize sublinear in t. If one examines the proof, then one can exclude OR-distillation into bitsize O(t 1 ɛ k c ), for any constants ɛ > 0 and c. Micha l Pilipczuk Kernelization lower bounds, part 2 24/33

104 Back to Fortnow and Santhanam OR-distillation of t = k 1000 instances of size k into one instance with bitsize k 7 is unlikely. Well, it should be unlikely even if we required any bitsize sublinear in t. If one examines the proof, then one can exclude OR-distillation into bitsize O(t 1 ɛ k c ), for any constants ɛ > 0 and c. Let s look again at the proof of the cross-composition Theorem. Micha l Pilipczuk Kernelization lower bounds, part 2 24/33

105 Cross-composition proof, recap k = max x i, log t = O(k) L Q Q compo compo compo poly(k) poly(k) poly(k) cmpr cmpr cmpr OR-R R Micha l Pilipczuk Kernelization lower bounds, part 2 25/33

106 Cross-composition proof, recap k = max x i, log t = O(k) L Q Q compo compo compo poly(k) poly(k) poly(k) cmpr K O(K d ɛ ) cmpr cmpr OR-R R Micha l Pilipczuk Kernelization lower bounds, part 2 25/33

107 Cross-composition proof, recap k = max x i, log t = O(k) L Q Q compo t O(t 1/d ) compo compo poly(k) poly(k) poly(k) cmpr K O(K d ɛ ) cmpr cmpr OR-R R Micha l Pilipczuk Kernelization lower bounds, part 2 25/33

108 Weak compositions, formally It seems that a composition with output bitsize dependence on t being O(t 1/d ) should exclude compression into bitsize O(k d ɛ ). Micha l Pilipczuk Kernelization lower bounds, part 2 26/33

109 Weak compositions, formally It seems that a composition with output bitsize dependence on t being O(t 1/d ) should exclude compression into bitsize O(k d ɛ ). Weak cross-composition An unparameterized problem Q weakly cross-composes into a parameterized problem L, if there exists a polynomial equivalence relation R, a real constant d 1, and an algorithm that, given R-equivalent strings x 1, x 2,..., x t, in time poly ( t + t i=1 x i ) produces one instance (y, k ) such that (y, k ) L iff x i Q for at least one i = 1, 2,..., t, k = t 1/d+o(1) poly (max t i=1 x i ). Micha l Pilipczuk Kernelization lower bounds, part 2 26/33

110 Weak cross-composition theorem Constant d will be called the dimension of the weak cross-composition. Micha l Pilipczuk Kernelization lower bounds, part 2 27/33

111 Weak cross-composition theorem Constant d will be called the dimension of the weak cross-composition. Weak cross-composition theorem Suppose some NP-hard problem Q admits a weak cross-composition into L with dimension d. Suppose further that L admits a polynomial compression with bitsize O(k d ɛ ), for some ɛ > 0. Then NP conp/poly. Micha l Pilipczuk Kernelization lower bounds, part 2 27/33

112 Weak cross-composition theorem Constant d will be called the dimension of the weak cross-composition. Weak cross-composition theorem Suppose some NP-hard problem Q admits a weak cross-composition into L with dimension d. Suppose further that L admits a polynomial compression with bitsize O(k d ɛ ), for some ɛ > 0. Then NP conp/poly. Note: Also called cross-composition of bounded cost by Bodlaender et al. (SIDMA, 2014). Micha l Pilipczuk Kernelization lower bounds, part 2 27/33

113 Lower bound for Vertex Cover Using weak cross-composition we now prove that Vertex Cover does not admit a kernel with bitsize O(k 2 ɛ ), for any ɛ > 0. (unless...) Micha l Pilipczuk Kernelization lower bounds, part 2 28/33

114 Lower bound for Vertex Cover Using weak cross-composition we now prove that Vertex Cover does not admit a kernel with bitsize O(k 2 ɛ ), for any ɛ > 0. (unless...) But it had a linear kernel!? Micha l Pilipczuk Kernelization lower bounds, part 2 28/33

115 Lower bound for Vertex Cover Using weak cross-composition we now prove that Vertex Cover does not admit a kernel with bitsize O(k 2 ɛ ), for any ɛ > 0. (unless...) But it had a linear kernel!? Vertex Cover has a kernel with 2k vertices, which therefore requires O(k 2 ) bits to be encoded. Micha l Pilipczuk Kernelization lower bounds, part 2 28/33

116 Lower bound for Vertex Cover Using weak cross-composition we now prove that Vertex Cover does not admit a kernel with bitsize O(k 2 ɛ ), for any ɛ > 0. (unless...) But it had a linear kernel!? Vertex Cover has a kernel with 2k vertices, which therefore requires O(k 2 ) bits to be encoded. Crux: choose an appropriate problem Q to start with. Micha l Pilipczuk Kernelization lower bounds, part 2 28/33

117 Multicolored Biclique Multicolored Biclique Input: Bipartite graph H with bipartition A B, where A = A 1... A k, B = B 1... B k. Question: Is there a biclique of size K k,k in H that contains one vertex from each A i and each B i? Micha l Pilipczuk Kernelization lower bounds, part 2 29/33

118 Multicolored Biclique Multicolored Biclique Input: Bipartite graph H with bipartition A B, where A = A 1... A k, B = B 1... B k. Question: Is there a biclique of size K k,k in H that contains one vertex from each A i and each B i? Multicolored Biclique is NP-hard even if each A i and B i has the same size. Micha l Pilipczuk Kernelization lower bounds, part 2 29/33

119 Multicolored Biclique Multicolored Biclique Input: Bipartite graph H with bipartition A B, where A = A 1... A k, B = B 1... B k. Question: Is there a biclique of size K k,k in H that contains one vertex from each A i and each B i? Multicolored Biclique is NP-hard even if each A i and B i has the same size. We provide a weak cross composition of dimension 2 from this version into Vertex Cover. Micha l Pilipczuk Kernelization lower bounds, part 2 29/33

120 Multicolored Biclique Multicolored Biclique Input: Bipartite graph H with bipartition A B, where A = A 1... A k, B = B 1... B k. Question: Is there a biclique of size K k,k in H that contains one vertex from each A i and each B i? Multicolored Biclique is NP-hard even if each A i and B i has the same size. We provide a weak cross composition of dimension 2 from this version into Vertex Cover. Assumptions: all the input instances have the same k, each color class has size n in every input instance, t = s 2. Micha l Pilipczuk Kernelization lower bounds, part 2 29/33

121 Composition Create s copies of the left side and s copies of the right side. N = s 2kn  1 1  1  1 2  2 1  2  2 2  3 1  3  3 2 ˆB 1 1 ˆB 2 ˆB 1 1 ˆB2 1 ˆB 2 ˆB 2 2 ˆB3 1 ˆB 2 ˆB 3 3 Micha l Pilipczuk Kernelization lower bounds, part 2 30/33

122 Composition Embed s 2 instances into s 2 pairs of the sides. Â 1 1 Â 1 Â 1 2 Â 2 1 Â 2 Â 2 2 Â 3 1 Â 3 Â 3 2 complement of one of the instances ˆB 1 1 ˆB 1 2 ˆB 1 ˆB 2 1 ˆB 2 2 ˆB 2 ˆB 3 1 ˆB 3 ˆB 3 2 Micha l Pilipczuk Kernelization lower bounds, part 2 30/33

123 Composition Ask for an independent set of size 2k; equivalently, a vertex cover of size N 2k. Â 1 1 Â 1 Â 1 2 Â 2 1 Â 2 Â 2 2 Â 3 1 Â 3 Â 3 2 complement of one of the instances ˆB 1 1 ˆB 1 2 ˆB 1 ˆB 2 1 ˆB 2 2 ˆB 2 ˆB 3 1 ˆB 3 ˆB 3 2 Micha l Pilipczuk Kernelization lower bounds, part 2 30/33

124 Composition Edges on the sides ensure choosing one instance and respecting colors. Edges originating in this instance ensure that the instance is solved. Â 1 1 Â 1 Â 1 2 Â 2 1 Â 2 Â 2 2 Â 3 1 Â 3 Â 3 2 complement of one of the instances ˆB 1 1 ˆB 1 2 ˆB 1 ˆB 2 1 ˆB 2 2 ˆB 2 ˆB 3 1 ˆB 3 ˆB 3 2 Micha l Pilipczuk Kernelization lower bounds, part 2 30/33

125 Wrapping up Parameter in the output instance is N 2k = t 1/2 2kn 2k, which means that the weak composition has dimension 2. Micha l Pilipczuk Kernelization lower bounds, part 2 31/33

126 Wrapping up Parameter in the output instance is N 2k = t 1/2 2kn 2k, which means that the weak composition has dimension 2. Hence, there is no kernel with O(k 2 ɛ ) bits for VC. Micha l Pilipczuk Kernelization lower bounds, part 2 31/33

127 Wrapping up Parameter in the output instance is N 2k = t 1/2 2kn 2k, which means that the weak composition has dimension 2. Hence, there is no kernel with O(k 2 ɛ ) bits for VC. Reduction from VC to FVS: add a degree-2 vertex to every edge, thus creating a triangle. Micha l Pilipczuk Kernelization lower bounds, part 2 31/33

128 Wrapping up Parameter in the output instance is N 2k = t 1/2 2kn 2k, which means that the weak composition has dimension 2. Hence, there is no kernel with O(k 2 ɛ ) bits for VC. Reduction from VC to FVS: add a degree-2 vertex to every edge, thus creating a triangle. Hence also FVS does not admit a O(k 2 ɛ ) kernel. Micha l Pilipczuk Kernelization lower bounds, part 2 31/33

129 Wrapping up Parameter in the output instance is N 2k = t 1/2 2kn 2k, which means that the weak composition has dimension 2. Hence, there is no kernel with O(k 2 ɛ ) bits for VC. Reduction from VC to FVS: add a degree-2 vertex to every edge, thus creating a triangle. Hence also FVS does not admit a O(k 2 ɛ ) kernel. For more O(k d ɛ ) lower bounds, see the book. Micha l Pilipczuk Kernelization lower bounds, part 2 31/33

130 Further perspectives Compression vs. Kernelization Micha l Pilipczuk Kernelization lower bounds, part 2 32/33

131 Further perspectives Compression vs. Kernelization VC has kernel with O(k) vertices and O(k 2 ) edges. What about FVS? Micha l Pilipczuk Kernelization lower bounds, part 2 32/33

132 Further perspectives Compression vs. Kernelization VC has kernel with O(k) vertices and O(k 2 ) edges. What about FVS? Is compositionality the only reason why polynomial kernelization is infeasible? Micha l Pilipczuk Kernelization lower bounds, part 2 32/33

133 Further perspectives Compression vs. Kernelization VC has kernel with O(k) vertices and O(k 2 ) edges. What about FVS? Is compositionality the only reason why polynomial kernelization is infeasible? Can we find a sensible problem where kernelization and compression are provable different? Micha l Pilipczuk Kernelization lower bounds, part 2 32/33

134 Further perspectives Compression vs. Kernelization VC has kernel with O(k) vertices and O(k 2 ) edges. What about FVS? Is compositionality the only reason why polynomial kernelization is infeasible? Can we find a sensible problem where kernelization and compression are provable different? Completeness theory for kernelization. Micha l Pilipczuk Kernelization lower bounds, part 2 32/33

135 Further perspectives Compression vs. Kernelization VC has kernel with O(k) vertices and O(k 2 ) edges. What about FVS? Is compositionality the only reason why polynomial kernelization is infeasible? Can we find a sensible problem where kernelization and compression are provable different? Completeness theory for kernelization. Hermelin, Kratsch, So ltys, Wahlström, Wu; IPEC Micha l Pilipczuk Kernelization lower bounds, part 2 32/33

136 Further perspectives Compression vs. Kernelization VC has kernel with O(k) vertices and O(k 2 ) edges. What about FVS? Is compositionality the only reason why polynomial kernelization is infeasible? Can we find a sensible problem where kernelization and compression are provable different? Completeness theory for kernelization. Hermelin, Kratsch, So ltys, Wahlström, Wu; IPEC Turing kernelization Micha l Pilipczuk Kernelization lower bounds, part 2 32/33

137 Further perspectives Compression vs. Kernelization VC has kernel with O(k) vertices and O(k 2 ) edges. What about FVS? Is compositionality the only reason why polynomial kernelization is infeasible? Can we find a sensible problem where kernelization and compression are provable different? Completeness theory for kernelization. Hermelin, Kratsch, So ltys, Wahlström, Wu; IPEC Turing kernelization Turing kernel is a polynomial-time algorithm with an access to an oracle that resolves kernels. Micha l Pilipczuk Kernelization lower bounds, part 2 32/33

138 Further perspectives Compression vs. Kernelization VC has kernel with O(k) vertices and O(k 2 ) edges. What about FVS? Is compositionality the only reason why polynomial kernelization is infeasible? Can we find a sensible problem where kernelization and compression are provable different? Completeness theory for kernelization. Hermelin, Kratsch, So ltys, Wahlström, Wu; IPEC Turing kernelization Turing kernel is a polynomial-time algorithm with an access to an oracle that resolves kernels. How to show infeasibility of Turing kernelization? Micha l Pilipczuk Kernelization lower bounds, part 2 32/33

139 Exercises Exercise 15.4, all the remaining points. Exercises 15.1 and Tikz faces based on a code by Raoul Kessels, under Creative Commons Attribution 2.5 license (CC BY 2.5) Micha l Pilipczuk Kernelization lower bounds, part 2 33/33