Bart Jansen Independent Set Kernelization for a Refined Parameter: Upper and Lower bounds

1. Bart JansenIndependent Set Kernelization for a Refined Parameter: Upper and Lower bounds Joint work with Hans Bodlaender Algorithm seminar, Bergen January 7th, 2011

2. Independent Set Kernelization for a Refined Parameter: Upper and Lower bounds • Introduction • Independent Set • Parameters • Kernelization • Upper bounds • Small kernel for parameter P3 cover • Reduction rules • Analysis • Ideas for for parameter Feedback Vertex Set • Lower bounds • Effect of introducing vertex weights • Conclusion

3. Our target problem independent set

4. Independent Set • Input: Graph G, integer q • Question: Is there a set S of ≥ q vertices which are pairwise non-adjacent? • NP-complete, even on planar graphs max degree 3 • Not approximable • We show how to attack the problem if some measure of “graph complexity” is low • Data reduction

5. Solutions to vertex deletion problems as complexity measures Parameters

6. Vertex Cover Edgeless Graphs Vertex Deletion Problems • Vertex Cover • Input: Graph G, integer q • Question: Is there a set S of ≤ q vertices such that G-S is edgeless? Equivalent question: Is there an Independent Set of size ≥ n – q?

7. Vertex Cover Edgeless Graphs P3 cover Paths ≤2 nodes Vertex Deletion Problems • P3 Cover • Input: Graph G, integer q • Question: Is there a set S of ≤ q vertices such that G-S is a collection of paths on at most 2 vertices?

8. Vertex Cover Edgeless Graphs P3 cover Paths ≤2 nodes Feedback vtx Set Forests Vertex Deletion Problems • Feedback Vertex Set • Input: Graph G, integer q • Question: Is there a set S of ≤ q vertices such that G-S is a forest? (Acyclic)

9. Vertex Cover Edgeless Graphs P3 cover Paths ≤2 nodes Feedback vtx Set Forests Graph Complexity Measures • We can use the minimum sizes of these vertex deletion sets as measures of the complexity of a graph • Every edgeless graph is a collection of paths on ≤ 2 nodes • Every collection of paths on ≤ 2 nodes is a forest • Difference between the parameters can be unbounded ≤ ≤

10. Graph families

11. Attacking hard problems with small parameters kernelization

12. Graph problems with structural parameters • Consider a computational decision problem on graphs • Input: encoding x of a question about graph G, integer k. • Question: does graph G have a (…)? • Parameter:k • Parameter value k expresses some measure of the complexity of the graph • size of a minimum Vertex Cover, • P3 Cover, • Feedback Vertex Set, • etc.

13. Kernelization for graph problems • A kernelization algorithm takes (x, k) as input and computes an instance (x’, k’) of same problem in polynomial time, such that • Answer to x is YES  answer to x’ is YES • k’ ≤ k • |x’| ≤ f(k) for some function f • The function f is the size of the kernel • We want f to be a (small) polynomial • Kernelization reduces the size of the graph to something which depends • only on the complexity measure of the input, • not on the size of the input • Afterwards solve the smaller instances by some other method

14. Perspective for this talk • We want to solve the Independent Set problem • We use the solution values of the vertex deletion problems as complexity measures (parameters) of the input instances • Previous state of the art: • “Does graph G with vertex cover of size k have an independent set of size q?” • can be transformed in polynomial time into: • “Does graph G’ with vertex cover of size k’ have an independent set of size q’ ?” • where |G’| ≤ 2 k, • and k’ ≤ k. • Complexity-theoretic evidence that the factor 2 is optimal

15. Our results: upper bounds • “Does graph G with feedback vertex set of size k have an independent set of size q?” • can be transformed in polynomial time into: • “Does graph G’ with feedback vertex set of size k’ have an independent set of size q’ ?” • where |G’| ≤ O(k3), • and k’ ≤ k. • Our new bound uses more units of a smaller measure • |G’| ≤ O(|MinFVS|3)  |G’| ≤ 2 |MinVC| • Compare: “1000 ants weigh less than 3 horses” • Refined parameter • For simplicity we present the following result: • Transformation such that |G’| ≤ O(|MinP3Cover(G)|3). • The Independent Set problem parameterized by the size of a feedback vertex set admits a cubic-vertex kernel

16. Cubic-vertex kernel for parameter p3cover

17. Independent Set with P3-cover • Input: Graph G, modulator X such that G – X is a collection of paths on at most 2 vertices, integer q. • Question: Does G have an Independent Set of size q? • Parameter: k := |X|. G - X X

18. Canonical solution structure G - X • The maximum independent set (MIS) of G – X contains 1 vertex from each path in G – X • We call this a canonical solution for graph G • It uses no vertices of X • Poly-time computable • Vertices from X are only useful if they allow for a larger IS than the canonical solution X

19. Conflicts induced by a vertex in X G - X • Consider vertex v in X • Compute a maximum independent set in G-X which avoids neighbors of v • Compare to the canonical solution (MIS in G-X) • Call the difference cf(v) the number of conflicts induced by v • Intuitively: the price we pay in G-X for using vertex v in an independent set • We can only improve on the canonical solution if the number of vertices we gain in X, is more than the number we lose in G-X X

20. Reduction rule 1Deleting single vertices in X G - X • If cf(v) ≥ |X| then delete v • There is always an optimal IS without v • Consider an IS using v • Might use |X| within X • Solution inside G-X at least |X| worse than canonical • Compare to: • Don’t use anything in X • Use optimum in G – X (Canonical solution) X

21. Conflicts induced by pairs of vertices in X G - X • Consider non-adjacent vertices {u,v} in X • Compute a maximum independent set in G-X which avoids neighbors of {u,v} • Compare to canonical solution • Call the difference cf({u,v}) the number of conflicts induced by{u,v} • Intuitively: the price we pay in G-X for using vertices {u,v} in an independent set X

22. Reduction rule 2Adding edges in X G - X • If cf({u,v})≥|X| then add edge {u,v} • There is always an optimal IS that avoids one of {u,v} • Consider an IS using {u,v} • Compared to the canonical solution it uses at least |X| less in G-X • So the canonical solution is at least as large • Does not use any vertices from X X

23. Reduction rule 3Deleting P1 components from G-X G - X • If there is an isolated vertex v in G – X which does not have any neighbors in X, • then delete v and decrease q by 1 • We can always use v in an independent set • “Does G have an independent set of size q?” now reduces to“Does G – v have an independent set of size q-1?” X

24. Reduction rule 4Deleting P2 components from G-X G - X • If there is a P2 in G-X on vertices {x,y} such that • no single vertex in X sees {x,y}, • no pair of non-adjacent vertices in X together sees {x,y} • then delete {x,y} and decrease q by 1 • We can always use one of {x,y} in an independent set • No independent set in X contains neighbors of x and y simultaneously • “Does G have an independent set of size q?” • now reduces to • “Does G - {x,y} have an independent set of size q-1?” X Observe: P2’s in G – X that survive this rule have restricted structure!

25. Analysis • To prove: after exhausting these reduction rules we have |V| ≤ |X| + 2|X|2 + 2|X|3. • Count how many paths we have in G – X. • Type 1: All P1 and the P2 whose vertices have a common neighbor • Type 2: The P2 whose vertices have no common neighbor • Claim: # Type 1 ≤ |X|2. • Claim: # Type 2 ≤ |X|3. G - X X

26. G - X Type 1 ≤ |X|2 X • Type 1: All P1 and the P2 whose vertices have a common neighbor • A P1 path of Type 1 must be adjacent to a vertex in X (Rule 3) • A P2 path of Type 1 must be adjacent to a vertex in X (definition) • Claim: no vertex in X is adjacent to more than |X| paths of Type 1 • If v in X is adjacent to |X| paths of Type 1, then cf(v) ≥ |X| • If we use v in IS, then we cannot use any vertices on adjacent Type 1 paths • IS size in G-X decreases by at least |X| if we use v • But by Rule 1 there are no vertices in X with cf(v) ≥ |X| • So we can charge all Type 1 paths to a (common) neighbor in X • We charge less than |X| to each vertex in X • Total charge = number of Type 1 paths ≤ |X|2

27. Type 2 ≤ |X|3 G - X X G - X • Type 2: The P2 whose vertices have no common neighbor • Claim: for Type 2 paths on {x,y} there are non-adjacent u,v in X such that u sees x, and v sees y • From definition of Type 2: no vertex in X sees both • If there is no such pair u,v then the path is deleted by Rule 4 • Claim: if u,v in X are non-adjacent, then there are less than |X| P2’s in G – X such that u sees the left endpoint and v sees the right endpoint • If there are at least |X| such P2’s then cf({u,v}) ≥ |X| and we would add the edge {u,v} by Rule 2 • So we can charge each P2 of Type 2 to some non-adjacent pair in X which sees the endpoints of this P2 • We charge less than |X| to each pair • Total charge = number of Type 2 paths ≤ |X|2 • |X| = |X|3 X

28. Summing it up • To prove: after exhausting these reduction rules we have |V| ≤ O(|X|3). • We proved bounds on the number of paths: • # Type 1 ≤ |X|2. • # Type 2 ≤ |X|3. • Number of vertices on a path is at most 2 • Besides vertices on paths, graph contains only X. • |V|=|X| + |V(Type 1)| + |V(Type 2)| ≤ |X|+2(|X|2+|X|3). • Reduction rules can be applied in polynomial time • What is left of X forms a P3 Cover for the resulting graph • Complexity of final instance is not greater than of input instance

29. A sketch of the general result Cubic-vertex kernel for parameter FEEDBACK VERTEX SET

30. Independent Set with Feedback Vertex Set • Input: Graph G, modulator X such that G – X is a forest, integer q. • Question: Does G have an Independent Set of size q? • Parameter: k := |X|. • Solve in 2|X|(|V| + |E|) time • Try all subsets S of X • Skip if S is not independent • Otherwise compute MIS in G-X which avoids neighbors of S • Solve MIS in G – X – N(S) • This is a forest! • Return maximum value of |S| + MIS G - X X

31. Outline • We can still compute a canonical solution (MIS of G – X) in polynomial time since G – X is a forest • As before, number of conflicts induced by vertex v in X, or a non-adjacent pair {u,v} in X, is the decrease in the size of the solution within G – X, when using those vertices • Rule 1: Delete v in G – X with cf(v) ≥ |X| • Rule 2: Add edge between non-adjacent u,v in X if cf({u,v}) ≥ |X| • Rule 3: Delete a tree T in G – X if there are no non-adjacent vertices {u,v} in X which induce a conflict on T • Decrease q by MIS(T) • Not obvious that checking for pairs is enough • Rule 4, 5: Simplify structure of trees in G – X • Analysis: • charge vertices in a tree to neighbors in X • total charge cannot be too big without triggering reduction rules • 20 pages of proof for the analysis

32. The modulator X in the input • We have assumed that we get the modulator X (the deletion set) as part of the input • Might not be the case in practice • Kernelization claims do not rely on X being a minimum set; the size of the reduced instance is bounded in |X| • So we compute a 2-approximation X, use it instead • |G’| is bounded in O(|X|3) • |X| is bounded by 2 |MinFVS(G)| • Hence |G’| is bounded by O(|MinFVS(G)|3)

33. The weighted variant of the problem no polynomial kernel for parameter p3cover

34. Weighted Independent Set with P3-cover • Input: Vertex-weighted graph G, modulator X such that G – X is a collection of paths on at most 2 vertices, integer q. • Question: Does G have an Independent Set of total weight at least q? • Parameter: k := |X|. G - X X Weight 12 Weight 30

35. Contrasting result • Weighted Independent Set with P3-cover does not admit a polynomial kernel • (assuming a widely-believed conjecture from complexity theory) • Proof uses a variation of many-one reductions • Intuition: • There is no answer-preserving polynomial-time procedure that reduces an instance of Weighted Independent Set to some instance whose size is bounded by the size of a P3 cover • Independent Set parameterized by P3 cover is the first example where the use of vertex weights does not affect fixed-parameter tractability, but does affect kernelizability • Compare: for Independent Set with parameter Vertex Cover both the weighted and unweighted problem admit small kernels!

36. Why vertex weights make the problem harder to kernelize • Main idea: • Build a graph G which contains adjacent pairs of vertices inside the modulator X • If you select exactly one from each pair, then the rest of the independent set behaves in some nice way • But any maximum cardinality independent set would not use any vertices from X at all • Give the vertices in these pairs high weight! G - X X

37. conclusion and discussion

38. Summary of kernelization results • Table shows number of vertices in reduced graphs • * marks existing results • Our results can be combined with existing kernelization • Ensures reduces instances using new technique are not bigger than using old technique

39. Vertex Cover Edgeless Graphs P3 cover Paths ≤2 nodes Feedback vtx Set Forests Kernelizability of (Unweighted) Independent Set ≤ ≤

40. Kernelizability of (Unweighted) Independent Set Vertex Cover Increasing size P3 Cover Cluster Deletion Distance Feedback Vertex Set ? ? Bipartite Deletion Distance Outerplanar Deletion Distance Treewidth

41. Kernel lower bounds for Unweighted Independent Set with structural parameters • Consider some graph class F such that • F is hereditary (closed under vertex deletion) • F contains all complete graphs • Maximum Independent Set can be solved in polynomial time for graphs in F • The independent set problem parameterized by the minimum number of vertices which have to be deleted to obtain a graph in class F, is in FPT • (assuming the deletion set S is given) • BUT: There is no polynomial kernel for this parameterized problem (unless …) • Proof using cross-composition

42. Implications • The Maximum Independent Set problem parameterized by the number k of vertices which have to be deleted to obtain a • Perfect graph, • Chordal graph, • Interval graph, • Cograph, • Etc …, • is in the class FPT but does not admit a polynomial kernel (unless …)

43. Conclusion • We have studied Independent Set parameterized by different measures of graph complexity • Size of a Vertex Cover, P3 Cover, Feedback Vertex Set • Usage of vertex weights affects kernelizability • Hierarchy of parameters (complexity measures) which we can explore • Open problems • Deletion distance to bipartite/outerplanar graphs • Improve the degree of the polynomial: cubic to quadratic?