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Parameterized Complexity Part I – Basics, Kernels and Branching

Parameterized Complexity Part I – Basics, Kernels and Branching. Daniel Lokshtanov. Basics / Motivation. If L is NP-hard then there is no algorithm which solves all instances of L in polynomial time. What about the easy instances?. How do we capture easy ?. Example: Vertex Cover.

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Parameterized Complexity Part I – Basics, Kernels and Branching

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  1. Parameterized ComplexityPart I – Basics, Kernels and Branching Daniel Lokshtanov

  2. Basics / Motivation If L is NP-hard then there is no algorithm which solves all instances of L in polynomial time. What about the easy instances? How do we capture easy?

  3. Example: Vertex Cover In:G, k Question: such that everyedgein G has an endpoint in S? Such a set S is called a vertex cover.

  4. Algorithms for Vertex Cover Naive: O(nkm). Can we do it in linear time for k=10? Or linear time for any fixed integer k?

  5. Pre-processing for Vertex Cover If any vertex v has degree k+1 it must be a part of any vertex cover of size k Pick it in to solution. Removev and decrease k by 1.

  6. Pre-processing If no vertices of degree k+1 and k2 edges left say NO. k2 edges left. Remove vertices of degree 0, then there are 2k2 vertices left. In linear time, we made n 2k2 and m k2. Brute force now takes time O((2k2)kk)

  7. Running time Total running time is: O(n+m + (2k2)kk) Linear for any fixed k Pretty slow even for k = 10 

  8. Parameterized Complexity Every instance comes with a parameter k. The problem is fixed parameter tractable (FPT) if exists algorithm with running time f(k)nc. Often k is solution size, but could be many other things So Vertex Cover parameterized by solution size is fixed parameter tractable.

  9. Kernelization For vertex cover we first reduced the instance to size O(k2) in polynomial time, then we solved the instance. Let’s give this approach a name – kernelization.

  10. Kernels A f(k)-kernel is a polynomial time algorithm that takes an instance I with parameter k and outputs an equivalent instance I’ with parameter k’ such that: |I’| ≤ f(k) k’ ≤ f(k) (but typically k’ ≤ k)

  11. Kernelizable = FPT A problem Π is solvable in f(k)nctime for some f. ⇔ Π is decidable and has a g(k) kernel for some g. • Kernelize and solve.  If n ≤ f(k), done. If n ≥ f(k) solve in time nc+1

  12. Kernel: Point-line cover In: n points in the plane, integer k Question:Can you hit all the points with k straight lines? Fact:Point-Line cover is NP-Complete.

  13. Take them out with only 3 shots? Alien invasion! Victory!

  14. Reduction rules R1: If some line covers k+1 points use it (and reduce k by one). R2: If no line covers n/k points, say NO. If neither R1 nor R2 can be applied then n ≤ k2. KPR’13:No kernel with O(k2-) points unless coNP NP/poly Kernel with k2 points!

  15. Kernelization Initially thought of as a technique for designing FPT algorithms. Interesting in its own right, because it allows us to analyze polynomial time pre-processing.

  16. Better kernel for Vertex Cover? Next:kernel for Vertex Cover with 2k vertices, using linear programming relaxation.

  17. Vertex Cover (I)LP Minimize Z If OPTLPOPT > k  output ``no’’

  18. Nemhauser Trotter Theorem • There is always an optimal solution to Vertex Cover LP that sets variables to . • For any –solution there is a matching from the 1-vertices to the 0-vertices, saturating the 1-vertices.

  19. Nemhauser Trotter Proof + + - - -

  20. Reduction Rule If exists optimal LP solution that sets xv to 1, then exists optimal vertex cover that selects v. Remove v from G and decrease k by 1. Correctness follows from Nemhauser Trotter Polynomial time by LP solving.

  21. Kernel Suppose reduction rule can not be applied and consider any optimal solution to LP. No vertex is 0(remove isolated vertices) No vertex is 1. All vertices are . OPTLPk. n 2k

  22. Vertex Cover Algorithm First apply the simple O(k2) kernel. Then reduce to 2k vertices (in poly(k) time) Try all subsets of the vertices. Running time: = O(n + m + kc + 22k) = O(n+m+4k)

  23. Branching A simple and powerful technique for designing FPT algorithms.

  24. Vertex Cover (again) Let uv At least one of u and v must be in the solution. G has a vertex cover of size k G\vhas a vertex cover of size k-1 G\uhas a vertex cover of size k-1 OR Recursive algorithm!

  25. Running time k k-1 k-1 k-2 k-2 k-3 k-3 ... Total running time is O(2knc) O(n + m + 2kkc) if we run kernel first.

  26. 3-Hitting Set Input: Family S1...Smof sets of size 3 over a universe U = v1...vn, integerk. Question: Is there a set X V such that |X| k and every set Si intersects with X? Parameter:k

  27. Branching for 3-Hitting Set Pick a set Si = {va,vb,vc}. At least one of them must be in the solution X. Branch on which one, decrease k by one. Remove all sets that are hit. Total running time: O(3k (n+m))

  28. Even Better Branching for Vertex Cover (Going below 2k) If all vertices have degree then G is a set of paths and cycles, so we can solve Vertex Cover in polynomial time.

  29. Even Better Branching Let v , degree(v) 3. If v is not in the solution, then N(v) is. G has a vertex cover of size k G\N(v)has a vertex cover of size k-degree(v) G\vhas a vertex cover of size k-1 OR Recursive algorithm!

  30. Running time T(n, k) = Running time on a graph on at most n vertices and parameter at most k. N(k) = Number of nodes in a recursion tree if parameter is at most k. L(k) = Number of leavesin a recursion tree if parameter is at most k. T(n, k) O(N(k) (n+m))

  31. Recurrence If existsvertex of degree3. L(k-1) + L(k-3) L(k) 1 otherwise. Will prove L(k) 1.47kby induction. (recurrence) L(k) L(k-1) + L(k-3)1.47k-1 + 1.47k-3 1.47k (1.47-1 + 1.47-3) 1.47k (induction hypothesis) (choice of 1.47)

  32. Running time analysis Number of leaves in the recursion tree is at most 1.47k, so total running time is O(1.47k(n+m)). Fastest known algorithm for Vertex Cover has running time 1.27k (Chen, Kanj, Xia, 2010).

  33. Alternative Parameters So far we have only seen the solution size as the parameter. Often other parameters also make sense, or even make more sense than solution size.

  34. k-Coloring A proper k-coloring is a funcion f : V(G)  {1...k} such that no edge has same colored endpoints. Input: G, k Question: Does G have a proper k-coloring? Parameter: k Can not have FPT algorithm – NP-hard for k=3!

  35. k-Coloring parameterized by VC Input: G, integer k, set X V(G) such that X is a vertex cover of G, integer x = |X|. Question: Does G have a proper k-coloring? Parameter: x FPT now means f(x)nO(1).

  36. k-Coloring parameterized by VC If x+1 k say YES X Thus, assume k x. Branch on kx colorings of X. I = V(G) \ X For each guess, color I greedily. Total running time: O(kx(n+m)) = O(xx(n+m)). LMS’11: No xo(x)nO(1)time algorithm assuming Exponential Time Hypothesis.

  37. Above Guarantee Vertex Cover So far we have only seen the solution size as the parameterfor vertex cover. Alternative parameter solution size – OPTLP Note that can be very small even if solution size is big!

  38. Vertex Cover Above LP In:G, k. Question:Does there exist a vertex cover S of size at most k? Parameter: where OPTLPis the value of an optimum LP solution. Now FPT means f()nc time!

  39. Reduction Rule Recall the reduction rule from the kernel for Vertex Cover: • If exists optimal LP solution that sets xv to 1, then exists optimal vertex cover that selects v. • Remove v from G and decrease k by 1. Now the unique LP optimum sets all vertices to

  40. Branching Pick an edge uv. Solve (G\u, k-1) and (G\v, k-1). since otherwise there is an optimal LP solution for G that sets u to 1. Then

  41. Branching - Analysis OPTLP– kdrops by ½ ... in both branches! Total time: 4k-OPTLP nO(1) Caveat: The reduction does not increase the measure!

  42. Part II – Parameterized Reductions

  43. Reductions between Problems It is often useful to reduce A to B such that an algorithm for A gives an algorithm for B. • ifB is ``easy’’ then A is ``easy’’. • If A is ``hard’’ then B is ``hard’’. Examples: NP-hardness, reductions to 2-SAT, hardness of approximation, reductions to Set Cover.

  44. Parameterized Reductions Time:T(I,k) Equivalentinstance (I’,k’) of problem B Instance (I,k) of problem A h(k)|I|O(1) when does a reduction + f(k’)|I’|O(1) time algorithm for B givesomethingmeaningful for A? Need: k’ g(k) for some function g of k only!

  45. Algorithms by Reduction Recall, we saw an algorithm for Vertex Cover above LP with running time 4k-OPTLP nO(1). Now we give an algorithm for Almost2-SAT by reduction to Vertex Cover above LP.

  46. Almost 2-SAT In:2-SAT formula, integer k Question: Can we remove k variables from and make it satisfiable?Parameter:k 2-SAT formula example: = (x y)

  47. Almost 2-SAT Vertex Cover/k-LP

  48. Reduction analysis Time taken by reduction: polynomial. n variables in 2-SAT formula 2n vertices in output graph, OPTLP = n. Vertex Covers of size n+k = OPTLP + k Almost 2-SAT solutions of size k Vertex Cover 4k-OPTLP nO(1) algorithm  Almost 2-SAT4k nO(1) algorithm!

  49. Odd Cycle Transversal In:G, k Question:Does there exist a set of size at most ksuch that G\S is bipartite? Parameter: k Exercise: Give a 4knO(1) time algorithm for Odd Cycle Transversal.

  50. Odd Cycle Transversal Almost 2-Sat x y x y z z

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