1. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DE-AC04-94AL85000. Reconnect ‘04 Problem-Specific Preprocessing (and a hint at FPT) Cynthia Phillips Sandia National Laboratories

2. Slide 2 Preprocessing Sets of little “tricks” that make the problem smaller while still allowing you to extract the optimal solution –e.g. There exists an optimal solution for which this choice holds, or this choice cannot hold. If you know problem structure this can be quite powerful

3. Slide 3 A Formalization Suppose your problem input size is naturally described by two values –n describes most of the problem –k a single parameter that can be “big”, but is usually small for practical instances Examples –Graphs with bounded degree or bounded diameter –Nesting level in databases –Strings with small alphabet size –Cases where the value of the solution is small (cleaning up data)

4. Slide 4 Fixed-Parameter Tractability A problem with size (n,k) is fixed-parameter tractable (FPT) if it can be solved in time Where f(k) is an arbitrary function of the parameter k A problem is FPT if and only if it is kernelizable: there is a polynomial- time transformation from the problem of size (n,k) to another instance of the problem (n s,k s ) such that From here an enumerative method has a better chance

5. Slide 5 Example: Vertex Cover is FPT Find a minimum-size set of vertices such that each edge has at least one endpoint in the set. k = size of the vertex cover 2 1 4 7 5 3 6

6. Slide 6 Kernelizing Kernelization usually is a set of (local) independent transformation rules Rules cascade Apply till the problem is irreducible

7. Slide 7 Kernelizing Vertex Cover We’ll see 4 rules (not counting trivial degree-0 one). There are more (even LP-based), but these are really simple! Make this a decision problem: Does graph G=(V,E) have a vertex cover of size k? Each rule transforms to a new question: Does graph G s =(V s,E s ) have a vertex cover of size k s ? where

8. Slide 8 Rule 1 If vertex v has degree 1, then put its neighbor u in the optimal VC and delete all edges (u,w): u v yx u v yx

9. Slide 9 Rule 2 If vertex v has degree 2, and its two neighbors u and w are connected, then put its neighbors u and w in the optimal VC: v wu v wu

10. Slide 10 Rule 3 If vertex v has degree 2, and its two neighbors u and w are not connected, then shrink the (u,v,w) triangle to a single node v 1. Use k s = k - 1 v wu v wu v1v1

11. Slide 11 Rule 3 Justification If vertex v has degree 2, and its two neighbors u and w are not connected, then shrink the (u,v,w) triangle to a single node If either u or w is necessary for the VC, then effectively v becomes degree one and the other is selected by rule 1 v wu

12. Slide 12 Rule 3 Justification If vertex v has degree 2, and its two neighbors u and w are not connected, then shrink the (u,v,w) triangle to a single node v 1 Use k s = k - 1 Find the optimal VC of the graph with v 1 If v 1 is not selected, just cover the triple independently. Choose v. The total VC has size at most k s + 1 = k v wu

13. Slide 13 Rule 3 Justification If vertex v has degree 2, and its two neighbors u and w are not connected, then shrink the (u,v,w) triangle to a single node v 1 Use k s = k - 1 Find the optimal VC of the graph with v 1 If v 1 is selected, must take either u or w, so have to take both. Size of the VC is at most k s - 1 + 2 = k v wu

14. Slide 14 Rule 4 Take all the nodes of degree > k If we didn’t take them, we’d have to take all (> k) neighbors (too many for a VC of size k)

15. Slide 15 Kernelization Results If no more rules apply, the graph has

16. Slide 16 Proof Graph with n s nodes and a VC of size k s Satisfies all rules so degree of v (number of neighbors) for all v satisfies: The nodes not in the VC are an independent set of size n s -k s

17. Slide 17 Proof F is the set of edges adjacent to nodes in the VC The n-k nodes in the independent set have degree at least 3 The k nodes in the VC have degree at most k

18. Slide 18 Proof

19. Slide 19 Comments These rules are local and independent (any order) –Could parallelize this