Solving Hard Problems Incrementally

Transcription

1 Rod Downey, Judith Egan, Michael Fellows, Frances Rosamond, Peter Shaw May 20, 2013

2 Introduction Dening the Right Problem Incremental Local Search Finding the right FPT subroutine INC-DS is in FPT INC DS has no Poly(k) kernel Conclusion

3 Joint work with Rod Downey Judith Egan Michael Fellows Frances Rosamond Peter Shaw

4 History of Inc FPT Asking the right Problem is Not Always Obvious! Initially Michael Fellows and Rod Downey posed the k-neighborhood questions for problem X Given: Instance I and current solution S Parameter: k Question: Is there a better solution S within k steps of the usual local search algorithm for Problem X Jiong Guo at APCAC 2012 showed this approach resulted in W [1] hardness proof based on a really bad current solutions S

5 History of Incremental FPT rst attempt Simultaneously Rolf Niedermeier and Sepp Hartung and Rod Downey and Michael Fellows came up with similar ideas. The latter resulted by a suggestion by Anil-Nerode for cross-fertilization between parameterized complexity and incremental computing. First Attempt: Given: Instance I and defective solution S Parameter: k Question: Is there a solution S with d H (S, S ) k But this turns out to be W [2] hard for Dominating Set

6 An Example Some Examples Inertia Data currently unknown Speedup - Heuristic Subroutine

7 An Example An FPT Turbo-charged heuristic for Graph Coloring Hartung and Niedermeirer show how to improve the Graph Coloring heuristic via an incremental FPT subroutine Shows an FPT subroutine for the graph coloring problem. When Applied it the heuristic algorithm obtains on average an 11% reduction in the number of colors to the Iterated Greedy algorithm (A current best approach) Average 170 times seed improvement k 117 and c 8.

8 An Example The Incremental List Coloring Subroutine Incremental Conservativek-List Coloring (INC k-list Coloring) Given: A graph G = (V, E ), a k-list coloring f for G [V \ x] and c N Parameter: (c, k) Question: Is there a k-list coloring f for G such that {v V \ {x} : f (v) = f (v)} c NP-Hard for k 3 W [1]-hard for parameter c Incremental version parameterized by (k, c) is FPT O (k(k 1) c ) Exponential kernel

9 An Example A meta approach for local search subroutines Read back to front Hartun and Niedermeier's paper reveals a meta approach for Turbo-Charging local search algorithms with an FPT subroutines. the greedy heuristic can be viewed as a inductive route that leads eventually to the input you were assumed to deal with. From one step to the next, carry along a solution to be, hopefully modied Think of as iterative compression but abstractly only In the case of iterative compression: compressed, exactly

10 The inductive route A Greedy heuristic - Version A Observations: 1. Start with a complete graph G k with n vertices 2. Order the vertices of the input graph G from small degree to large degree 3. At each step (add the next vertex) on our way to G (the inductive route) 4. If the new vertex is not already dominated, then add the highest degree vertex in N[v] 5. Apply the FPT Incremental subroutine to nd the best vertices to add to the solution.

11 An Alternative Greedy heuristic An Alternative Greedy Heuristic for Dominating Set Plan - B Observations: 1. Start with a complete graph G k with n vertices 2. Order the vertices of the input graph G from small degree to large degree 3. gradually delete edges, eventually to obtain G (the inductive route) 4. Initially the solution has size 1 5. When the size of S becomes too large call the incremental FPT subroutine 6. Apply the FPT Incremental subroutine to nd the best vertices to add to the solution.

12 An Alternative Greedy heuristic Inc Dominating Set (INC-DS) Version B INC-DS-II Given: G, e, S where e is an edge and S is a dominating set of G but not G e = G Parameter: k Question: Can we nd S such that d v (S, S ) k, S = S and S is a dominating set of G

13 An Alternative Greedy heuristic Incremental Dominating Set (INC-DS) INCREMENTAL DOMINATING SET (INC-DS) Given: A graph G and a dominating set S for G, a graph G obtained from G with hamming distance d e (G, G ) k, positive integers k, r. Parameter: (k, r) Question: Does there exist S such that the hamming distance d v (S, S ) r and S is a dominating set for G?

14 INC-DS is in FPT Inc Dominating Set (INC-DS) is in FPT Observations: 1. Edge deletions (u, v) are irrelevant if both u and v are in S. 2. A vertex v is still dominated if v itself, or a neighbour u, is in S. 3. Adding edges does not harm S. 4. Only trouble is when we delete edges such that v no longer has a neighbour in S.

15 Reduction to DOM-A-VC Reduction to DOM-A-VC Outline of proof of Theorem 1. The original instance can be visualized in three parts A, B, C. Part C can be removed because there are no edges between A an C. A k or fewer non-dominated vertices. Each connected to one or more already dominated vertex. Note A is a vertex cover for A B B Already dominated vertices. May have edges between them. C Vertices S. May have edges between them.

16 Reduction to DOM-A-VC Inc Dominating Set is in FPT 1. Reset G. Do all edge additions and all edge deletions until we have a problem. 2. Finish all edge deletions, at most k of them, to get G. These deletions necessarily leave some vertices not dominated. 3. There are at most k edge deletions so there are at most k vertices that are no longer dominated in G. 4. Picture the input instance after S is removed. Part A consists of the k vertices that need to be dominated. Part B is an independent set. After removing edges between vertices of B, (they do nothing to dominate A) and these vertices are already dominated by S. Every vertex in B is connected to a vertex in A. So A is a vertex cover of size k. If k r then we are done, take all of A. Otherwise k > r and the problem is now DOM-A-VC.

17 Proof Overview Proof INC-DS in FPT Theorem The problem INC-DS is in FPT. The proof of Theorem 1 involves a reduction to the following problem of possible independent interest: DOMINATING A VERTEX COVER (DOM-A-VC) Given: A graph G and a vertex cover C for G of size k, and a positive integer r k Parameter: (k,r) Question: Is there a set D of at most r vertices of G such that every vertex in C either belongs to D or has a neighbour in D?

18 DOM-A-VC is FPT DOM-A-VC FPT A k Hex These 2 vertices have the same type as they have the same neighbourhood exactly in A. Triangle both of these reach vertices a, b, c only. Call this TYPE a-b-c

19 DOM-A-VC is FPT DOM-A-VC is FPT Note there are only 2 2k dierent types by their neighborhoods in the eected A (left hand side) No need to add to S more than one vertex of any given type This gives us an (exponential) FPT kernel for the problem INC DS has no Poly(k) kernel INC Independent Set is still W [2] hard

20 What we need to show Lower Bounds Techniques - 3 Steps To show that a problem has no polynomial kernel there are 3 steps: 1. Show that the parameterized problem Q is FPT. 2. Find a parameterized problem P that is known to have no polynomial kernel. 3. Show the unparameterized versions of both P and Q are NP-complete. 4. Show PPT transform from P to Q 5. Using contrapositive of Lemma [DLS]If P has no polynomial kernel then neither does Q.

21 INC DS no Poly(k) Part I A parameterized problem is Compositional if there is a composition algorithm for it. Lemma [DLS] Let L be a compositional parameterized problem whose unparameterized version L is NP-complete then unless PH = 3 P there is no polynomial kernel for L. Theorem The problem INC-DS is does not have a polynomial kernel unless conp NP. PPT reduce π to INC-DS where π is a non-poly(k) FPT problem.

22 INC DS no Poly(k) Part I Incremental DS is NP-Hard If G has a dominating set of size s, then in G we can delete k 1 and G will have a dominating set S of size k + 1. Suppose G has a dominating set of size s and G such that d v (S S ) r. Then, H has a dominating set of size s.

23 Reduction from DOM-A-VC to Incremental Dominating Set Reduction from DOM-A-VC to Incremental Dominating Set Assume we have a VC of size k in G 1 and it is dominated by r vertices from either the boxed vertices of the independent set or both. In G 2 we can delete any k vertices and dominate the new graph with the same vertices and a and b, hence r + 2 vertices. In G 2 we have the graph dominated by {a, b}. Modify the graph by deleting k edges. If the edges are deleted

24 Reduction from DOM-A-VC to Incremental Dominating Set Reduction from RBDS to DOM-A-VC If there is a dom set in B that dominates R then in DOM-A-VC use the same vertices to dominate the VC If there is a small r Dom set among the independent set and some from VC that dominate the VC, then use those same vertices in B to dominate R. Replacing the vertices in VC to their neighbours in B

25 Reduction from DOM-A-VC to Incremental Dominating Set Inc Dominating Set (INC-DS) version II INC-DS-II Given: G, e, S where e is an edge and S is a dominating set of G but not G e = G Parameter: k Question: Can we nd S such that d v (S, S ) k, S = S and S is a dominating set of G

26 Reduction from DOM-A-VC to Incremental Dominating Set Another experiment: Feedback Vertex Set A simple heuristic Step 1 Order the vetices from high degree t o low degree Step 1b (alt) Start with spanning tree Step 3 Use this as an initial move to build G Step 4 Turn this into an edge-addition move; start with the empty graph on n vertices an add eges as directed by M to build G. This is the inductive route Step 5 If adding the edge e creates a unique uncovered cycle, choose the verte of the cycle of highest degree. If multiple cycles involving e are created, then choose the endpoint of highest degree. This leads to the incremental problem: INC-FVS Given: G, e, S where e is an edge and S is a Feedback Vertex Set G but not G = G + e Parameter: k

27 Future Work Conclusion Central thesis: Almost all practical heuristics can be improved by appropriately targeted FPT subroutines Applying parameterized complexity and incremental computing to heuristics Improving heuristics FPT subroutines