Hitting and harvesting pumpkins. Gwenaël Joret Christophe Paul Ignasi Sau Saket Saurabh Stéphan Thomassé

Transcription

1 Hitting and harvesting pumpkins Gwenaël Joret Christophe Paul Ignasi Sau Saket Saurabh Stéphan Thomassé

2 Pumpkins c-pumpkin: c NB: graph = multigraph

3 Graphs with no c-pumpkin minor c = 1: empty graphs c = 2: forests c = 3: no two cycles share an edge etc.

4 Pumpkin transversals For the whole talk: c 1 fixed G input graph n := G c-pumpkin transversal: vertex subset X s.t. G X has no c-pumpkin minor c = 3 X Transversal number: min. size of a c-pumpkin transversal

5 c-pumpkin Transversal problem NP-hard c Interested in FPT algorithms and approximation algorithms In parameterized version: extra parameter k goal: decide if c-pumpkin transversal of size k

6 Some special cases c = 1: Vertex Cover 2-approximation algorithm O( k + kn)-time FPT algorithm [Chen, Kanj, Xia 10]

7 Some special cases c = 1: Vertex Cover 2-approximation algorithm O( k + kn)-time FPT algorithm [Chen, Kanj, Xia 10] c = 2: Feedback Vertex Set 2-approximation algorithms [Bafman, Berman, Fujito 99, Becker & Geiger 96] O(3.83 k k n 2 )-time FPT algorithm [Cao, Chen, Liu 10]

8 Some special cases c = 1: Vertex Cover 2-approximation algorithm O( k + kn)-time FPT algorithm [Chen, Kanj, Xia 10] c = 2: Feedback Vertex Set 2-approximation algorithms [Bafman, Berman, Fujito 99, Becker & Geiger 96] O(3.83 k k n 2 )-time FPT algorithm [Cao, Chen, Liu 10] c = 3: Diamond Hitting Set 9-approximation algorithm [Fiorini, J., Pietropaoli 10]

9 c: [Fomin, Lokshtanov, Misra, Philip, Saurabh 11] O(log 3/2 n)-approximation algorithm 2 O(k log k) n O(1) -time FPT algorithm

10 c: [Fomin, Lokshtanov, Misra, Philip, Saurabh 11] O(log 3/2 n)-approximation algorithm 2 O(k log k) n O(1) -time FPT algorithm Theorem (J., Paul, Sau, Saurabh, Thomassé) There is a single-exponential FPT algorithm c (2 O(k) n O(1) running time) NB: no 2 o(k) n O(1) -time FPT algorithm unless ETH fails

11 Pumpkin packings c-pumpkin packing: collection of vertex-disjoint subgraphs of G, each containing a c-pumpkin minor c = 2 Packing number: max. cardinality of a c-pumpkin packing

12 c-pumpkin Packing problem NP-hard c 2 c = 1 : Matching c = 2 : Cycle Packing O(log n)-approximation algorithm [Krivelevich, Nutov, Salavatipour 07] Ω(log 1/2 ε n)-inapproximability [Friggstad, Salavatipour 11]

13 Approximate min-max relation for packings and transversals Theorem (J., Paul, Sau, Saurabh, Thomassé) There exist a c-pumpkin packing M, and a c-pumpkin transversal X s.t. X O c (log n) M O c (log n)-approximation algorithm for c-pumpkin Transversal and c-pumpkin Packing

14 FPT algorithm Goal: Single-exponential FPT algorithm for c-pumpkin Transversal Tool: Kernel of size (k 2 log 3/2 k) [Fomin, Lokshtanov, Misra, Philip, Saurabh 11]

15 FPT algorithm Goal: Single-exponential FPT algorithm for c-pumpkin Transversal Tool: Kernel of size (k 2 log 3/2 k) [Fomin, Lokshtanov, Misra, Philip, Saurabh 11] Switch to Disjoint c-pumpkin Transversal problem: Input: G, transversal X with X k + 1 Question: is there a transversal X with X k that is disjoint from X? c = 3 k = 4 X

16 FPT algorithm Goal: Single-exponential FPT algorithm for c-pumpkin Transversal Tool: Kernel of size (k 2 log 3/2 k) [Fomin, Lokshtanov, Misra, Philip, Saurabh 11] Switch to Disjoint c-pumpkin Transversal problem: Input: G, transversal X with X k + 1 Question: is there a transversal X with X k that is disjoint from X? c = 3 k = 4 X X'

17 Lemma d k n O(1) algorithm for Disjoint c-pumpkin Transversal (d + 1) k n O(1) algorithm for c-pumpkin Transversal Ingredients of FPT algorithm: ad-hoc reduction rules protrusion-based reduction rule branching rule linear kernel in special case

18 Approximation algorithms Goal: Finding a c-pumpkin packing M and a c-pumpkin transversal X s.t. X O c (log n) M

19 Reduction rules Reduction rules: produce a smaller graph with same packing and transversal numbers u v u v

20 Small pumpkins Subgraph small if of size h(c) log n computable function) (where h some fixed, d (G) := average degree of underlying simple graph of G If d (G) 2 t then G G with G = O t (log n) s.t. G contains a K t -minor [Fiorini, J., Theis, Wood 10] if d (G) 2 2 c+1 then G has a small subgraph containing a c-pumpkin minor

21 Theorem Either G has a small c-pumpkin minor or some reduction rule can be applied Proof idea: k W C { r P

22 Approximation algorithm: M ; X If G not reduced: Apply reduction rule on G Call algorithm on resulting graph Else: Compute a small c-pumpkin minor M Call algorithm on G \ V (M), giving a packing M aaa and a transversal X M M {M} X X V (M)

23 Thank You!