Dynamic Parameterized Problems - Algorithms and Complexity

Dynamic Parameterized Problems

R. Krithika The Institute of Mathematical Sciences, Chennai, India

NMI Workshop on Complexity TheoryIIT Gandhinagar

Nov 2016

(Joint work with Abhishek Sahu and Prafullkumar Tale)

Dynamic Graphs

Dynamic Graphs

Current graph is at a distance 8 from the initial graph

Dynamic Graphs

Dynamic GraphsG

Dynamic Graphs

vertex cover of G

G

Dynamic Graphs

vertex cover of G

G H

Dynamic Graphs

edit distance 8

vertex cover of G

G H

Dynamic Graphs

edit distance 8

vertex cover of G

G H

vertex cover of H

Dynamic Graphs

edit distance 8

Hamming distance 1

vertex cover of G

G H

vertex cover of H

Dynamic Graphs

edit distance 8

G H

Dynamic Graphs

edit distance 8

G H

vertex cover of G

Dynamic Graphs

edit distance 8

G H

Hamming distance 2

vertex cover of G

Dynamic Graphs

edit distance 8

G H

Hamming distance 2

vertex cover of G vertex cover of H

Dynamic Problem Template

Dynamic Problem TemplateInstance:

• Graphs G, H on same vertex set s.t de(G,H) ≤ k

• A solution S of G

• An integer r

Question: Does H have a solution T s.t dv(S,T) ≤ r?

Parameter(s): k, r

Dynamic Problem TemplateInstance:

• Graphs G, H on same vertex set s.t de(G,H) ≤ k

• A solution S of G

• An integer r

Question: Does H have a solution T s.t dv(S,T) ≤ r?

Parameter(s): k, r

k- edit parameter

r- distance parameter

Dynamic Dominating Set


D


D D


D D

edit distance kG H

Dynamic Dominating SetS

G

S is a DS for G

Dynamic Dominating SetS

G

S is a DS for G

TH

T is a DS for H


SS

G

S is a DS for G

TH

T is a DS for H


S

(S ∪ T) is a DS for H

SG

S is a DS for G

TH

T is a DS for H


S

d(S,S ∪ T) = |T-S| ≤ |T-S| + |S-T| = d(S,T)


SG

S is a DS for G

TH

T is a DS for H


S

d(S,S ∪ T) = |T-S| ≤ |T-S| + |S-T| = d(S,T)


SG

S is a DS for G

TH

T is a DS for H

Suffices to search for a DS of H that contains S

Dynamic Dominating SetD

Dynamic Dominating SetD D

C


C

Addition of edges do not affect the solution


D


D D

C


D D

C

Deletion of edges within C or within D do not affect the solution


C


C

Deletion of edges between C and D could result in undominated vertices!


D


D D

C


D

At most k vertices are not dominated by D

D

C


C


C

A set cover instance on k-element universe


C

A set cover instance on k-element universe

O*(2k) algorithm

Dynamic Connected Dominating Set


D


D D


D D

edit distance kG H

k1 del k2 add


SG

S is a CDS for G


SG

S is a CDS for G

TH

T is a CDS for H


SS

G

S is a CDS for G

TH

T is a CDS for H


S


SG

S is a CDS for G

TH

T is a CDS for H


S

d(S,S ∪ T) = |T-S| ≤ |T-S| + |S-T| = d(S,T)


SG

S is a CDS for G

TH

T is a CDS for H


S

d(S,S ∪ T) = |T-S| ≤ |T-S| + |S-T| = d(S,T)


SG

S is a CDS for G

TH

T is a CDS for H

S T


S

d(S,S ∪ T) = |T-S| ≤ |T-S| + |S-T| = d(S,T)


SG

S is a CDS for G

TH

T is a CDS for H

S T (S ∪ T) is a CDS for H


S

d(S,S ∪ T) = |T-S| ≤ |T-S| + |S-T| = d(S,T)


SG

S is a CDS for G

TH

T is a CDS for H

Suffices to search for a CDS of H that contains S

S T (S ∪ T) is a CDS for H


D


D D


D D

At most k1 vertices are not dominated by D D has at most k2 components

Dynamic Connected Dominating Setcomponents of D

Dynamic Connected Dominating SetD1 D2 D3 . . . Dk2 components of D


undominated vertices

D1 D2 D3 . . . Dk2 components of D


v1 v2 v3 . . . vk1 undominated vertices




N(v1) N(v2) N(v3) . . . N(vk1)




N(v1) N(v2) N(v3) . . . N(vk1)


D1 D2 D3 . . . Dk2



N(v1) N(v2) N(v3) . . . N(vk1)


D1 D2 D3 . . . Dk2

A group Steiner tree problem with parameter k1+k2 = k



N(v1) N(v2) N(v3) . . . N(vk1)


D1 D2 D3 . . . Dk2

O*(2k) algorithm

A group Steiner tree problem with parameter k1+k2 = k

Set Cover Conjecture

Set Cover ConjectureSet Cover


U = { u1, u2, u3, . . . un }


U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }


U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm


U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm No faster algorithm known


U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm No faster algorithm knownNo lb under SETH known


U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm

SCC: No O*((2-c)n) algorithm for any c > 0

No faster algorithm knownNo lb under SETH known


U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm

SCC: No O*((2-c)n) algorithm for any c > 0

No faster algorithm knownNo lb under SETH known

Cygan, Dell, Lokshtanov, Marx, Nederlof, Okamoto, Paturi, Saurabh, Wahlstrom (2012)

Tightness Under Set Cover Conjecture


Set Cover (l)


Set Cover (l)

U = { u1, u2, u3, . . . un }


Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }


Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

ui

U


Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

ui

U

Fj

F


Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

ui

U

Fj

F


Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

ui

U

Fj

F

G

CDS


Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

ui

U

Fj

F

uiFj

x

H

G

CDS


Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

ui

U

Fj

F

uiFj

x

H

k=nr=l

G

CDS


Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

ui

U

Fj

F

uiFj

x

H

k=nr=l

(2-c)k algo DDS

(2-c)n algo SC

G

CDS

Concluding RemarksDynamic Problem Parameterized Complexity

k r

Dominating Set 2k2 [DEFRS14], 2k (tight) W[2]-hard [DEFRS14]

Connected Dominating Set

4k [AEFRS15], 2k (tight) W[2]-hard [AEFRS15]

Vertex Cover1.174k, 1.1277k (expo space), O(k) kernel

1.2738r, O(r2) kernel

Connected Vertex Cover 4k [AEFRS15], 2k W[2]-hard [AEFRS15]

Feedback Vertex Set1.6667k (randomized),

O(k) kernel3.592r, O(r2) kernel

∏-DeletionFixed-parameter (in)tractability related to that

of non-dynamic version

Concluding Remarks

• Maintain k-size solution?

• Other interesting parameters

• =k1 edge additions and =k2 edge deletions

• treewidth, vertex cover

• Relation to reconfiguration problems and online problems

• Interesting data structures

Thank you :)

Questions?

ReferencesDynamic Problems in Parameterized Complexity Framework• F. N. Abu-Khzam, J. Egan, M. R. Fellows, F. A. Rosamond, and P. Shaw. On the parameterized complexity

of dynamic problems. Theoretical Computer Science, 607, Part 3:426–434, 2015.• R.G. Downey, J. Egan, M.R. Fellows, F.A. Rosamond, and P. Shaw. Dynamic dominating set and turbo-

charging greedy heuristics. J. Tsinghua Sci. Technol, 19(4):329–337, 2014.• S. Hartung and R. Niedermeier. Incremental list coloring of graphs, parameterized by conservation.

Theoretical Computer Science, 494:86–98, 2013.

Vertex Cover• J. Chen, I. A. Kanj, and W. Jia. Vertex cover: further observations and further improvements. Journal

of Algorithms, 41(2):280–301, 2001.• J. Chen, I. A. Kanj, and G. Xia. Improved upper bounds for vertex cover. Theoretical Computer Science,

411(40-42):3736–3756, 2010.

Treewidth Bounds• F. V. Fomin, S. Gaspers, S. Saurabh, and A. A. Stepanov. On two techniques of combining branching and

treewidth. Algorithmica, 54(2):181–207, 2009.• J. Kneis, D. Mölle, S. Richter, and P. Rossmanith. A bound on the pathwidth of sparse graphs with

applications to exact algorithms. SIAM Journal on Discrete Mathematics,23(1):407–427, 2009.

ReferencesFeedback Vertex Set• F. V. Fomin, S. Gaspers, D. Lokshtanov, and S. Saurabh. Exact algorithms via monotone local

search. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing, STOC ’16. ACM, 2016.

• T. Kociumaka and M. Pilipczuk. Faster deterministic feedback vertex set. Information Processing Letters, 114(10):556–560, 2014.

• S. Thomassé. A quadratic kernel for feedback vertex set, chapter 13, pages 115–119. 2009.

Set Cover and Group Steiner Tree• M. Cygan, H. Dell, D. Lokshtanov, D. Marx, J. Nederlof, Y. Okamoto, R. Paturi, S. Saurabh, and M.

Wahlstrom. On problems as hard as CNF-SAT. ACM Transactions on Algorithms, 12(3), Article No. 41, 2016.

• F. V. Fomin, D. Kratsch, and G. J. Woeginger. Exact (exponential) algorithms for the dominating set problem. In Graph-Theoretic Concepts in Computer Science, pages 245–256. Springer, 2004.

• N. Misra, G. Philip, V. Raman, S. Saurabh, and S. Sikdar. FPT algorithms for connected feedback vertex set. Journal of combinatorial optimization, 24(2):131–146, 2012.

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