Dynamic Parameterized Problems - Algorithms and Complexity

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  • 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

  • Dynamic Graphs

  • Dynamic Graphs

  • Dynamic Graphs

  • Dynamic Graphs

  • Dynamic Graphs

  • Dynamic Graphs

  • 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

  • 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

  • Dynamic Dominating Set

    D

  • Dynamic Dominating Set

    D D

  • Dynamic Dominating Set

    D D

    edit distance kG H

  • Dynamic Dominating Set

  • 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

  • Dynamic Dominating Set

    SS

    G

    S is a DS for G

    TH

    T is a DS for H

  • Dynamic Dominating Set

    S

    (S T) is a DS for H

    SG

    S is a DS for G

    TH

    T is a DS for H

  • Dynamic Dominating Set

    S

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

    (S T) is a DS for H

    SG

    S is a DS for G

    TH

    T is a DS for H

  • Dynamic Dominating Set

    S

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

    (S T) is a DS for H

    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 Set

  • Dynamic Dominating SetD

  • Dynamic Dominating SetD D

    C

  • Dynamic Dominating SetD D

    C

    Addition of edges do not affect the solution

  • Dynamic Dominating Set

  • Dynamic Dominating Set

    D

  • Dynamic Dominating Set

    D D

    C

  • Dynamic Dominating Set

    D D

    C

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

  • Dynamic Dominating Set

  • Dynamic Dominating SetD

  • Dynamic Dominating SetD D

    C

  • Dynamic Dominating SetD D

    C

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

  • Dynamic Dominating Set

  • Dynamic Dominating Set

    D

  • Dynamic Dominating Set

    D D

    C

  • Dynamic Dominating Set

    D

    At most k vertices are not dominated by D

    D

    C

  • Dynamic Dominating Set

  • Dynamic Dominating SetD

  • Dynamic Dominating SetD D

    C

  • Dynamic Dominating SetD D

    C

    A set cover instance on k-element universe

  • Dynamic Dominating SetD D

    C

    A set cover instance on k-element universe

    O*(2k) algorithm

  • Dynamic Connected Dominating Set

  • Dynamic Connected Dominating Set

    D

  • Dynamic Connected Dominating Set

    D D

  • Dynamic Connected Dominating Set

    D D

    edit distance kG H

    k1 del k2 add

  • Dynamic Connected Dominating Set

  • Dynamic Connected Dominating Set

    SG

    S is a CDS for G

  • Dynamic Connected Dominating Set

    SG

    S is a CDS for G

    TH

    T is a CDS for H

  • Dynamic Connected Dominating Set

    SS

    G

    S is a CDS for G

    TH

    T is a CDS for H

  • Dynamic Connected Dominating Set

    S

    (S T) is a DS for H

    SG

    S is a CDS for G

    TH

    T is a CDS for H

  • Dynamic Connected Dominating Set

    S

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

    (S T) is a DS for H

    SG

    S is a CDS for G

    TH

    T is a CDS for H

  • Dynamic Connected Dominating Set

    S

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

    (S T) is a DS for H

    SG

    S is a CDS for G

    TH

    T is a CDS for H

    S T

  • Dynamic Connected Dominating Set

    S

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

    (S T) is a DS for H

    SG

    S is a CDS for G

    TH

    T is a CDS for H

    S T

  • Dynamic Connected Dominating Set

    S

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

    (S T) is a DS for H

    SG

    S is a CDS for G

    TH

    T is a CDS for H

    S T

  • Dynamic Connected Dominating Set

    S

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

    (S T) is a DS for H

    SG

    S is a CDS for G

    TH

    T is a CDS for H

    S T

  • Dynamic Connected Dominating Set

    S

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

    (S T) is a DS for H

    SG

    S is a CDS for G

    TH

    T is a CDS for H

    S T

  • Dynamic Connected Dominating Set

    S

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

    (S T) is a DS for H

    SG

    S is a CDS for G

    TH

    T is a CDS for H

    S T (S T) is a CDS for H

  • Dynamic Connected Dominating Set

    S

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

    (S T) is a DS for H

    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

  • Dynamic Connected Dominating Set

  • Dynamic Connected Dominating Set

    D

  • Dynamic Connected Dominating Set

    D D

  • Dynamic Connected Dominating Set

    D D

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

  • Dynamic Connected Dominating Set

  • Dynamic Connected Dominating Setcomponents of D

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

  • Dynamic Connected Dominating Set

    undominated vertices

    D1 D2 D3 . . . Dk2 components of D

  • Dynamic Connected Dominating Set

    v1 v2 v3 . . . vk1 undominated vertices

    D1 D2 D3 . . . Dk2 components of D

  • Dynamic Connected Dominating Set

    v1 v2 v3 . . . vk1 undominated vertices

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

    D1 D2 D3 . . . Dk2 components of D

  • Dynamic Connected Dominating Set

    v1 v2 v3 . . . vk1 undominated vertices

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

    D1 D2 D3 . . . Dk2 components of D

    D1 D2 D3 . . . Dk2

  • Dynamic Connected Dominating Set

    v1 v2 v3 . . . vk1 undominated vertices

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

    D1 D2 D3 . . . Dk2 components of D

    D1 D2 D3 . . . Dk2

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

  • Dynamic Connected Dominating Set

    v1 v2 v3 . . . vk1 undominated vertices

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

    D1 D2 D3 . . . Dk2 components of D

    D1 D2 D3 . . . Dk2

    O*(2k) algorithm

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

  • Set Cover Conjecture

  • Set Cover ConjectureSet Cover

  • Set Cover ConjectureSet Cover

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

  • Set Cover ConjectureSet Cover

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

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

  • Set Cover ConjectureSet Cover

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

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

    O*(2n) algorithm

  • Set Cover ConjectureSet Cover

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

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

    O*(2n) algorithm No faster algorithm known

  • Set Cover ConjectureSet Cover

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

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

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

  • Set Cover ConjectureSet Cover

    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

  • Set Cover ConjectureSet Cover

    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

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

    uiU

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

    uiU

    Fj

    F

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

    uiU

    Fj

    F

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

    x

    uiU

    Fj

    F

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

    x

    uiU

    Fj

    F

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

    x

    uiU

    Fj

    F

    G

    CDS

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

    x

    uiU

    Fj

    F

    uiFj

    x

    H

    G

    CDS

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

    x

    uiU

    Fj

    F

    uiFj

    x

    H

    k=nr=l

    G

    CDS

  • Tightness Under Set Cover Conjecture

    Set Cover (l)

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

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

    x

    uiU

    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

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    Theoretical Computer Science, 494:8698, 2013.

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

    of Algorithms, 41(2):280301, 2001. J. Chen, I. A. Kanj, and G. Xia. Improved upper bounds for vertex cover. Theoretical Computer Science,

    411(40-42):37363756, 2010.

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

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    applications to exact algorithms. SIAM Journal on Discrete Mathematics,23(1):407427, 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):556560, 2014.

    S. Thomass. A quadratic kernel for feedback vertex set, chapter 13, pages 115119. 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 245256. 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):131146, 2012.

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