a polynomial-space exact algorithm for tsp in degree-5 graphs

Download A Polynomial-Space Exact Algorithm for TSP in Degree-5 Graphs

Post on 16-Apr-2017

303 views

Embed Size (px)

TRANSCRIPT

  • A Polynomial-Space Exact Algorithmfor TSP in Degree-5 Graphs

    Norhazwani Md Yunos, Aleksandar Shurbevski, Hiroshi Nagamochi

    Graduate School of InformaticsKyoto University, Japan

    The 12th International Symposium on Operations Research and Its Applicationsin engineering, technology and management (ISORA 2015)

    Luoyang, China21-24 August 2015

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 1 / 17

  • 28

    () ()

    ,http://www.amp.i.kyoto-u.ac.jp

    http://www.amp.i.kyoto-u.ac.jp/

  • Traveling Salesman Problem

    One of the most widely studied problems in combinatorial optimization.

    A famous and important NP-hard optimization problem.

    Input:An undirected edge-weighted graphG = (V,E).

    Output:The minimum cost/length of a tour inG that passes all vertices of V exactlyonce; or

    A message for the infeasibility of G.

    5 2

    3

    6

    2

    1

    4 3

    2 2

    G

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 2 / 17

  • Traveling Salesman Problem

    One of the most widely studied problems in combinatorial optimization.

    A famous and important NP-hard optimization problem.

    Input:An undirected edge-weighted graphG = (V,E).

    Output:The minimum cost/length of a tour inG that passes all vertices of V exactlyonce; or

    A message for the infeasibility of G.

    5 2

    3

    6

    2

    1

    4 3

    2 2

    G

    11

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 2 / 17

  • Traveling Salesman Problem

    One of the most widely studied problems in combinatorial optimization.

    A famous and important NP-hard optimization problem.

    Input:An undirected edge-weighted graphG = (V,E).

    Output:The minimum cost/length of a tour inG that passes all vertices of V exactlyonce; or

    A message for the infeasibility of G.

    5 2

    3

    6

    2

    1

    4 3

    2 2

    G

    2

    1

    21

    1

    infeasible

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 2 / 17

  • TSP in Degree-k Graphs

    Input:

    An undirected edge-weighted degree-k graph G = (V,E).

    Degree-k graphs = graphs in which vertices have maximum degree at most k.

    Output:The minimum cost of a tour in G that passes all vertices of V exactlyonce; or

    A message for the infeasibility of G.

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 3 / 17

  • Previous Result

    Graphs Time Space Method Authors (Year)

    General 2n 2nDynamic

    ProgrammingBellman (1960)

    General 4nnlog n Poly.Divide andConquer

    Gurevich & Shelah(1987)

    Degree-3 1.2312n Poly.BranchingAlgorithm

    Xiao & Nagamochi(2013)

    Degree-4 1.692n Poly.BranchingAlgorithm

    Xiao & Nagamochi(2015)

    Degree-5 2.4531n Poly.BranchingAlgorithm

    This presentation(2015)

    Degree-k,k 6 open Poly.

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 4 / 17

  • Forced TSP

    Input:

    An undirected edge-weighted graph G = (V,E),

    Set of forced edges F E.

    Output:

    The minimum cost of a tour in (G,F) that passes all vertices of V exactlyonce, and all forced edges of F; or

    A message for the infeasibility of (G,F).

    Design a polynomial-space branching algorithmReduction procedure.

    Branching operations.

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 5 / 17

  • A Variety Type of Vertices

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 6 / 17

  • Type of Vertices and their Weight, w

    Forced

    vertices:

    f3-vertex f4-vertex f5-vertexw3 = 0.1567 w4 = 0.3134 w5 = 0.4701

    Unforced

    vertices:

    u3-vertex u4-vertex u5-vertexw3 = 0.2769 w4 = 0.6075 w5 = 1

    : unforced edges : forced edges

    w(v) = 0

    otherwise

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 7 / 17

  • Measure-and-Conquer

    Measure for a given instance I = (G,F) of forced TSP:

    (I) =

    vV(G)

    (w(v))

    u3=0.27690

    f3=0.1567u4=0.6075

    u3=0.2769 u4=0.6075

    f5=0.4701

    (I) = w3 + w3 + w3 + w5 + w4 + w4 + 0

    = 2.3956

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 8 / 17

  • Reduction Procedure

    Infeasibility conditions:

    i)

    ii)

    or

    Reduction Rules:

    i)

    ii)

    : unforced edges : forced edges : deleted edges

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 9 / 17

  • Branching Operation

    Ge

    force(e) delete(e)

    Instance I with size

    G

    Instance I

    with size -a

    G

    Instance I

    with size -b

    eG

    (I) = 2.3956

    Choose edge e

    and branch on

    force(e) delete(e)

    eG

    (I) = 1.9620

    eG

    (I) = 1.8053

    : unforced edges : forced edges : deleted edges

    (a, b) is a branching vector of the branching rules.This implies the linear recurrence: T() T( a) + T( b)

    T() = O(c)Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 10 / 17

  • Branching Operation

    Ge

    force(e) delete(e)

    Instance I with size

    G

    Instance I

    with size -a

    G

    Instance I

    with size -b

    eG

    (I) = 2.3956

    Choose edge e

    and branch on

    force(e) delete(e)

    eG

    (I) = 0.6268

    eG

    (I) = 0

    : unforced edges : forced edges : deleted edges

    (a, b) is a branching vector of the branching rules.This implies the linear recurrence: T() T( a) + T( b)

    T() = O(c)Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 10 / 17

  • How to Choose an Edge to Branch On

    Branching rules applied to an edge e = vt:

    v

    t

    e

    While there is a vertex of degree 5,

    For the choice of a vertex v of degree-5:

    High Priority Less Priority

    f5-vertex u5-vertex

    v v

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 11 / 17

  • How to Choose an Edge to Branch On

    For the choice of a vertex t:

    High Priority Less Priority

    v

    t1

    t2 t3

    t4

    e

    t5

    v

    t1t2 t3

    t4

    e

    f3-vertex

    v

    t

    u3-vertex

    v

    t

    f4-vertex

    v

    t

    u4-vertex

    v

    t

    f5-vertex

    v

    t

    u5-vertex

    v

    t

    There are 14 cases which make our branching rules.

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 12 / 17

  • Switching to TSP in Degree-4 Graphs

    When the graph has no degree-5 vertices, switch and use theO(1.69193n)-time algorithm for TSP in degree-4 graphsby Xiao & Nagamochi (2015).

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 13 / 17

  • Analysis (Example for c-3)

    force(vt1) delete(vt1)

    : unforced edges

    : forced edges

    : newly deleted edges : newly forced edges

    v

    t1

    t2 t3

    t4

    e

    t5 t6

    v

    t1

    t2 t3

    t4

    e

    t5 t6

    v

    t1

    t2 t3

    t4

    e

    t5 t6

    Branching vector:

    (w5 + w3 w3 + 3m2, w5 w4 + w3 + 2m3)

    wherem2 =min{w3, (w4 w3), (w4 w3), (w5 w4), (w5 w4)}.m3 =min{w3 , (w3 w3),w4 , (w4 w4),w5 , (w5 w5)}.

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 14 / 17

  • Analysis

    When there exist degree-5 vertices:Each of the 14 branching vectors has a branching factor 2.453051.

    For switching to TSP in degree-4 graphs:Measure is calculated based on the maximum ratio of vertex weights forTSP in degree-4 graphs and TSP in degree-5 graphs.The running bound for TSP in degree-4 graphs is:

    T() O(1.69193z)

    where z = max{0.21968w3 ,0.45540

    w3, 0.59804w4

    , 1w4}

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 15 / 17

  • Conclusion and Future Works

    Result:

    The TSP in an n-vertex graph G with maximum degree 5 can be solvedin O(2.4531n)-time and polynomial-space.

    Future Work:It is interesting to obtain a polynomial-space algorithm with a runningtime of O(2n) or less.

    Modified analysis technique.Re-examination of the branching rules.

    Work on TSP in higher degree graphs.

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 16 / 17

  • Thank you

    Norhazwani et al. A Polynomial-Space Algorithm for TSP5 ISORA 2015 17 / 17

Recommended

View more >