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A Max-Cut approximation using a graph based MBO scheme

We would like to thank the EPSRC for supporting this work through the DTP grant EP/M50810X/1

Abstract / Introduction Full Text(HTML) Figure(16) / Table(15) Related Papers Cited by
  • The Max-Cut problem is a well known combinatorial optimization problem. In this paper we describe a fast approximation method. Given a graph $ G $, we want to find a cut whose size is maximal among all possible cuts. A cut is a partition of the vertex set of $ G $ into two disjoint subsets. For an unweighted graph, the size of the cut is the number of edges that have one vertex on either side of the partition; we also consider a weighted version of the problem where each edge contributes a nonnegative weight to the cut.

    We introduce the signless Ginzburg–Landau functional and prove that this functional $ \Gamma $-converges to a Max-Cut objective functional. We approximately minimize this functional using a graph based signless Merriman–Bence–Osher (MBO) scheme, which uses a signless Laplacian. We derive a Lyapunov functional for the iterations of our signless MBO scheme. We show experimentally that on some classes of graphs the resulting algorithm produces more accurate maximum cut approximations than the current state-of-the-art approximation algorithm. One of our methods of minimizing the functional results in an algorithm with a time complexity of $ \mathcal{O}(|E|) $, where $ |E| $ is the total number of edges on $ G $.

    Mathematics Subject Classification: Primary: 05C85, 35R02, 35Q56; Secondary: 49K15, 68R10.

    Citation:

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  • Figure 1.  The value $ f_{\varepsilon}^+(\mu^j) $ as a function of the iteration number $ j $ in the (MBO+) scheme on AS8Graph, using the spectral method and $ \Delta_1^+ $, with $ K = 100 $, and $ \tau = 20 $. The left hand plot shows the initial condition and all iterations of the (MBO+) scheme on AS8Graph, where as the right hand plot displays the 3rd to the final iterations of the (MBO+) scheme on AS8Graph

    Figure 2.  The value $ f_{\varepsilon}^+(\mu^j) $ as a function of the iteration number $ j $ in the (MBO+) scheme on the GNutella09 graph, using the spectral method and $ \Delta_1^+ $, with $ K = 100 $, and $ \tau = 20 $. The left hand plot shows the initial condition and all iterations of the (MBO+) scheme on GNutella09, where as the right hand plot displays all iterations of the (MBO+) scheme on GNutella09, without the initial condition

    Figure 3.  Visualisation of maximum cut approximations (best viewed in colour)

    Figure 4.  Bar chart of Max-Cut approximations on 100 realisations of $ G(1000, 0.01) $

    Figure 5.  Bar chart of Max-Cut approximations on 100 realisations of $ G(2500, 0.4) $

    Figure 6.  Bar chart of Max-Cut approximations on 100 realisations of $ G(5000, 0.001) $

    Figure 7.  Average degree distribution of 100 realisations of a random graph and the degree distribution of a scale free graph

    Figure 8.  A Max-Cut approximation on a random 4-modular graph (best viewed in colour)

    Figure 9.  Bar chart of Max-Cut approximations on 100 realisations of $ R(2500, 2, 0.009, 0.8) $

    Figure 10.  Bar chart of Max-Cut approximations on 100 realisations of $ R(4000, 20, 0.01, 0.7) $

    Figure 11.  Bar chart of Max-Cut approximations on 100 realisations of $ R(10000, 10, 0.01, 0.8) $

    Figure 12.  Cut size as function of $ K $ for three graphs (best viewed in colour)

    Figure 13.  Comparison of cut size approximation vs $ K $ on Amazon0302 graph

    Figure 14.  Cut size as function of $ \tau $ for three graphs (best viewed in colour)

    Figure 15.  Bar chart of Max-Cut approximations on 100 realisations of $ G(1000, 0.01) $ using the implicit Euler method and the explicit Euler method

    Figure 16.  Bar chart of Max-Cut approximations on 100 realisations of $ R(4000, 20, 0.01, 0.7) $

    Table 1.  Average (MBO+) and (GW) run-times for each realisation of $ G(n, p) $, time in seconds

    Graph $ \Delta_1^+ $ (S) $ \Delta_1^+ $ (E) $ \Delta_s^+ $ (S) $ \Delta_s^+ $ (E) $ \Delta_0^+ $ (S) $ \Delta_0^+ $ (E) GW
    $ G(1000, 0.01) $ 0.20 1.58 0.34 1.52 0.56 1.06 5.25
    $ G(2500, 0.4) $ 8.04 172.91 13.33 181.40 6.40 172.73 55.36
    $ G(5000, 0.001) $ 4.38 16.96 6.37 14.95 24.99 6.97 257.09
     | Show Table
    DownLoad: CSV

    Table 6.  (GW) cut approximations on graphs with a scale free structure, time in seconds

    Graph GW Best GW Avg GW Least GW Time
    AS1 22864 22346.26 20546 2324.98
    AS2 13328 13039.10 12048 594.29
    AS3 15240 14961.56 14050 826.65
    AS4 15328 15015.34 14072 832.28
    AS5 14190 13810.82 12922 721.51
    AS6 18191 17851.24 16483 1368.35
    AS7 22901 22421.80 21244 2321.34
    AS8 23170 22593.10 21110 2613.62
    GNutella09 20658 20242.02 18815 1095.04
    Wiki-Vote 73363 71510 62886 1074.98
     | Show Table
    DownLoad: CSV

    Table 2.  Properties of $ G(n, p) $ graph realisations vs scale free graphs

    Graph $ |V| $ $ |E| $ $ d_{-} $ $ d_{+} $
    $ G(1000, 0.01) $(1) 1000 4919 1 21
    $ G(1000, 0.01) $(2) 1000 4939 2 21
    $ G(2500, 0.4) $(1) 2500 1248937 910 1079
    $ G(2500, 0.4) $(2) 2500 1251182 904 1081
    $ G(5000, 0.001) $(1) 4962 12646 1 16
    $ G(5000, 0.001) $(2) 4969 12642 1 16
    Graph $|V|$ $|E|$ $d_{-}$ $d_{+}$
    AS1126942655912566
    AS276901541311713
    AS386891770911911
    AS489041765311921
    GNutella098114260131102
    Wiki-Vote711510076211065
     | Show Table
    DownLoad: CSV

    Table 3.  (MBO+) cut approximations using $ \Delta_1^+ $ on graphs with a scale free structure, time in seconds

    Graph $ \Delta_1^+ $ (S) Best $ \Delta_1^+ $ (S) Avg $ \Delta_1^+ $ (S) Least $ \Delta_1^+ $ (S) Time
    AS1 22744 22542.20 22183 $ {\bf{15.85}} $
    AS2 13249 13153.72 13054 $ {\bf{3.55}} $
    AS3 15118 15027.22 14907 4.73
    AS4 15194 15143.44 15042 $ {\bf{5.67}} $
    AS5 14080 13988.90 13928 $ {\bf{4.82}} $
    AS6 18053 17964.74 17876 10.06
    AS7 22741 22535.00 22150 17.82
    AS8 22990 22720.36 22334 17.22
    GNutella09 20280 20143.74 19983 8.16
    WikiVote 72981 72856.40 72744 2.46
    Graph $\Delta_1^+$ (E) Best $\Delta_1^+$ (E) Avg $\Delta_1^+$ (E) Least $\Delta_1^+$ (E) Time
    AS122798 22670.762226823.62
    AS213281 13199.72 131208.76
    AS315175 15095.46 150079.95
    AS415270 15202.70 1511710.88
    AS514120 14020.62 139449.50
    AS618134 18034.10 1793316.50
    AS722826 22696.42 2252525.78
    AS823070 22951.54 2255025.38
    GNutella0920437 20361.92 2029517.14
    WikiVote73159 73126.34 730869.06
     | Show Table
    DownLoad: CSV

    Table 4.  (MBO+) cut approximations using $ \Delta_s^+ $ on graphs with a scale free structure, time in seconds

    Graph $ \Delta_s^+ $ (S) Best $ \Delta_s^+ $ (S) Avg $ \Delta_s^+ $ (S) Least $ \Delta_s^+ $ (S) Time
    AS1 22809 22620.8 $ {\bf{22325}} $ 17.83
    AS2 13271 13178.86 13103 4.12
    AS3 15166 15082.1 14992 $ {\bf{4.66}} $
    AS4 15237 15166.24 15077 5.78
    AS5 14075 14011.96 13911 5.47
    AS6 18088 17968.04 17859 $ {\bf{9.14}} $
    AS7 22822 22629.66 22218 $ {\bf{15.73}} $
    AS8 23061 22884.8 22547 $ {\bf{15.46}} $
    GNutella09 20282 20186.32 20101 $ {\bf{6.82}} $
    WikiVote 73169 73003.44 72917 $ {\bf{2.25}} $
    Graph $\Delta_s^+$ (E) Best $\Delta_s^+$ (E) Avg $\Delta_s^+$ (E) Least $\Delta_s^+$ (E) Time
    AS12278922629.622226127.63
    AS21325613176.64130949.09
    AS31513915059.541496710.24
    AS41523415159.761507911.57
    AS51409614011.91393010.47
    AS61808817994.661787616.12
    AS72282322639.582223724.5
    AS823036228652244025.08
    GNutella092039720332.282017018.75
    WikiVote7299372772.26725499.00
     | Show Table
    DownLoad: CSV

    Table 5.  (MBO+) cut approximations using $ \Delta_0^+ $ on graphs with a scale free structure, time in seconds

    Graph $ \Delta_0^+ $ (S) Best $ \Delta_0^+ $ (S) Avg $ \Delta_0^+ $ (S) Least $ \Delta_0^+ $ (S) Time
    AS1 22578 22303.10 21844 297.79
    AS2 13081 12935.80 12763 62.41
    AS3 14995 14869.52 14702 90.32
    AS4 15097 14994.92 14885 88.53
    AS5 13952 13795.24 13561 70.81
    AS6 17836 17672.50 17527 149.60
    AS7 22571 22328.18 21932 294.26
    AS8 22824 22585.88 22075 287.79
    GNutella09 19079 18419.36 17951 72.03
    WikiVote 65504 60599.74 56917 46.11
     | Show Table
    DownLoad: CSV

    Table 7.  Average (MBO+) and (GW) run-times for each realisation of $ R(n, c, p, r) $, time in seconds

    Graph $ \Delta_1^+ $ (S) $ \Delta_1^+ $ (E) $ \Delta_s^+ $ (S) $ \Delta_s^+ $ (E) $ \Delta_0^+ $ (S) $ \Delta_0^+ $ (E) GW
    $ R(2500, 2, 0.009, 0.8) $ 0.80 10.43 0.79 10.26 4.36 6.13 56.30
    $ R(4000, 20, 0.01, 0.7) $ 4.05 30.46 4.49 29.52 16.26 18.19 248.25
    $ R(10000, 10, 0.01, 0.8) $ 49.98 266.10 52.85 266.40 210.94 194.52 3893.87
     | Show Table
    DownLoad: CSV

    Table 10.  (MBO+) cut approximations using $ \Delta_0^+ $ on randomly weighted graphs, time in seconds

    Graph $ \Delta_0^+ $ (S) Best $ \Delta_0^+ $ (S) Avg $ \Delta_0^+ $ (S) Least $ \Delta_0^+ $ (S) Time
    W1 3413.96 3345.32 3276.63 0.61
    W2 34784.30 34304.33 33627.16 0.51
    W3 1602346.52 1600022.33 1595791.12 $ {\bf{6.97}} $
    W4 320251.52 319940.38 319663.40 $ {\bf{6.25}} $
    W5 4793.44 4761.72 4715.51 18.66
    W6 71219.49 70427.83 69643.31 18.93
    W7 239991.72 237647.45 235617.15 19.17
    W8 134097.55 131088.97 126123.70 272.56
    W9 27528.99 26554.77 25501.34 69.63
    W10 90271.70 88031.84 83130.60 264.89
    Graph $\Delta_0^+$ (E) Best $\Delta_0^+$ (E) Avg $\Delta_0^+$ (E) Least $\Delta_0^+$ (E) Time
    W13524.243456.553406.931.03
    W235664.1835040.7134383.571.03
    W31605419.971602251.821597064.59203.27
    W4320321.63319809.73319237.08192.51
    W55017.664983.904954.637.76
    W674195.8773688.9773231.677.33
    W7251330.73249754.88248091.067.51
    W8----
    W9----
    W10----
     | Show Table
    DownLoad: CSV

    Table 11.  (GW) cut approximations on randomly weighted graphs, time in seconds

    Graph GW Best GW Avg GW Least GW Time
    W1 3585.17 3535.63 3494.26 5.74
    W2 36101.30 35698.47 35151.60 6.07
    W3 1620705.80 1618813.52 1616502.33 43.58
    W4 323573.40 323275.84 322795.83 44.09
    W5 5038.00 5000.74 4953.71 265.27
    W6 74372.75 73852.36 73293.27 241.33
    W7 251802.56 250316.08 248098.85 263.44
    W8 138159.14 135899.20 129576.95 2629.60
    W9 28705.35 28169.25 26422.54 689.16
    W10 93547.26 91571.68 87487.99 2646.94
     | Show Table
    DownLoad: CSV

    Table 8.  (MBO+) cut approximations using $ \Delta_1^+ $ on randomly weighted graphs, time in seconds

    Graph $ \Delta_1^+ $ (S) Best $ \Delta_1^+ $ (S) Avg $ \Delta_1^+ $ (S) Least $ \Delta_1^+ $ (S) Time
    W1 3612.00 3569.08 3537.10 0.47
    W2 36487.51 36082.58 35687.87 $ {\bf{0.30}} $
    W3 1622125.53 $ {\bf{1620885.77}} $ $ {\bf{1619371.25}} $ 8.09
    W4 323926.34 323639.05 $ {\bf{323321.92}} $ 8.59
    W5 5054.26 5033.54 5010.38 $ {\bf{4.00}} $
    W6 74560.24 74218.26 73776.17 $ {\bf{3.90}} $
    W7 252448.52 251045.03 249459.89 4.18
    W8 137202.14 135952.94 133480.08 16.17
    W9 28351.01 28194.96 28009.15 $ {\bf{3.99}} $
    W10 92376.49 91570.35 90172.90 17.02
    Graph $\Delta_1^+$ (E) Best $\Delta_1^+$ (E) Avg $\Delta_1^+$ (E) Least $\Delta_1^+$ (E) Time
    W1 3622.58 3580.533548.821.41
    W2 36530.25 36191.16 35928.561.67
    W31603390.761600505.431596558.94185.03
    W4320347.01319612.93318849.26195.66
    W5 5104.45 5081.95 5063.6415.31
    W6 75499.50 75175.73 74833.8015.70
    W7 255793.23 254569.97 253091.9115.71
    W8137569.32 136896.1 136094.6023.83
    W928545.45 28369.43 28141.769.24
    W1093021.06 92489.04 91626.9925.37
     | Show Table
    DownLoad: CSV

    Table 12.  Properties of our large datasets we are testing on

    Graph $ |V| $ $ |E| $ $ d_{-} $ $ d_{+} $
    Amazon0302 262111 899792 1 420
    Amazon0601 403394 2443408 1 2752
    GNutella31 62586 147892 1 95
    PA RoadNet 1088092 1541898 1 9
    Email-Enron 36692 183831 1 1383
    BerkStan-Web 685230 6649470 1 84290
    Stanford 281904 1992636 1 38625
    WWW1999 325729 1090108 1 10721
     | Show Table
    DownLoad: CSV

    Table 13.  Results of (MBO+) using $ \Delta_1^+ $ and the Euler method on large datasets, time in hours

    Graph $ \Delta_1^+ $ (E) Best $ \Delta_1^+ $ (E) Avg $ \Delta_1^+ $ (E) Min $ \Delta_1^+ $ (E) Time
    Amazon0302 618942 618512.18 618030 0.49
    Amazon0601 1580070 1576960.80 1571089 1.90
    GNutella31 116552 116213.74 115916 0.06
    PA RoadNet 1380131 1379797.90 1379416 0.64
    Email-Enron 112665 111680.24 110279 0.02
    BerkStan-Web 5335813 5319662.06 5281630 0.83
    Stanford 1585802 1580445.14 1570469 0.47
    WWW1999 813000 809329.52 806130 0.21
     | Show Table
    DownLoad: CSV

    Table 9.  (MBO+) cut approximations using $ \Delta_s^+ $ on randomly weighted graphs, time in seconds

    Graph $ \Delta_s^+ $ (S) Best $ \Delta_s^+ $ (S) Avg $ \Delta_s^+ $ (S) Least $ \Delta_s^+ $ (S) Time
    W1 3601.29 3569.23 3545.85 $ {\bf{0.33}} $
    W2 36192.09 36059.80 35867.83 0.49
    W3 $ {\bf{1622372.91}} $ 1620484 1618809.76 8.40
    W4 $ {\bf{323933.40}} $ $ {\bf{323642.4}} $ 323114.45 7.65
    W5 5068.19 5041.94 5015.16 4.50
    W6 74844.37 74505.45 73963.79 4.67
    W7 253043.96 251668.30 250600.35 $ {\bf{4.12}} $
    W8 137195.52 136360.17 134856.06 $ {\bf{15.38}} $
    W9 28389.38 28227.09 28067.66 4.12
    W10 92439.42 91952.98 90488.33 $ {\bf{15.33}} $
    Graph $\Delta_s^+$ (E) Best $\Delta_s^+$ (E) Avg $\Delta_s^+$ (E) Least $\Delta_s^+$ (E) Time
    W13614.373577.563542.191.40
    W236321.8036150.0535910.901.53
    W31604257.121600145.681597577.4187.88
    W4320691.88319596.27318900.13199.01
    W55096.555072.365041.8915.9
    W675456.8775089.7374745.1718.09
    W7255316.85253821.64252527.1315.48
    W8137282.02136475.24134333.124.51
    W928445.9428258.6428101.229.18
    W1092731.6292093.0590448.6124.36
     | Show Table
    DownLoad: CSV

    Table 14.  Cut sizes obtained by (MBO+) using the implicit Euler scheme on scale free graphs, time in seconds

    Graph $ \Delta_1^+ $ Best $ \Delta_s^+ $ Best $ \Delta_0^+ $ Best
    AS4 15276 15279 9259
    AS8 23083 23033 13725
    W9 28553.66 28485.28 17146.69
    Graph $\Delta_1^+$ Avg $\Delta_s^+$ Avg $\Delta_0^+$ Avg
    AS4 15196.5215175.529124.68
    AS8 22934.3022844.1613585.56
    W9 28360.4628294.4016847.92
    Graph $\Delta_1^+$ Least $\Delta_s^+$ Least $\Delta_0^+$ Least
    AS4 15124150568964
    AS8 225212245413477
    W9 28103.2828075.6216521.43
    Graph $\Delta_1^+$ Time $\Delta_s^+$ Time $\Delta_0^+$ Time
    AS447.8350.94 7.47
    AS8105.22114.74 11.48
    W938.6142.57 5.26
     | Show Table
    DownLoad: CSV

    Table 15.  Average (MBO+) run-times for each realisation of $ G(1000, 0.01) $ and $ R(4000, 20, 0.01, 0.7) $, time in seconds. (I) indicates the implicit Euler method and (E) indicates the explicit Euler method

    Graph $ \Delta_1^+ $ (I) $ \Delta_1^+ $ (E) $ \Delta_s^+ $ (I) $ \Delta_s^+ $ (E) $ \Delta_0^+ $ (I) $ \Delta_0^+ $ (E)
    $ G(1000, 0.01) $ 3.36 1.82 3.28 1.79 2.20 1.19
    $ R(4000, 20, 0.01, 0.7) $ 62.97 44.23 62.16 44.18 41.53 24.01
     | Show Table
    DownLoad: CSV
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