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Applying topological data analysis to local search problems

  • * Corresponding author: John Gunnar Carlsson

    * Corresponding author: John Gunnar Carlsson 
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  • We present an application of topological data analysis (TDA) to discrete optimization problems, which we show can improve the performance of the 2-opt local search method for the traveling salesman problem by simply applying standard Vietoris-Rips construction to a data set of trials. We then construct a simplicial complex which is specialized for this sort of simulated data set, determined by a stochastic matrix with a steady state vector $ (P,\pi) $. When $ P $ is induced from a random walk on a finite metric space, this complex exhibits similarities with standard constructions such as Vietoris-Rips on the data set, but is not sensitive to outliers, as sparsity is a natural feature of the construction. We interpret the persistent homology groups in several examples coming from random walks and discrete optimization, and illustrate how higher dimensional Betti numbers can be used to classify connected components, i.e. zero dimensional features in higher dimensions.

    Mathematics Subject Classification: 55N31, 60J10, 90C27.

    Citation:

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  • Figure 1.  A 2-opt exchange on a tour of $ 8 $ points. The edges $ (3,7) $ and $ (4,5) $ in (1a) are replaced with $ (3,4) $ and $ (7,5) $ in (1b)

    Figure 2.  The ratio of objective values obtained with and without TDA on the 41 TSPLIB benchmark instances; the horizontal position corresponds to alphabetical order of the name of the instance. Ratios less than $ 1 $ are marked with a circle, indicating that TDA-assisted search is superior. Ratios greater than 1 are marked with a diamond, indicating that TDA-assisted search is inferior. Ratios equal to 1 are marked with a square, indicating that the two methods result in the same objective value. The average ratio over all 42 instances is marked with a dashed line

    Figure 3.  Using higher order homology to obtain new local minimizers

    Figure 4.  The "best-so-far" objective values reported for the $ k $-means example from Figure 3, with $ k=3 $ and $ n=1000 $; local search does not improve between iterations 39 and 250, at which point we begin seeding using convex combinations of the representative $ \beta_{1} $ cycle as described. Using seeding of this kind rapidly leads to several new local minimizers between iterations 250 and 300

    Figure 5.  The barcode diagrams for Construction 1 applied to the "jeu de taquin" Markov chain. The 100 longest bars are shown. Despite having many shorter ones, one can see three lasting bars in the $ \beta_1 $ diagram

    Figure 6.  barcode diagrams built from the data set in Figure 6a. Figure 6b shows the results of applying Vietoris-Rips, which shows some $ \beta_1 $ features reflecting the two voids in the data set. Figures 6c and 6d are built using a random walk, and show more distinguishable features, because outliers produce high persistence values

    Figure 7.  A Beale function $ F(x,y) $ on a region shown in 7a, then modeled on a $ 35\times 35 $ grid. There are 4 types of path $ \varphi(x) $ with low values of $ F(x,\varphi(x)) $, upper left/lower left through upper right/lower right. The values of $ f(p) $ on $ 0 $-simplices $ p\in X $ for the complex from Section 3.3 are shown with $ m\in \{1,...,k\} $, for various values of $ k $. Notice that the diagram for $ k=8 $ is already quite similar to the $ k=\infty $ case, which is the default in Construction 1. The major bars for the relative first Betti numbers correspond to the four paths in 7f. Notice there is a second bar in 7e corresponding to two dense regions

    Table 1.  The sequence of vertices in a representative cycle for the jeu de taquin game, generated by Javaplex [1]

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