The Buss Reduction for the k-Weighted Vertex Cover Problem

The presentation slides of the paper "Hong Xu, Xin-Zeng Wu, Cheng Cheng, Sven Koenig, and T. K. Satish Kumar. The Buss reduction for the k-weighted vertex cover problem. In Proceedings of the 15th International Symposium on Artificial Intelligence and Mathematics (ISAIM). 2018."

More details: http://www.hong.me/papers/xu2018.html
Link to published paper: http://isaim2018.cs.virginia.edu/papers/ISAIM2018_Xu_etal.pdf.

Transcript

1. The Buss Reduction for
   the k-Weighted Vertex Cover Problem
   Hong Xu Xin-Zeng Wu Cheng Cheng Sven Koenig T. K. Satish Kumar
   {hongx, xinzengw, chen260, skoenig}@usc.edu tkskwork@gmail.com
   January 5, 2018
   University of Southern California, Los Angeles, California 90089, the United States of America
   The 15th International Symposium on Artificial Intelligence and Mathematics (ISAIM 2018)
   Fort Lauderdale, Florida, the United States of America
   

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2. Summary
   • For an NP-hard problem, it is desirable to have an algorithm that
   reduces problem sizes in polynomial time (but does not necessarily
   solve the problem). This is called a kernelization method.
   • The Buss reduction has been known as a kernelization method for the
   k-vertex cover (k-VC) problem.
   • We explicitly generalize it to the k-weighted vertex cover (k-WVC)
   problem and empirically study its properties.
   Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 1 / 21
   

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3. Agenda
   Motivation
   The Buss Reduction for the k-Weighted Vertex Cover Problem
   Experimental Results
   Analysis
   Conclusion
   Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 2 / 21
   

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4. Agenda
   Motivation
   The Buss Reduction for the k-Weighted Vertex Cover Problem
   Experimental Results
   Analysis
   Conclusion
   

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5. Motivation: the k-Weighted Vertex Cover (k-WVC) Problem
   The k-WVC problem: Find a vertex cover with a weight no more than k on a
   vertex-weighted undirected graph.
   Applications:
   • Combinatorial auctions (Sandholm 2002)
   • Kidney exchange (McCreesh et al. 2017)
   • Error correcting code (McCreesh et al. 2017)
   • Solving and understanding weighted constraint satisfaction
   problems (Kumar 2008a, 2016, 2008b)
   Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 3 / 21
   

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6. Motivation: Kernelization and the Buss Reduction
   • The k-WVC problem is known to be NP-hard.
   • To solve such a problem, an algorithm that reduces the size of the
   problem in polynomial time is desirable.
   • A kernelization method is one such algorithm.
   • The Buss reduction is one kernelization method for the k-VC problem.
   • Can we generalize the Buss reduction to the k-WVC problem?
   Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 4 / 21
   

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7. Agenda
   Motivation
   The Buss Reduction for the k-Weighted Vertex Cover Problem
   Experimental Results
   Analysis
   Conclusion
   

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8. The k-WVC Problem
   Given a vertex-weighted undirected graph G = V, E, w ,
   • A vertex cover is a set S ⊆ V such that every edge in G has at least one
   endpoint vertex in S.
   • The k-WVC problem asks for a vertex cover S with a weight no more
   than k on G, i.e.,
   v∈S
   w(v) ≤ k.
   • The k-VC problem is equivalent to the k-WVC problem with all weights
   equal to 1.
   Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 5 / 21
   

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9. The k-WVC Problem: Example
   Example: (k = 4)
   1
   2
   2 0
   1
   1
   (a)
   1
   2
   2 0
   1
   1
   (b)
   1
   2
   2 0
   1
   1
   (c)
   1
   2
   2 0
   1
   1
   (d)
   Red vertices are those vertices in the vertex cover.
   Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 6 / 21
   

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10. The Buss Reduction for the k-VC problem
    Intuition: If a vertex has a degree larger than k, it has to be
    in the vertex cover. Otherwise, all its neighbors have to be
    in the vertex cover and result in a vertex cover larger than k.
    k = 3
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 7 / 21
    

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11. The Buss Reduction for the k-VC problem
    The Buss reduction for the k-VC problem on G (Buss et al. 1993):
    • Find a vertex v with a degree larger than k and add it to the vertex
    cover.
    • Remove vertex v from G, and the remaining problem is the (k − 1)-VC
    problem on the resulting graph.
    • Repeat the steps above until k < 0 or no vertex can be added to the
    vertex cover.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 8 / 21
    

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12. The Buss Reduction for the k-WVC problem
    Intuition: If a vertex whose neighbors have a total weight
    larger than k, it has to be in the vertex cover. Otherwise, all
    its neighbors have to be in the vertex cover and result in a
    vertex cover with weight larger than k.
    4 1
    1
    1
    1
    k = 5
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 9 / 21
    

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13. The Buss Reduction for the k-WVC problem
    The Buss reduction for the k-WVC problem on G:
    • Find a vertex v whose neighbors have a total weight larger than k and
    add it to the vertex cover.
    • Remove vertex v from G, and the remaining problem is the
    (k − w(v))-WVC problem on the resulting graph.
    • Repeat the steps above until k < 0 or no vertex can be added to the
    vertex cover.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 10 / 21
    

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14. Agenda
    Motivation
    The Buss Reduction for the k-Weighted Vertex Cover Problem
    Experimental Results
    Analysis
    Conclusion
    

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15. Benchmark Instances
    We generated 18 benchmark instance sets with 1,000 benchmark instances
    each, by using one from each of the following properties:
    • Random graph model: Erdős-Rényi (ER) and Barabási-Albert (BA)
    • ER: Connectivity c = 8
    • BA: m = m0 = 2
    • Probabilistic distribution of vertex weights: constant, exponential with
    λ = 1 and λ = 100
    • “Constant distribution” can be somehow viewed as the exponential
    distribution with λ → +∞ since both of them have zero variance.
    • Note that the exponential distribution with λ = 100 has a very low
    variance.
    • Number of vertices: 1,000, 500, and 100
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 11 / 21
    

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16. No Benchmark Instance Solved
    • Experiments showed that no benchmark instance was solved directly
    using the Buss reduction.
    • The Buss reduction typically does not reduce ER or BA graphs to
    empty kernels if a k-WVC exists.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 12 / 21
    

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17. 0.000 0.005 0.010 0.015 0.020 0.025 0.030
    k/W
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    Fraction
    constant
    exponential-1
    exponential-100
    (a) The ER Instances
    0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
    k/W
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    Fraction
    constant
    exponential-1
    exponential-100
    (b) The BA Instances
    The fraction of instances η for which the Buss reduction outputs “NO” (no vertex
    cover with weight not larger than k exists) versus k/W for different weight
    distributions, where W is the total weight of the vertices in the graph. Only
    instances with 1,000 vertices are used here. The blue, orange, and green curves
    represent graphs that have constant, exponential-1, and exponential-100 weights,
    respectively.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 13 / 21
    

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18. 0.000 0.005 0.010 0.015 0.020 0.025 0.030
    k/W
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    Fraction
    constant
    exponential-1
    exponential-100
    (a) The ER Instances
    0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
    k/W
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    Fraction
    constant
    exponential-1
    exponential-100
    (b) The BA Instances
    Observations:
    • By changing constant weights to weights sampled by exponential
    distributions, the critical range (where the phase transition takes
    place) shifts to larger k’s for the ER model and broadens for the BA
    model.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 14 / 21
    

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19. 0.00 0.05 0.10 0.15 0.20 0.25 0.30
    k/W
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    Fraction
    100
    500
    1000
    (a) The ER Instances
    0.00 0.05 0.10 0.15 0.20 0.25 0.30
    k/W
    0.0
    0.2
    0.4
    0.6
    0.8
    1.0
    Fraction
    100
    500
    1000
    (b) The BA Instances
    The fraction of instances η for which the Buss reduction outputs “NO” (no vertex
    cover with weight not larger than k exists) versus k/W for graphs of different
    sizes. Only instances with exponential-1 weights are used here.
    Observation: As the graph size increases, the critical range narrows and
    shifts to smaller k’s.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 15 / 21
    

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20. Average reduction rate measures how much the problem size has been
    reduced. (average number of vertices removed divided by number of
    vertices in the input graph)
    10 1 100
    k/W
    0.0000
    0.0002
    0.0004
    0.0006
    0.0008
    0.0010
    Average Reduction Rate
    constant
    exponential-1
    exponential-100
    (a) The ER Instances
    10 1 100
    k/W
    0.00
    0.02
    0.04
    0.06
    0.08
    0.10
    Average Reduction Rate
    constant
    exponential-1
    exponential-100
    (b) The BA Instances
    The average reduction rate as a function of k/W for different weight distributions.
    Only instances with 1,000 vertices are used here.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 16 / 21
    

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21. Average component reduction rate measures how much the hardness of
    the problem has been reduced. (one minus the ratio of the number of
    vertices in the largest connected component of the kernel to the number
    of vertices in the input graph)
    10 1 100
    k/W
    0.0000
    0.0005
    0.0010
    0.0015
    0.0020
    0.0025
    0.0030
    Average Component Reduction Rate
    constant
    exponential-1
    exponential-100
    (a) The ER Instances
    10 1 100
    k/W
    0.00
    0.02
    0.04
    0.06
    0.08
    0.10
    0.12
    Average Component Reduction Rate
    constant
    exponential-1
    exponential-100
    (b) The BA Instances
    Shows the average component reduction rate as a function of k/W for different
    weight distributions. Only instances with 1,000 vertices are used here.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 17 / 21
    

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22. Agenda
    Motivation
    The Buss Reduction for the k-Weighted Vertex Cover Problem
    Experimental Results
    Analysis
    Conclusion
    

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23. Analytical Approximation of Average Reduction Rate
    Consider the k-VC problem on a graph with N vertices.
    • The Buss reduction starts with k = k0
    .
    • The Buss reduction stops when k = k1
    .
    If we assume that the number of neighbors of vertices does not change
    from iteration to iteration, then we have
    1 − Fd
    (k1
    ) =
    k0
    − k1
    N
    ,
    where Fd
    (k1
    ) is the fraction of vertices that have degrees less or equal to k1
    .
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 18 / 21
    

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24. Analytical Approximation of Average Reduction Rate
    Similar argument for the k-WVC problem on a graph with N vertices, we
    have
    1 − FΩ
    (k1
    ) =
    k0
    − k1
    N w
    ,
    where
    • w is the average vertex weight, and
    • FΩ
    (·) is the CDF of Ω(v) =
    u∈∂v
    w(u).
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 19 / 21
    

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25. 10-2 10-1 100
    k
    0
    /N
    0
    2
    4
    6
    8
    10
    Reduction Rate
    10-4
    ER (Poisson)
    BA (Power-law)
    The plots of the reduction rates as a function of k0/N obtained by solving
    1 − Fd
    (k1) = k0−k1
    N
    , for N = 1, 000 numerically for ER graphs with parameter c = 8
    and BA graphs with parameters m0 = m = 2.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 20 / 21
    

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26. Agenda
    Motivation
    The Buss Reduction for the k-Weighted Vertex Cover Problem
    Experimental Results
    Analysis
    Conclusion
    

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27. Conclusion
    • We generalized the Buss reduction to the k-WVC problem.
    • We empirically studied it on ER and BA random graphs:
    • The Buss reduction typically does not reduce ER or BA graphs to empty
    kernels if a k-WVC exists.
    • By changing constant weights to weights sampled by exponential
    distributions, the critical range shifts to larger k’s for the ER model and
    broadens for the BA model.
    • As the graph size increases, the critical range narrows and shifts to
    smaller k’s.
    • The reduction rate and the component reduction rate drop to near zero
    quickly for ER instances and more gradually for BA instances.
    Xu et al. (University of Southern California) The Buss Reduction forthe k-Weighted Vertex Cover Problem 21 / 21
    

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28. References I
    J. F. Buss and J. Goldsmith. “Nondeterminism within P∗”. In: SIAM Journal on Computing 22.3 (1993),
    pp. 560–572.
    T. K. S. Kumar. “A Framework for Hybrid Tractability Results in Boolean Weighted Constraint Satisfaction
    Problems”. In: the International Conference on Principles and Practice of Constraint Programming. 2008,
    pp. 282–297.
    T. K. S. Kumar. “Kernelization, Generation of Bounds, and the Scope of Incremental Computation for
    Weighted Constraint Satisfaction Problems”. In: the International Symposium on Artificial Intelligence
    and Mathematics. 2016.
    T. K. S. Kumar. “Lifting Techniques for Weighted Constraint Satisfaction Problems”. In: the International
    Symposium on Artificial Intelligence and Mathematics. 2008.
    C. McCreesh, P. Prosser, K. Simpson, and J. Trimble. “On Maximum Weight Clique Algorithms, and How
    They Are Evaluated”. In: the International Conference on Principles and Practice of Constraint
    Programming. 2017, pp. 206–225. doi: 10.1007/978-3-319-66158-2_14.
    

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29. References II
    T. Sandholm. “Algorithm for Optimal Winner Determination in Combinatorial Auctions”. In: Artificial
    Intelligence 135.1 (2002), pp. 1–54. doi: 10.1016/S0004-3702(01)00159-X.
    

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