Computing convex hulls and counting integer points with polymake

Abstract

The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes. Using the polymake system we explore various algorithms and implementations. Our experience in this area is summarized in ten “rules of thumb”.

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Acknowledgments

We thank Thomas Opfer for contributing his implementation of the dual simplex method, originally written for his Master’s Thesis [53], to the polymake project, and this includes the code maintenance until today. Moreover, we are very grateful to the developers of cdd, lrs, normaliz and ppl as they gave us various kind of valuable feedback. The comments by David Avis and Winfried Bruns were particularly detailed. What we found most rewarding is the fact that, partially in reaction to the 2014 preprint version of this paper, the teams of lrs and normaliz published new releases of their codes. Throughout these show improvements which are sometimes very substantial, e.g., for normaliz ’ handling of non-symmetric cut polytopes. The interested reader may find it worth-while to compare the results below with that preprint version (which is still available as arXiv:1408.4653v1).

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Correspondence to Michael Joswig.

Additional information

M. Joswig, B. Lorenz and A. Paffenholz are partially supported by the DFG within the Priority Program 1489. M. Joswig is additionally supported by Einstein Foundation Berlin.

Appendices

Appendix 1: Experimental setup

Everything was calculated on identical Linux machines with the memory limit set to 4 GB (via ulimit). All tests in one section were done on the same machine. Any test exceeding this bound is marked as

figurep

in the respective tables. All timings were measured in CPU seconds, except for tests on non-symmetric cut polytopes where we used wallclock time to show the performance of the multithreaded version of normaliz. Those entries marked with a \({}^*\) ran only one iteration. The hardware for all tests was:

  • CPU: AMD Phenom(tm) II X6 1090T

  • bogomips: 6421.34

  • MemTotal: 8191520 kB

All tests were done on openSUSE 13.1 (x86_64), with Linux kernel 3.11.10-25, gcc 4.9.3 and perl 5.18.1.

All tests were run through polymake version 2.15-beta3 via the respective interfaces. This creates some overhead, for instance, due to data conversion. While bb is the only implementation which is actually part of polymake, this does not constitute a principal technical advantage over the other convex hull codes tested. The libraries cddlib 0.94h, lrslib 6.0 and libnormaliz 2.99.4 (which contains the same code as version 3.0) are shipped with polymake under the GNU General Public License (GPL). As far as ppl is concerned the polymake distribution only comes with a bare interface, i.e., without the ppl code. We used the ppl library version 1.1.

The external software packages used via a file based interfaces are 4ti2 version 1.6.6, azove version 2.0, LattE version 1.7.3 and porta version 1.4.1-20090921.

The GMP was configured to use the standard memory allocator malloc. Employing a different memory allocator, such as TCMalloc [37], can have a great impact, in particular, in a multi-threaded setting. However, the precise behavior depends on numerous factors. For instance, we observed that TCMalloc was clearly superior to malloc in an openSUSE 12.2 environment, while the difference is only marginal in our setup with openSUSE 13.1, on the same hardware.

The histogram in Fig. 6 was created with MATLAB [41].

Appendix 2: Tables with detailed computational results

Table 3 Timings (in seconds) for convex hull computations of non-symmetric cut polytopes \(\mathsf {Cut}(G_k)\), see Sect. 3.3
Table 4 Running times (in seconds) for integer hull computations of Fibonacci knapsack polytopes \(F_d(b)\), see Sect. 3.4.
Table 5 Running times (in seconds) for Voronoi diagrams of random point sets, see Sect. 3.5
Table 6 Timings (in sec.) for counting lattice points in \(F_5(b)\); Sect. 4.3
Table 7 Timings (in sec.) for counting lattice points in \(F_d(60)\); Sect. 4.3
Table 8 Running times (in second) for counting lattice points in random polytopes; Sect. 4.4
Table 9 Timings (in seconds) for counting lattice points in the matching polytope of \(K_n\), see Sect. 4.5

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Assarf, B., Gawrilow, E., Herr, K. et al. Computing convex hulls and counting integer points with polymake . Math. Prog. Comp. 9, 1–38 (2017). https://doi.org/10.1007/s12532-016-0104-z

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Keywords

  • Convex hull computation
  • Lattice point enumeration
  • Facets of integer hulls

Mathematics Subject Classification

  • 90-08
  • 52-04