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Improved Bounds of Integer Solution Counts via Volume and Extending to Mixed-Integer Linear Constraints

Authors Cunjing Ge , Armin Biere

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Author Details

Cunjing Ge
  • National Key Laboratory for Novel Software Technology, Nanjing University, China
  • School of Artificial Intelligence, Nanjing University, China
Armin Biere
  • University of Freiburg, Germany

Funding

  • Ge, Cunjing: Cunjing Ge is supported by the National Natural Science Foundation of China (62202218), and is sponsored by CCF-Huawei Populus Grove Fund (CCF-HuaweiFM202309).

Cite AsGet BibTex

Cunjing Ge and Armin Biere. Improved Bounds of Integer Solution Counts via Volume and Extending to Mixed-Integer Linear Constraints. In 30th International Conference on Principles and Practice of Constraint Programming (CP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 307, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CP.2024.13

Abstract

Solution counting and solution space integration over linear constraints are important problems with many applications. Previous works addressed either only counting integer points in polytopes (integer counting) with integer variables or alternatively only computing the volume of polytopes (solution space integration) on variables over the reals, including approximating the integer count via a polytope’s volume. We are not aware of a non-trivial algorithm which addresses the mixed case, where linear constraints are over mixed integer and real variables. In this paper, we propose a new randomized algorithm to approximate guarantees (bounds) of integer solution counts. Then we present upper and lower bounds for solution space integration over mixed-integer linear constraints. Thus, proposed algorithms can be extended to mixed-integer cases as well. The experiments show that approximations are often very close to exact results in practice, and bounds approximated by our algorithm are often tight and useful.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automated reasoning
Keywords
  • Integer Solution Counting
  • Mixed-Integer Linear Constraints
  • #SMT(LA) Problems
  • Volume of Polytopes

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References

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