Approximating Integer Solution Counting via Space Quantification for Linear Constraints

Ge, Cunjing; Ma, Feifei; Ma, Xutong; Zhang, Fan; Huang, Pei; Zhang, Jian

doi:10.24963/ijcai.2019/235

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…However, counterexamples show that approximating integer counts via volume with arbitrary precision is impossible. [23] proposed and proved a bound for approximation, i.e., the difference between the volume and the integer count. The overhead of computing the bound is negligible.…”

Section: Approximate Countingmentioning

confidence: 99%

Counting the Number of Solutions to Constraints

Zhang¹,

Ge²,

Ma³

2020

Preprint

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Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various forms, including, formulas in the propositional logic, linear inequalities over the reals or integers, Boolean combination of linear constraints. We describe some techniques and tools for solving the counting problems, as well as some applications (e.g., applications to automated reasoning, program analysis, formal verification and information security).

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Section: Approximate Countingmentioning

confidence: 99%

Counting the Number of Solutions to Constraints

Zhang¹,

Ge²,

Ma³

2020

Preprint

Self Cite

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“…In practice, it often still has difficulties when the number of variables is greater than 10 (preventing many applications). The relation between the number of lattice points inside a polytope and the volume of a polytope has been studied for approximate integer counting (Ge et al 2019). However, it is inevitable that the approximation bounds may far off from exact counts.…”

Section: Introductionmentioning

confidence: 99%

Approximate Integer Solution Counts over Linear Arithmetic Constraints

2024

AAAI

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Counting integer solutions of linear constraints has found interesting applications in various fields. It is equivalent to the problem of counting lattice points inside a polytope. However, state-of-the-art algorithms for this problem become too slow for even a modest number of variables. In this paper, we propose a new framework to approximate the lattice counts inside a polytope with a new random-walk sampling method. The counts computed by our approach has been proved approximately bounded by a (epsilon, delta)-bound. Experiments on extensive benchmarks show that our algorithm could solve polytopes with dozens of dimensions, which significantly outperforms state-of-the-art counters.

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Decomposition Strategies to Count Integer Solutions over Linear Constraints

Biere

2021

Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence

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Counting integer solutions of linear constraints has found interesting applications in various fields. It is equivalent to the problem of counting integer points inside a polytope. However, state-of-the-art algorithms for this problem become too slow for even a modest number of variables. In this paper, we propose new decomposition techniques which target both the elimination of variables as well as inequalities using structural properties of counting problems. Experiments on extensive benchmarks show that our algorithm improves the performance of state-of-the-art counting algorithms, while the overhead is usually negligible compared to the running time of integer counting.

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Approximating Integer Solution Counting via Space Quantification for Linear Constraints

Cited by 3 publications

References 6 publications

Counting the Number of Solutions to Constraints

Counting the Number of Solutions to Constraints

Approximate Integer Solution Counts over Linear Arithmetic Constraints

Decomposition Strategies to Count Integer Solutions over Linear Constraints

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