Lower Bounds on the Complexity of Mixed-Integer Programs for Stable Set and Knapsack
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2023
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| _version_ | 1866914712366612480 |
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| author | Schade, Jamico Sinha, Makrand Weltge, Stefan |
| author_facet | Schade, Jamico Sinha, Makrand Weltge, Stefan |
| contents | Standard mixed-integer programming formulations for the stable set problem on $n$-node graphs require $n$ integer variables. We prove that this is almost optimal: We give a family of $n$-node graphs for which every polynomial-size MIP formulation requires $Ω(n/\log^2 n)$ integer variables. By a polyhedral reduction we obtain an analogous result for $n$-item knapsack problems. In both cases, this improves the previously known bounds of $Ω(\sqrt{n}/\log n)$ by Cevallos, Weltge & Zenklusen (SODA 2018).
To this end, we show that there exists a family of $n$-node graphs whose stable set polytopes satisfy the following: any $(1+\varepsilon/n)$-approximate extended formulation for these polytopes, for some constant $\varepsilon > 0$, has size $2^{Ω(n/\log n)}$. Our proof extends and simplifies the information-theoretic methods due to Göös, Jain & Watson (FOCS 2016, SIAM J. Comput. 2018) who showed the same result for the case of exact extended formulations (i.e. $\varepsilon = 0$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16711 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lower Bounds on the Complexity of Mixed-Integer Programs for Stable Set and Knapsack Schade, Jamico Sinha, Makrand Weltge, Stefan Discrete Mathematics Data Structures and Algorithms Optimization and Control Standard mixed-integer programming formulations for the stable set problem on $n$-node graphs require $n$ integer variables. We prove that this is almost optimal: We give a family of $n$-node graphs for which every polynomial-size MIP formulation requires $Ω(n/\log^2 n)$ integer variables. By a polyhedral reduction we obtain an analogous result for $n$-item knapsack problems. In both cases, this improves the previously known bounds of $Ω(\sqrt{n}/\log n)$ by Cevallos, Weltge & Zenklusen (SODA 2018). To this end, we show that there exists a family of $n$-node graphs whose stable set polytopes satisfy the following: any $(1+\varepsilon/n)$-approximate extended formulation for these polytopes, for some constant $\varepsilon > 0$, has size $2^{Ω(n/\log n)}$. Our proof extends and simplifies the information-theoretic methods due to Göös, Jain & Watson (FOCS 2016, SIAM J. Comput. 2018) who showed the same result for the case of exact extended formulations (i.e. $\varepsilon = 0$). |
| title | Lower Bounds on the Complexity of Mixed-Integer Programs for Stable Set and Knapsack |
| topic | Discrete Mathematics Data Structures and Algorithms Optimization and Control |
| url | https://arxiv.org/abs/2308.16711 |