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On residual approximation in solution extension problems

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Abstract

The solution extension variant of a problem consists in, being given an instance and a partial solution, finding the best solution comprising the given partial solution. Many problems have been studied with a similar approach. For instance the Pre-Coloring Extension problem, the clustered variant of the Travelling Salesman problem, or the General Routing Problem are in a way typical examples of solution extension variant problems. Motivated by practical applications of such variants, this work aims to explore different aspects around extension on classical optimization problems. We define residue-approximations as algorithms whose performance ratio on the non-prescribed part can be bounded, and corresponding complexity classes. Using residue-approximation, we classify problems according to their residue-approximability, exhibit distinct behaviors and give several examples and first interesting results.

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Notes

  1. Steps 1, 2, 3 are equivalent to contract the connected component of the forest \({\mathcal {F}}\) by taking the minimum of all edges with a common end-point in \(\mathcal {{\bar{F}}}\).

  2. The Exponential-time hypothesis (Impagliazzo and Paturi 2001; Impagliazzo et al. 2001) states that there is a constant \(c>1\) such that no algorithm solves \(\ell \)-variable 3-Satisfiability in \(O(c^\ell )\) time.

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Authors and Affiliations

  1. LIRMM - CNRS UMR 5506, Montpellier, France

    Mathias Weller, Annie Chateau, Rodolphe Giroudeau, Jean-Claude König & Valentin Pollet

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  1. Mathias Weller
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  2. Annie Chateau
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  3. Rodolphe Giroudeau
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  4. Jean-Claude König
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  5. Valentin Pollet
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Correspondence to Annie Chateau.

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Weller, M., Chateau, A., Giroudeau, R. et al. On residual approximation in solution extension problems. J Comb Optim 36, 1195–1220 (2018). https://doi.org/10.1007/s10878-017-0202-5

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