The min-knapsack problem with compactness constraints extends the classical
knapsack problem, in the case of ordered items, by introducing a restriction
ensuring that they cannot be too far apart. This problem has applications in
statistics, particularly in the detection of change-points in time series. In
this paper, we propose a semidefinite programming approach for this problem,
incorporating compactness in constraints or in objective. We study and compare
the different relaxations, and argue that our method provides high-quality
heuristics and tight bounds. In particular, the single hyperparameter of our
penalized semidefinite models naturally balances the trade-off between
compactness and accuracy of the computed solutions. Numerical experiments
illustrate, on the hardest instances, the effectiveness and versatility of our
approach compared to the existing mixed-integer programming formulation.

Questo articolo esplora i giri e le loro implicazioni.

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