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Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds

Authors Cornelius Brand , Martin Koutecký , Alexandra Lassota , Sebastian Ordyniak

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ACM Subject Classification
  • Theory of computation → Convex optimization
Keywords
  • Mixed-Integer Programming
  • Separable Convex Optimization
  • Parameterized Algorithms
  • Parameterized Complexity

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Abstract

We provide several novel algorithms and lower bounds in central settings of mixed-integer (non-)linear optimization, shedding new light on classic results in the field. This includes an improvement on record running time bounds obtained from a slight extension of Lenstra’s 1983 algorithm [Math. Oper. Res. '83] to optimizing under few constraints with small coefficients. This is important for ubiquitous tasks like knapsack-, subset sum- or scheduling problems [Eisenbrand and Weismantel, SODA'18, Jansen and Rohwedder, ITCS'19].
Further, we extend our algorithm to an intermediate linear optimization problem when the matrix has many rows that exhibit 2-stage stochastic structure, which adds to a prominent line of recent results on this and similarly restricted cases [Jansen et al. ICALP'19, Cslovjecsek et al. SODA'21, Brand et al. AAAI'21, Klein, Reuter SODA'22, Cslovjecsek et al. SODA'24]. We also show that the generalization of two fundamental classes of structured constraints from these works (n-fold and 2-stage stochastic programs) to separable-convex mixed-integer optimization are harder than their mixed-integer, linear counterparts. This counters a widespread belief popularized initially by an influential paper of Hochbaum and Shanthikumar, namely that "convex separable optimization is not much harder than linear optimization" [J. ACM '90].
To obtain our algorithms, we employ the mixed Graver basis introduced by Hemmecke [Math. Prog. '03], and our work is the first to give bounds on the norm of its elements. Importantly, we use these bounds differently from how purely-integer Graver bounds are exploited in related approaches, and prove that, surprisingly, this cannot be avoided.

Cite As Get BibTex

Cornelius Brand, Martin Koutecký, Alexandra Lassota, and Sebastian Ordyniak. Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.32

Author Details

Cornelius Brand
  • Chair of Algorithms and Complexity Theory, Faculty for Informatics and Computer Science, University of Regensburg, Germany
Martin Koutecký
  • Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Alexandra Lassota
  • Eindhoven University of Technology, The Netherlands
Sebastian Ordyniak
  • University of Leeds, UK

Funding

  • Koutecký, Martin: Partially supported by Charles Univ. project UNCE 24/SCI/008 and by the project 22-22997S of GA ČR.
  • Ordyniak, Sebastian: Supported by the Engineering and Physical Sciences Research Council (EPSRC, project EP/V00252X/1).

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