, Volume 81, Issue 9, pp 3494–3518 | Cite as

The Complexity of Optimization on Grids

  • Luis Barba
  • Malte MilatzEmail author
  • Jerri Nummenpalo
  • Xiaoming Sun
  • Antonis Thomas
  • Jialin Zhang
  • Zhijie Zhang
Original Research


Unique-sink orientations of grids are models for linear optimization problems. Vertices of the grid represent possible solutions to the optimization problem; edge orientations indicate improving directions. The computational goal is to find the unique sink, representing the optimal solution. We study the query complexity of this model, where we consider two natural types of queries, vertex queries and edge queries. We describe a deterministic algorithm showing that the vertex query complexity of d-dimensional grids is \(O(n^{ \lceil d/2 \rceil })\). For the edge query complexity we obtain nearly the same bound, incurring only an \(n^{o(1)}\) overhead. Our algorithms rely on structural results with potential further applications in optimization theory.


Unique-sink orientation Optimization Linear programming 



We would like to thank Bernd Gärtner for introducing the problem at the 2015 Gremo Workshop on Open Problems (GWOP), June 1–5, 2015, Feldis (GR), Switzerland. This work was supported in part by the National Natural Science Foundation of China Grants No. 61433014, 61832003, 61761136014, 61872334, 61502449, the 973 Program of China Grant No. 2016YFB1000201.

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Authors and Affiliations

  1. 1.Department of Computer ScienceETH ZürichZurichSwitzerland
  2. 2.ZurichSwitzerland
  3. 3.CAS Key Lab of Network Data Science and Technology, Institute of Computing TechnologyChinese Academy of Sciences (CAS)BeijingChina
  4. 4.University of Chinese Academy of SciencesBeijingChina

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