Irregular polyomino tiling via integer programming with application in phased array antenna design

Abstract

A polyomino is a generalization of the domino and is created by connecting a fixed number of unit squares along edges. Tiling a region with a given set of polyominoes is a hard combinatorial optimization problem. This paper is motivated by a recent application of irregular polyomino tilings in the design of phased array antennas. Specifically, we formulate the irregular polyomino tiling problem as a nonlinear exact set covering model, where irregularity is incorporated into the objective function using the information-theoretic entropy concept. An exact solution method based on a branch-and-price framework along with novel branching and lower-bounding schemes is proposed. The developed method is shown to be effective for small- and medium-size instances of the problem. For large-size instances, efficient heuristics and approximation algorithms are provided. Encouraging computational results including phased array antenna simulations are reported.

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Acknowledgments

The first two authors were supported by AFOSR Grant FA9550-08-1-0268. The third author was supported by AFOSR Grant FA9550-12-1-0105. The authors thank Dr. Osman Y. Özaltın and Gabriel L. Zenarosa for their valuable comments on the earlier draft of the paper and Dr. Scott Santarelli for his assistance with antenna simulation software. The first two authors also acknowledge Dr. Arje Nachman and Dr. Donald W. Hearn from AFOSR for introducing them to the considered application. Finally, the authors thank the reviewers and the AE for their helpful comments.

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Correspondence to Oleg A. Prokopyev.

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Robert J. Mailloux: USAF Senior Scientist, Retired.

Appendix: Pseudocodes of the algorithms

Appendix: Pseudocodes of the algorithms

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Proposition 9

Algorithm 1 returns a valid lower bound.

Proof

Proof. To show its correctness, i.e., that it returns a feasible solution, assume \(j'\) is the last subset considered by Algorithm 1. For \(j'\), \(\hat{y}_i = \frac{r_{j}}{\ell } \le \frac{r_{j'}}{\ell }\) for every \(i\in I^-\) as value of \(\hat{y}_i\) must have been set by a previous subset j and \(r_{j} \le r_{j'}\). Therefore, \(\sum _{i\in \hat{f}_{j'}}\hat{y}_i \le r_{j'}\). This is true for all \(0\le j\le j'\). If \(j'<|\bar{N}|-1\), then \(L=m_1\) and \(\sum _{i\in \hat{f}_{j}}\hat{y}_i \le r_{j}\) for all \( j > j'\). \(\square \)

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Karademir, S., Prokopyev, O.A. & Mailloux, R.J. Irregular polyomino tiling via integer programming with application in phased array antenna design. J Glob Optim 65, 137–173 (2016). https://doi.org/10.1007/s10898-015-0354-8

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Keywords

  • Polyomino
  • Entropy
  • Set partitioning
  • Phased array antenna