Efficient algorithms for three‐dimensional axial and planar random assignment problems1. Issue 1 (11th November 2013)

• Record Type: Journal Article
• Authors: Frieze, Alan; Sorkin, Gregory B.
• Is Part Of: Random structures & algorithms. Volume 46:Issue 1(2015)
• Page Start: 160
• Page End: 196
• Publication Date: 2013-11-11
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20525
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Beautiful formulas are known for the expected cost of random two‐dimensional assignment problems, but in higher dimensions even the scaling is not known. In three dimensions and above, the problem has natural "Axial" and "Planar" versions, both of which are NP‐hard. For 3‐dimensional Axial random assignment instances of size <italic>n</italic>, the cost scales as Ω(1/ <italic>n</italic>), and a main result of the present paper is a linear‐time algorithm that, with high probability, finds a solution of cost <italic>O</italic>(<italic>n</italic>^{–1+<italic>o</italic>(1)}). For 3‐dimensional Planar assignment, the lower bound is Ω(<italic>n</italic>), and we give a new efficient matching‐based algorithm that with high probability returns a solution with cost <italic>O</italic>(<italic>n</italic> log <italic>n</italic>). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 160–196, 2015</p> </abstract>