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Improved Bounds for the Symmetric Rendezvous Value on the Line

School of Engineering and Management, Nanjing University, Nanjing 210093, China
Faculty of Business Administration, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3
Department of Management Sciences, Faculty of Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Faculty of Business Administration, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3
qmhan{at}nju.edu.cn
ddu{at}unb.ca
jvera{at}utwaterloo.edu
lzuluaga{at}unb.ca

A notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line, when the initial distance between the two players is two. We show that the symmetric rendezvous value is within the interval (4.1520, 4.2574), which considerably improves the previous best-known result (3.9546, 4.3931). To achieve the improved bounds, we call upon results from absorbing Markov chain theory and mathematical programming theory—particularly fractional quadratic programming and semidefinite programming. Moreover, we also establish some important properties of this problem, which could be of independent interest and useful for resolving this problem completely. Finally, we conjecture that the symmetric rendezvous value is asymptotically equal to 4.25 based on our numerical calculations.

Subject classifications: search and surveillance; rendezvous search; games/group decisions; teams; analysis of algorithms; suboptimal algorithms; symmetric rendezvous search; game theory; approximation algorithm; semidefinite programming relaxation.
History: Received June 2006; revision received January 2007; accepted February 2007.