Quasi-elementary Landscapes and Superpositions of Elementary Landscapes

Abstract

Abstract. There exists local search landscapes where the evaluation function is an eigenfunction of the graph Laplacian that corresponds to the neighborhood structure of the search space. Problems that display this structure are called "Elementary Landscapes" and they have a number of special mathematical properties. The term "Quasi-elementary landscapes" is introduced to describe landscapes that are "almost" elementary; in quasi-elementary landscapes there exists some efficiently computed "correction" that captures those parts of the neighborhood structure that deviate from the normal structure found in elementary landscapes. The "shift" operator, as well as the "3-opt" operator for the Traveling Salesman Problem landscapes induce quasi-elementary landscapes. A local search neighborhood for the Maximal Clique problem is also quasi-elementary. Finally, we show that landscapes which are a superposition of 2 elementary landscapes are also quasi-elementary in structure.

How to exploit fitness landscape properties of timetabling problem: A new operator for quantum evolutionary algorithm

“…During the last two decades many researchers have studied the landscape of optimisation problems including Travelling Salesman Mathias and Whitley (1992); Stadler and Schnabl (1992); Boese (1995); Whitley and Ochoa (2011); Whitley and Chicano (2012); Tayarani-N. and Prugel-Bennett (2016), Quadratic Assignment Merz and B.Freisleben (2000); Angel and Zissimopoulos (2001); Chicano et al (2010); Tayarani-N. and Prügel-Bennett (2015), Knapsack Yoshizawa and Hashimoto (2000); Tavares et al (2008), Max-Sat Weixiong and Zhang (2004); Sutton et al (2009Sutton et al ( , 2010; Qasem and Prügel-Bennett (2010); Prugel-Bennett and Tayarani-Najaran (2011), graph drawing Lehn and Kuntz (2001), Graph-Colouring Hertz et al (1994a); Hamiez and Hao (2001); Culberson and Gent (2001); Bouziri et al (2009); Hertz et al (1994b); Bouziri et al (2011); ; Tayarani-N. and Prugel-Bennett (2014), evolutionary antenna design Alander et al (2002), flow-shop scheduling Czogalla and Fink (2011); Zhao et al (2019) and Bayesian network structure Wu et al (2011) problems.…”