Almost all analyses of time complexity of evolutionary algorithms (EAs) have been conducted for (1+1) EAs only. Theoretical results on the average computation time of population-based EAs are few. However, the vast majority of applications of EAs use a population size that is greater than one. The use of population has been regarded as one of the key features of EAs. It is important to understand in depth what the real utility of population is in terms of the time complexity of EAs, when EAs are applied to combinatorial optimization problems. This paper compares (1 + 1) EAs and (N + N ) EAs theoretically by deriving their first hitting time on the same problems. It is shown that a population can have a drastic impact on an EA's average computation time, changing an exponential time to a polynomial time (in the input size) in some cases. It is also shown that the first hitting probability can be improved by introducing a population. However, the results presented in this paper do not imply that population-based EAs will always be better than (1 + 1) EAs for all possible problems. [9] , which make theoretical comparison between EAs and other optimization algorithms difficult. It is necessary to gain a deeper understanding of the time complexity of EAs in order to understand whether an EA is expected to scale well with the input size and when an EA is expected to provide the most benefits to a given problem. There has been some work on the analysis of time complexity of (1 + 1) [8] . Because (1 + 1) EAs do not include recombination and population-based selection, the results on (1 + 1) EAs cannot be generalised to EAs with population size greater than one. It is important to understand the impact a population may have on an EA's average computation time. Such an understanding is expected to shed some light on the real utility of population-based EAs in combinatorial optimization [1], [2] . In this paper, we compare (1 + 1) and (N + N ) EAs theoretically on two families of problems. We derive the first hitting time for (1 + 1) and (N + N ) EAs, respectively. Such results enable us to observe when the time would be polynomial or exponential in input size. The mathematical techniques used in this paper follow those in the analytical approach to the passage time of Markov chains [18] . Unlike drift analysis [8] , which estimates the first hitting time from the drift of a Markov chain, these techniques calculate the first hitting time of a Markov chain directly from the transition matrix. The advantage over the drift analysis is that an exact expression of the first hitting time can be obtained for some EAs. The disadvantage is that such exact expressions are difficult, if not impossible, to derive from transition matrices if they are too complex. The rest of this paper is organized as follows. Section II introduces some notations, definitions and theorems about the first hitting time of a Markov chain. Section III contains our main results. Given typical problems, we derive the first hitting time of (1 + 1) and (N + N ) EAs, respectively, and try to answer the following questions. 1) Will an (N + N ) EA change the time complexity of a (1 + 1) EA for a given problem, e.g., from exponential to polynomial time or vice verse? 2) How much will an (N + N ) EA change the mean first hitting probability of a (1 + 1) EA for a given problem? 3) How much will an (N + N ) EA change the mean first hitting time of a (1 + 1) EA for a given problem? Even partial answers to the above questions will deepen our understanding of population's roles in EAs. Finally, Section IV concludes the paper with a brief summary of the paper and a few remarks.
As one of the earliest rigorous analysis, He and Yao REF investigated how the expected hitting time of EAs varies as the population size increases.
From an individual to a population: An analysis of the first hitting time of population-based evolutionary algorithms
IEEE Transactions on Evolutionary Computation
Mathematics Computer Science