Abstract

Randomized search heuristics are frequently applied to NP-hard combinatorial optimization problems. The runtime analysis of randomized search heuristics has contributed tremendously to our theoretical understanding. Recently, randomized search heuristics have been examined regarding their achievable progress within a fixed-time budget. We follow this approach and present a fixed-budget analysis for an NP-hard combinatorial optimization problem. We consider the well-known Traveling Salesperson Problem (TSP) and analyze the fitness increase that randomized search heuristics are able to achieve within a given fixed-time budget. In particular, we analyze Manhattan and Euclidean TSP instances and Randomized Local Search (RLS), (1+1) EA and (1+) EA algorithms for the TSP in a smoothed complexity setting, and derive the lower bounds of the expected fitness gain for a specified number of generations.

1  Introduction

Randomized search heuristics (RSH) such as randomized local search, evolutionary algorithms, and ant colony optimization have become very popular in recent years to solve hard combinatorial optimization problems such as the Traveling Salesperson Problem (TSP) (Beyer, 1992; Fischer and Merz, 2008; Merz and Huhse, 2008; Nürnberg and Beyer, 1997; Tao and Michalewicz, 1998).

Regarding RSH as classical randomized algorithms (Motwani and Raghavan, 1995), a lot of progress has been made in recent years on their theoretical understanding (Auger and Doerr, 2011; Jansen, 2013). Initially, most of the studies were focused on simple example functions. Gradually, the analysis on combinatorial optimization problems was also established. We refer to the textbook by Neumann and Witt (2010) and the survey article by Oliveto et al. (2007) for a comprehensive presentation on the runtime analysis of randomized search heuristics for problems from combinatorial optimization.

All these studies on analyzing the time complexity were based on a single perspective, the expected optimization time. There were slight variations on this considering the number of generations or fitness evaluations. Jansen and Zarges (2011) and Zhou et al. (2012) pointed out that there is a gap between the empirical results and the theoretical results obtained on the optimization time. Theoretical research most often yields asymptotic results on finding the global optimum while practitioners are more concerned about achieving some good result within a reasonable time budget. Furthermore, it is beneficial to know how much progress an algorithm can make given some additional time budget. Experimental studies on this topic have been carried out in the domain of algorithm engineering (Sanders and Wagner, 2013) and the term fixed-budget runtime analysis has been introduced by Jansen and Zarges (2012, 2014b).

So far, fixed-budget analysis has been conducted for very simple test functions, on which the considered randomized search heuristics, such as randomized local search and the (1+1) EA, follow a typical search trajectory (Doerr et al., 2013; Jansen and Zarges, 2014b) with high probability. This implies that on these functions the development of the best fitness over time forms an almost deterministic curve that describes the algorithm’s typical behavior. Given such a strong relation between fitness and time, and the availability of upper and lower tail bounds, it is then possible to derive tight upper and lower bounds on the expected fitness increase over any given period of time. More recent studies in this line of research include comparisons of fixed-budget results obtained by evolutionary algorithms to other RSH such as artificial immune systems (Corus et al., 2015; Jansen and Zarges, 2014c), fixed-budget analysis in dynamic optimization (Jansen and Zarges, 2014a), and introduction of more general methods of analysis (Lengler and Spooner, 2015).

The goal of obtaining tight upper and lower bounds is feasible only for functions where randomized search heuristics show a typical search trajectory and tail bounds are available to bound deviations from this trajectory. This usually does not apply to hard combinatorial problems like the Traveling Salesperson Problem, and currently no fixed-budget analysis is available for such problems. We argue that for these problems fixed-budget results can be obtained by relaxing the above goal towards only considering lower bounds on the expected fitness gain. Lower bounds can be determined based on the expected minimum improvement made in a generation. In this manner there is no requirement for obtaining tail bounds, which drastically widens the scope of problems that can be tackled with this approach. Even though lower bounds on the expected fitness gain may not be tight, they provide proven guarantees on the progress made by an RSH. The aim of this approach is to establish guarantees on the expected fitness gain for various kinds of RSH, hence providing guidance for choosing, designing, and tuning RSH such that they find high-fitness solutions in a short time.

This study provides a starting point for fixed-budget analysis of randomized search heuristics for combinatorial optimization problems. We consider randomized local search (RLS) and (1+1) Evolutionary Algorithm ((1+1) EA) on the famous Traveling Salesperson Problem (TSP). The dynamics of evolutionary algorithms on the TSP have been studied empirically (Beyer, 1992; Nürnberg and Beyer, 1997), but the authors are not aware of any rigorous theoretical works studying the fitness of evolutionary algorithms on this problem.

More specifically, here we analyze Manhattan and Euclidean TSP instances in the setting of smoothed complexity (Spielman and Teng, 2004). Smoothed analysis provides a generic framework to analyze algorithms like 2-Opt for TSP with the capability to interpolate between average and worst case analysis. This analysis was first proposed by Spielman and Teng (2004) focusing on the simplex algorithm to explain the discrepancy between its exponential worst case runtime and the fast performance in practice. The probabilistic model proposed by Englert et al. (2007) is reminiscent of the original smoothed analysis model. Later, these results were refined by Manthey and Veenstra (2013). Here, we will adhere to the initial analysis by Englert et al. (2007) as our major focus is on transferring these results to a fixed-budget analysis of RSH. Then we further generalize our results for (1+) RLS and (1+) EA as well.

We build on the analysis of Englert et al. (2014) for 2-Opt which allows us to get bounds on the expected progress of a 2-Opt operation in the smoothed setting. We consider 2-dimensional metric TSP instances. First, we obtain fixed-budget results based on the minimum improvement that RLS and (1+1) EA can make in one generation. We further improve these results, following Englert et al. (2014), by analyzing a sequence of consecutive 2-Opt steps together to identify linked pairs. Interestingly, considering only single improving steps gives a constant lower bound on the progress achievable in each of the generations whereas the analysis of a sequence of consecutive 2-Opt steps gives a larger than expected progress per step if is large. The analysis follows in a similar way for the population-based algorithms. Additionally, the analysis of population-based algorithms shows that there is a fitness gain improved by at least times of the lower bound of the fitness gain for RLS or (1+1) EA for both single step and consecutive step cases.

This article extends its conference version (Nallaperuma et al., 2014) by the investigations of the expected fitness gains for population-based algorithms. Furthermore, it extends the conference article by the analysis regarding the achieved approximation ratio given a fixed-time budget and experimental analysis on sample algorithm runs.

The organization of this article is as follows. Section 2 describes problem context and the considered algorithms. Sections 3 and 4 contain the analysis for Manhattan and Euclidean instances, respectively. Section 5 extends the results for population-based algorithms. Section 6 presents experimental results on sample algorithm runs. Finally, Section 7 concludes with highlights and future directions.

2  Preliminaries

Generally, an instance of the TSP consists of a set of vertices (depending on the context, synonymously referred to as points) and a distance function that associates a distance value with each pair . The goal is to find a tour of minimal length that visits each vertex exactly once and returns to the initial vertex, that is, to compute a permutation minimizing

A TSP instance is considered to be metric if its distance function is in a metric space. A metric space satisfies reflexivity, symmetry, and triangle inequality conditions. A pair of the set and a function is called a metric space if for all the following properties are satisfied:

  • if and only if ,

  • , and

  • .

We consider cities given by points , , in the plane. For a distance metric the distance of two points and is
formula
We study the Euclidean TSP and the Manhattan TSP which are two prominent cases of the metric TSP having the distance metric Euclidean () and Manhattan (), respectively.

2.1  RLS and a Simple Evolutionary Algorithm

We consider simple randomized search heuristics and analyze them with respect to the progress that they make within a given time budget. Randomized Local Search (RLS) (Algorithm 1) and (1+1) EA (Algorithm 2) are two randomized search heuristics that have won great popularity during recent years (Eiben and Smith, 2003). These algorithms work with a population size of one and produce one offspring in each generation. A basic mutation is given by the well-known 2-Opt operator. The usual effect of the 2-Opt is to delete the two edges and from the tour and reconnect the tour using edges and (see Figure 1).

formula
formula
Figure 1:

The effect of the 2-Opt operation on a TSP tour. Inverting a subsequence in the permutation representation corresponds to a 2-Opt move in which a pair of edges in the current tour is replaced by a pair of edges not in the tour.

Figure 1:

The effect of the 2-Opt operation on a TSP tour. Inverting a subsequence in the permutation representation corresponds to a 2-Opt move in which a pair of edges in the current tour is replaced by a pair of edges not in the tour.

RLS performs one mutation step in each generation to produce an offspring. In contrast, (1+1) EA chooses an integer variable drawn from the Poisson distribution with expectation 1 in each mutation step and performs sequentially mutation operations. In case , we speak of a singular mutation, or a singular generation. (1+1) EA can simulate a mutation step in singular generations that occurs with a probability of , . Moreover, it has a positive probability of generating a global optimum in every generation by executing the right number and sequence of mutation steps. Therefore, it is guaranteed to find a global optimum in finite time, though this time may be exponential in . Note that in our algorithms we consider the notion of fitness with regard to the minimization of the tour length. As evolutionary algorithms often maximize fitness, we use the term fitness gain to describe fitness improvements, that is, the decrease of the tour length.

We study these algorithms regarding the expected progress that they make within a given number of generations. We consider the algorithms on random instances in the setting of smoothed analysis (Englert et al., 2014; Spielman and Teng, 2004).

In this model, points are placed in a -dimensional unit hypercube for . Each point , , is chosen independently according to its own probability density function for some parameter defining the maximal density. For example, the uniform distribution has the probability density function
formula
This means that for any given point within the interval the only choice for the probability density is . Accordingly, for the unit hypercube this would be 1. In fact this value is the maximal density for the uniform distribution over the considered unit hypercube. To model worst-case instances, it is assumed that these densities are chosen by an adversary who is trying to create the most difficult random instances possible. By adjusting the parameter , one can tune the power of this adversary and hence interpolate between worst and average cases. The larger , the more concentrated the probability mass can be, the better the adversary can approximate worst-case instances by the distributions. On the other hand, when gets closer to 1, the probability mass gets less concentrated and thus the points are positioned more randomly. When , the points are positioned uniformly at random in the unit hypercube resulting in average case analysis. The two types of instances are called -perturbed Manhattan instances and -perturbed Euclidean instances (Englert et al., 2014). For our analysis we consider the points in a 2-dimensional unit hypercube .

This model also covers a smoothed analysis with a slight modification. There the adversary determines the initial distribution of points and then a slight perturbation is applied to each position, adding a Gaussian random variable with small standard deviation . There, has to be set as  (Englert et al., 2014). 1 This smoothed model is considered in the study by Manthey and Veenstra (2013).

Analyzing our algorithms in this setting, we may assume that any two different tours have different function values. Hence, both algorithms always accept strict improvements.

2.2  Minimum Improvement of a 2-Opt Step

We now summarize results by Englert et al. (2014) on the minimal improvement of a 2-Opt step. Later on, these results will be used in our analysis of the randomized search heuristics. We denote the random variable that describes the fitness gain obtained in one generation and the fitness gain in generations. We call a 2-Opt step improving if it decreases the tour length in TSP. A 2-Opt step is called singular if it is the only 2-Opt step executed in that generation. Based on the smallest improvement of any improving 2-Opt step we can find the expected improvement made in generations of (1+1) EA. We set the interval , describing a range of a 2-Opt improvement for an .2 Let us first consider a fixed 2-Opt step in which the edges and are exchanged with the edges and . This 2-Opt step decreases the length of the tour by .

Let denote the smallest possible improvement made by any improving 2-Opt step:
formula
Inspired by the original ideas of Kern (1989), Chandra et al. (1994) bounded the probability that this smallest improvement lies within the interval with a high probability for the uniform distribution.

We will make use of the following theorem by Englert et al. (2014), which gives an upper bound on the probability that an improving 2-Opt step gives an improvement of at most for the Manhattan metric. Throughout this analysis we consider the two-dimensional space.

Theorem 1 Manhattan metric (Englert et al., 2014, Theorem 7):
For the Manhattan metric and any , it holds
formula

Based on this result we get a lower bound on the probability that the smallest improvement is greater than any given  (see Theorem 5). It should be noted that here we consider the algorithms to accept non-strict improvements so that it may be possible to have . However, this is not probable with the considered smoothed setting because a necessary condition for this event is that there exist two edges with the same length, which for two fixed edges has probability 0 (if ). The union bound over all pairs of edges gives a probability upper bound of 0. Therefore, we almost surely have .

Similar to the Manhattan instances, for the Euclidean instances also, the minimum improvement per a 2-Opt step is inspired by the original ideas of Kern (1989). Based on this, the expected runtime was proved polynomial for the uniform distribution by Chandra et al. (1994). This was later extended for a more generalized setting having any probability distribution by Englert et al. (2014).

Lemma 2 Euclidean metric (Englert et al., 2014, Lemma 18):
For the Euclidean metric and any , it holds
formula

In case the considered algorithms reach a local optimum, we cannot guarantee a steady fitness gain. So instead we use the fact that local optima have a good approximation ratio. The approximation ratio for the worst local optimum with regard to 2-Opt was proven originally in Chandra et al. (1994) for the uniform distribution. This was later generalized by Englert et al. (2014) for any probability distribution with a given density function with maximal density .

Theorem 3 (Englert et al., 2014, Theorem 4):

Let . For -perturbed instances the expected approximation ratio of the worst tour that is locally optimal for 2-Opt is , where represents the number of dimensions.

3  Analysis for Manhattan Instances

In this section, we first present the analysis for RLS and (1+1) EA based on the minimum possible improvement for a single 2-Opt step. We later extend the analysis for the improvement in a sequence of consecutive 2-Opt steps. The fitness gain results hold for the considered algorithms if they have not reached a local optimum. Otherwise, we provide an upper bound on the approximation ratio. Some of our results are also stated for a variant of the (1+1) EA, called (1+1) EA*, which is defined later in Algorithm 3.

Theorem 4:

For -perturbed Manhattan instances and for RLS, (1+1) EA and (1+1) EA*, the approximation ratio for the worst local optimum is bounded above by .

Proof:

We consider Manhattan instances in a 2-dimensional unit hypercube . Then as a direct consequence from Theorem 3 the expected approximation ratio is at most .

3.1  Analysis of a Single 2-Opt Step

We start by showing a lower bound on the fitness gain achievable by RLS.

Theorem 5:

In generations, RLS achieves an expected fitness gain of unless a local optimum is reached.

Proof:
Based on Theorem 1, we get
formula
as a lower bound on the probability that the minimum improvement is at least .
Let denote the random variable describing the fitness gain obtained in a single improving 2-Opt step. This is obviously no less than the minimum possible improvement . For any fixed , the expected fitness gain per one improving 2-Opt step can be bounded from below as follows:
formula
Setting we get
formula
and accordingly
formula
The number of mutations occurring in one generation is 1 and the probability for an improving 2-Opt step is at least if the current solution is not locally optimal. Therefore, the expected value for the fitness gain in any 2-Opt step can be bounded from below by
formula
Hence, the expected value for the fitness gain in generations if no locally optimal solution has been obtained in between can be derived as .
Theorem 6:

In generations (1+1) EA achieves an expected fitness gain of unless a local optimum is reached.

Proof:

We account for the progress only in singular steps. The expectation for the fitness gain in a singular improving 2-Opt step can be derived following the above proof in Theorem 5. The probability of a singular mutation occurring in a generation is due to the Poisson distribution with expectation 1. The probability of a singular improving 2-Opt step is therefore at least . Accordingly, and following the steps in Theorem 5, the expected value for the fitness gain in generations can be derived as .

3.2  Analysis of Linked Steps for RLS

The lower bound for the expected fitness gain presented in the previous section is based on the minimum improvement in a single 2-Opt step. This bound can be improved further by considering the improvement made in a sequence of consecutive 2-Opt steps.

The analysis of consecutive steps in Englert et al. (2014) is based on the number of disjoint pairs of 2-Opt steps linked by an edge, such that in one step an edge is added and in the other it is removed. We call such a pair of 2-Opt steps a linked pair (see Figure 2). Different types of linked pairs of 2-Opt steps are considered as follows. Let and be the edges that are replaced by and in the first 2-Opt step, and and be replaced by and in the second 2-Opt step. This second step could occur any time after the first step in a sequence of 2-Opt steps.

Figure 2:

An example for a linked pair: The edges and are replaced by and in the first 2-Opt step, and and are replaced by and in the second 2-Opt step resulting in a linked pair formed by the edges and .

Figure 2:

An example for a linked pair: The edges and are replaced by and in the first 2-Opt step, and and are replaced by and in the second 2-Opt step resulting in a linked pair formed by the edges and .

Following Englert et al. (2014), we consider three different types of steps:

  1. .

  2. .

  3. .

Examples for the three types of the linked steps are shown in Figure 3. As explained in Englert et al. (2014), it is important to limit the number of occurrences of type 2 as no guarantee on the fitness gain made by type 2 steps is available. We need to show that there is a sufficient number of linked pairs of type 0 and 1 as for the linked pairs of type 0 and 1 a good progress can be guaranteed.

Figure 3:

Types of linked pairs: type 0 (top), type 1 (middle), and type 2 (bottom). The existing edges to be removed, the new edges to be added by the 2-Opt step, and the path segments connecting multiple edges are indicated in bold black lines, dotted lines, and dashed path segments, respectively. Note that for some cases (type 0 case 2, type 1 and 2 all cases), unrelated 2-Opt steps in between (represented by ) are necessary to alter the tour to enable the second 2-Opt step.

Figure 3:

Types of linked pairs: type 0 (top), type 1 (middle), and type 2 (bottom). The existing edges to be removed, the new edges to be added by the 2-Opt step, and the path segments connecting multiple edges are indicated in bold black lines, dotted lines, and dashed path segments, respectively. Note that for some cases (type 0 case 2, type 1 and 2 all cases), unrelated 2-Opt steps in between (represented by ) are necessary to alter the tour to enable the second 2-Opt step.

Due to Englert et al. (2014, Lemma 9) there are at least such pairs in a sequence of consecutive 2-Opt steps. The analysis in Englert et al. (2014) considers all 2-Opt steps in sequence and constructs (disjoint) linked pairs (of any type) in a greedy fashion. When processing some step , we search for steps and , where the two edges inserted by are being removed again, if such steps exist. If either or exist, the respective pair or is being added to a list of disjoint linked 2-Opt steps, and both and are being removed from the list to ensure disjointness of pairs. For an example of this process, we refer to Figure 6 in Englert et al. (2014). The proof of Englert et al. (2014, Lemma 9) shows that when removing all pairs of type 2 from this list, at least pairs of type 0 or 1 remain.

We further improve this bound, considering the fact that the possible number of pairs excluded is constrained by the number of edges in the final tour.

Lemma 7:

Let be the total number of linked pairs in an improving 2-Opt sequence. Then the number of linked pairs of type 0 or 1 in that sequence is at least .

Proof:
Consider a fixed pair of 2-Opt steps linked by an edge. Without loss of generality assume that in the first step the edges and are exchanged with the edges and , for distinct vertices . Also without loss of generality assume that in the second step the edges and are exchanged with the edges and . Consider the two steps and with that form a pair of type 2. Let us consider the next steps as with in which the edge is removed from the tour, if such a step exists, and with in which the edge is removed from the tour if such a step exists. Observe that neither nor can be a pair of type 2 because otherwise the improvement of one of the steps and or one of the steps and must be negative. In particular, we must have . Figure 4 illustrates this situation.
Figure 4:

Example scenario that the steps (top left), (top right), form a type 2 pair and then the steps , (bottom left), and the steps , (bottom right), form type 2 pairs. These steps lead to the original tour again (top right or top left) implying that some of the internal steps are not improving. Hence, these are not accepted.

Figure 4:

Example scenario that the steps (top left), (top right), form a type 2 pair and then the steps , (bottom left), and the steps , (bottom right), form type 2 pairs. These steps lead to the original tour again (top right or top left) implying that some of the internal steps are not improving. Hence, these are not accepted.

Thus, there cannot be a type 2 linked pair that associates with another type 2 linked pair. Therefore, each of the type 2 pairs can be associated with at most two different pairs and of type 0 or 1, unless the steps or are undefined. This happens if the edges added to the tour in are never removed. Since the final tour contains edges, at most pairs are excluded due to this. If we consider the number of type 2 pairs as , then the total number of pairs of type 0 or 1 must be at least . This implies and . The number of good (type 0 or 1) pairs is therefore .

Now we can estimate the number of good pairs in a sequence of consecutive 2-Opt steps. Following the above argument, we can improve Englert et al. (2014, Lemma 8) on the total number of (disjoint) linked pairs. The following Lemma 8 provides an improved bound for the number of good disjoint pairs in a sequence of 2-Opt steps.

Lemma 8:

In every sequence of consecutive 2-Opt steps, the number of disjoint pairs of 2-Opt steps of type 0 or 1 is at least .

Proof:

From Englert et al. (2014, Lemma 8), for each processed 2-Opt step at most two other steps are excluded from being processed if or is defined. Hence, for a sequence of steps there are at least pairs, from which we have to subtract the number of steps where neither nor is defined. This happens if they are currently in the tour. Therefore, this number is considering that the number of edges in the final tour is exactly , and can be excluded only if both edges inserted in the tour are never removed again. Therefore, the total number of disjoint pairs is at least . Combining the result of above Lemma 7 to this, we obtain the number of disjoint pairs of type 0 and 1.

Due to Lemma 8 there are at least such pairs in a sequence of consecutive 2-Opt steps. Here we consider the probability of both 2-Opt steps in a linked pair having improvement at least . We derive this probability from the following lemma from Englert et al. (2014) which bounds the probability that improvement falls within the interval of .

Lemma 9 (Englert et al., 2014, Lemma 10):

In a -perturbed instance with vertices, the probability that there exists a pair of type 0 or 1 in which both 2-Opt steps are improvements by at most is bounded by .

Based on Lemmas 8 and 9, we can bound the fitness gain for a given number of generations. Note that the theorem requires a lower bound on the number of generations because only for large enough can we guarantee that linked 2-Opt steps of type 0 or 1 do exist.

Theorem 10:

In generations, constant, RLS obtains an expected fitness gain of unless a local optimum is reached.

Proof:
Let denote the minimum possible improvement made by any pair of type 0 or 1. Using the result in Lemma 9,
formula
for a constant . Let be the random variable describing the fitness gain obtained in a pair of linked 2-Opt steps of type 0 or 1. For any , the expected fitness gain can be bounded as follows.
formula
formula
Setting we get and as a consequence
formula
The probability of making an improving 2-Opt step is at least , so long as no local optimum has been reached. Therefore, the expected number of improving 2-Opt steps made in generations is at least . Let be the number of improving steps. The improving steps are the steps accepted by RLS generating a sequence of consecutive steps. By Lemma 8 we know there are at least disjoint type 0 or 1 pairs in such a sequence of consecutive steps. Therefore,
formula
As for , we have . Then
formula
A lower bound for the expected fitness gain for generations is therefore
formula

3.3  Analysis of Linked Steps for (1+1) EA

The challenge for analyzing the (1+1) EA instead of RLS lies in the fact that the (1+1) EA can execute multiple 2-Opt steps in one generation. For RLS Englert et al. (2014) showed that certain pairs of improving 2-Opt steps yield a large fitness increase on perturbed instances, with high probability. Executing multiple 2-Opt steps in one generation complicates this argument, as some of these mutations may not be improving. As such, they might interfere with the mentioned pairs, and prohibit a large fitness increase. In the following, we show that there are sufficiently many linked 2-Opt operations that take place in generations where only one 2-Opt step is executed.

To this end, we consider a slightly modified variant of the (1+1) EA, which we call (1+1) EA* (see Algorithm 3). The (1+1) EA* will exclude generations containing multiple 2-Opt steps where an edge  is being inserted in one of these 2-Opt steps and being removed in a later 2-Opt step of the same generation.

The purpose of this modification is to enable a theoretical analysis as some of the excluded steps are difficult to handle. (1+1) EA and (1+1) EA* show identical behavior most of the time; it is easy to show that the probability of removing an edge that was inserted in the same generation is  . So (1+1) EA and (1+1) EA* are identical most of the time, apart from a small fraction of steps. However, even a small fraction can be harmful as it can take the algorithm on a different trajectory, with unforeseen consequences. Note that when the (1+1) EA* does behave differently from the (1+1) EA, it rejects a new offspring that would otherwise improve the current tour. We therefore believe that we are being pessimistic by considering the progress of the (1+1) EA* instead of that of the (1+1) EA. Experiments presented in Section 6 will further investigate the difference between the (1+1) EA and the (1+1) EA*.

The following lemma tries to estimate the number of disjoint linked pairs of 2-Opt steps within a sequence of generations. First, we estimate the probability of a 2-Opt step forming a linked pair if one of the matching 2-Opt steps exists. Therefore, inside the lemma we consider the internal 2-Opt steps to within a generation . Then we combine this result with the number of improving and singular steps in a sequence following Lemma 7 to obtain a lower bound on the expected number of good pairs having all singular 2-Opt steps.

Lemma 11:
In every sequence of generations of the (1+1) EA*, the expected number of disjoint pairs of 2-Opt steps, all of which are singular, is at least
formula
unless a local optimum is reached beforehand.
Proof:

Recall the definitions for improving and singular 2-Opt steps in Section 2.2. We adapt the proof of Lemma 8 in Englert et al. (2014) to take into account steps that are rejected by the (1+1) EA*, and the fact that the (1+1) EA* can accept non-improving 2-Opt steps in generations with multiple 2-Opt steps.

Let be a list of all 2-Opt steps executed in generations. Then we process this list to create a list of linked 2-Opt steps, all of which are singular.

The probability of the (1+1) EA* making an improving 2-Opt step is at least , so long as no local optimum has been reached. The probability of a singular generation is due to the Poisson distribution. So, the probability of having an improving and singular step is at least .

Following the proof of Lemma 9 we process steps , , and to create . The only difference is that here we consider to be both improving and singular.

We estimate the probability of a step occurring and being a singular step. Let denote the event that an accepted generation contains an improving 2-Opt step where is being removed from the tour.

Let denote the set of all edges  such that a 2-Opt move removing and  results in a strict fitness improvement. Let denote the search point of the (1+1) EA* at generation , 2-Opt step within that generation, and let us regard  as a set of edges in the tour. Note that then describes the number of improving 2-Opt moves where is being removed from the tour.

Let be the index of the first 2-Opt step in a new generation, and let be the random number of 2-Opt steps being executed in that generation. If , that is, only one 2-Opt step is executed, the conditional probability of is given by
formula
1
If operations are being executed in that generation, the probability of is bounded by the union bound:
formula
Note that might increase if edges from are being inserted into the tour. However, the additional selection criterion on the (1+1) EA* implies that, if a following step removes and one of the edges inserted previously, in the same generation, this sequence of 2-Opt steps will be rejected. Thus, only edges can cause and
formula
Note that, using the law of total probability,
formula
2
Combining (1) and (2) with Bayes’ Theorem, we get
formula
It follows that the probability of finding a linked pair or is at least , if one of the steps or exists.
Recall that the expected number of singular and improving steps in generations is at least . Following the same arguments in Lemma 7 provides an expected number of at most processed elements , each of which has a probability of for pairing with some or , if one of them exists. Hence, the resulting expected number of pairs is at least
formula

The following theorem now gives a lower bound on the expected fitness gain of the (1+1) EA*.

Theorem 12:

In generations, constant, (1+1) EA* obtains an expected fitness gain of unless a local optimum is reached.

Proof:
As in the proof of Theorem 10, we have for a constant 
formula
From Lemma 11 we know that the expected number of disjoint pairs of 2-Opt steps, both of which are singular, is at least
formula
Lemma 7 implies that among these there are at least
formula
type 0 or 1 pairs. The expected fitness gain in generations, , is therefore at least
formula

3.4  The Approximation Ratio over the Algorithm Run

We further interpret our fitness gain results in terms of the expected approximation ratio. The approximation ratio of an iterative algorithm is defined as
formula
where is the tour length of the algorithm at time , and is the length of an optimal tour. In order to bound , we state the following result as a consequence of Theorem 4 of Englert et al. (2014).
Corollary 13:
For metric TSP instances in with probability
formula
Proof:
The statement was implicitly shown in the proof of Theorem 4 of Englert et al. (2014). Their result applies for unit hypercubes of arbitrary dimensions , but here we specialise their result to . They show for an integer that the length of an optimal tour is at least (Englert et al., 2014, p. 236)
formula
in a case that occurs with probability  (Englert et al., 2014, (5.2)).

As the bound on the length of the optimal tour holds with overwhelming probability, the following lemma shows that the expected approximation ratio after steps is bounded by the expected tour length after steps, divided by the upper bound on , modulo an exponentially small additive term.

Lemma 14:
The expected approximation ratio after steps is at most
formula
Proof:
By the law of total expectation,
formula
We estimate the remaining conditional expectation by , using the following argument from Englert et al. (2014, p. 236) that applies to all tours. Consider the longest edge in and let be its length and denote its end points. We have as is the length of its longest edge. Each optimal tour also has to connect and . By the triangle inequality, any path between and is no shorter than the direct edge of length . Hence and .
The expected approximation ratio is thus at most
formula
proving the claim as .
Theorem 15:

Based on single step improvements, for Manhattan instances and for RLS, the expected approximation ratio after generations is at most if the current solution is not already locally optimal.

Proof:
Based on Lemma 14, the expected approximation ratio at iteration is
formula
where the expected fitness gain for iterations is . As the initial tour is generated uniformly at random, is bounded as follows. By linearity of expectation, equals times the expected Manhattan distance between two points chosen uniformly at random from the unit square. The latter is twice the expected absolute difference of -coordinates:
formula
Hence . Therefore, substituting the values for the starting tour and the expected fitness gain from the proof of Theorem 5 this yields
formula
to the final result of the theorem if not locally optimal.

Similarly, we can derive upper bounds on the expected approximation ratio after generations for the considered algorithms and the Manhattan metric as presented in Table 1.

Table 1:
Upper bounds on the expected approximation ratio in generations for RLS and (1+1) EA or (1+1) EA* for Manhattan instances due to single-step and consecutive-steps analysis if the current solution is not locally optimal.
Manhattan Metric
AlgorithmSingle Step (any )Consecutive Steps
RLS  ,  
(1+1) EA /(1+1) EA*  ,  
Manhattan Metric
AlgorithmSingle Step (any )Consecutive Steps
RLS  ,  
(1+1) EA /(1+1) EA*  ,  

The term is omitted for clarity. Otherwise, the approximation ratio is . The results for RLS single step, consecutive steps, and (1+1) EA single step and consecutive steps are based on Theorems  5,  10,  6, and  12, respectively. The constants and are the hidden constants in the expected fitness gain from Theorems 10 and 12, respectively.

4  Analysis for Euclidean Instances

We now turn our attention to Euclidean instances. First, we obtain the expected progress based on a single 2-Opt step for RLS and (1+1) EA, and later improve these results by analyzing a sequence of consecutive 2-Opt steps. The fitness gain results presented in this section hold for the considered algorithms if they have not reached a local optimum. Otherwise, we provide an upper bound on the approximation ratio.

Theorem 16:

For -perturbed Euclidean instances and for RLS, (1+1) EA and (1+1) EA*, the approximation ratio for the worst local optimum is bounded above by .

Proof:

We consider Euclidean instances in a 2-dimensional unit hypercube . Then, as a direct consequence from Theorem 3, the expected approximation ratio is at most .

4.1  Analysis of a Single 2-Opt Step

Theorem 17:

In generations RLS achieves an expected fitness gain of unless a local optimum is reached.

Proof:
Due to Lemma 2, we have
formula
Let denote the random variable that describes the fitness gain in an improving 2-Opt step. Then similar to the proof for the Manhattan instances (Theorem 5), we get
formula
Let us set , , a small enough constant such that , which is established in the following calculation. Let us consider the formula . By substituting the value for the this gives,
formula
is less than . Therefore, the above formula is
formula
Subsequently, we get
formula
The number of mutations occurring in one generation is 1 and the probability for an improving 2-Opt step is at least . Therefore, the expected value for the fitness gain in any generation is
formula
The expected value for the fitness gain in generations is
formula
Theorem 18:

In generations (1+1) EA achieves an expected fitness gain of unless a local optimum is reached.

Proof:
The expected fitness gain for an improving singular generation can be derived following the proof of RLS in Theorem 17 with the exception that here the probability of a single step mutation will occur in a generation is . Hence, the expected fitness gain for any generation is
formula
Accordingly, the expected value for the fitness gain in generations is .

4.2  Analysis of Linked Steps for RLS

The previous lower bounds are based on the minimum possible improvement a single 2-Opt step can make. We can further improve this bound considering the improvement made in a sequence of consecutive steps. Similar to the analysis on the consecutive steps for Manhattan instances in Section 3, here we also consider the set of linked pairs of type 0 and 1. In a sequence of generations, there are at least such pairs due to Lemma 8. Here we consider Lemma 14 in Englert et al. (2014) related to the probability of existence of each of the two types of linked pairs in a sequence of consecutive steps for Euclidean instances. Based on these, we can bound the expected fitness gain made in generations.

Lemma 19 (Englert et al., 2014, Lemma 14):

For -perturbed instances, the probability that there exists a pair of type 0 and 1 in which both 2-Opt steps are improvements by at most is bounded by .

Theorem 20:

In generations, constant, RLS achieves an expected fitness gain of unless a local optimum is reached.

Proof:
Using Lemma 19, the probability that the improvement in a linked 2-Opt step of type 0 or 1 is less than is at most
formula
Following the proof ideas of Theorem 10 on the consecutive 2-Opt steps for Manhattan instances, the expected fitness gain for a pair of linked 2-Opt steps of type 0 or 1 can be bounded from below as
formula
We set for a small enough constant  such that , justified in the following. When plugging in ,
formula
since . Likewise, when plugging in ,
formula
since For both terms, the implicit constants can be made arbitrarily small by choosing small, thus establishing .
This implies
formula
Following the same argument in Theorem 10 the expected number of type 0 or 1 pairs in a sequence of steps is . A lower bound for the expected fitness gain for generations is therefore
formula

4.3  Analysis of Linked Steps for (1+1) EA

We improve the current results for (1+1) EA with the analysis for consecutive 2-Opt steps in a similar way to the analysis presented in the previous section. Again, we consider the (1+1) EA* but conjecture that the expected fitness gain in the (1+1) EA is no smaller than that for the (1+1) EA*. Based on our arguments on the number of type 0 or 1 linked pairs from Lemmas 11 and 8 and the stated Lemma 19 of Englert et al. (2014) on the probability of the existence of a pair of both improving steps we can bound the expected fitness gain in generations.

Theorem 21:

In generations, constant, (1+1) EA* achieves an expected fitness gain of unless a local optimum is reached.

Proof:
Following the proof ideas in above Theorem 20 we get for a
formula
From Lemma 11 we know that the number of disjoint pairs of 2-Opt steps, both of which are singular, is at least
formula
Lemma 7 implies that among these there are at least
formula
type 0 or 1 pairs.
The expected fitness gain in generations, , is therefore at least
formula

The lower bounds on the expected fitness gain for RLS, (1+1) EA and (1+1) EA* are presented in Table 2.

Table 2:
Expected fitness gain in generations for RLS and (1+1) EA for Manhattan and Euclidean instances due to single-step and consecutive-steps analysis. The former applies for any time span ; the latter requires .
RLS(1+1) EA /(1+1) EA*
MetricSingle Step (any )Consecutive ()SingleConsecutive ()
Man     
Euc     
RLS(1+1) EA /(1+1) EA*
MetricSingle Step (any )Consecutive ()SingleConsecutive ()
Man     
Euc     

The consecutive-steps analysis was formally proven for the (1+1) EA* and transfers to the (1+1) EA if, as conjectured, the latter does not perform worse. All fitness gains assume that no local optimum is reached. Otherwise, the expected approximation ratio is . The results for RLS single step, consecutive steps and (1+1) EA single step and consecutive steps are based on Theorems  5,  10,  6, and  12 for the Manhattan metric and Theorems  17,  20,  18, and  21, for the Euclidean metric, respectively.

4.4  The Approximation Ratio over the Algorithm Run

We again interpret our fitness gain results in terms of the expected approximation ratio, similar to the previous analysis on Manhattan instances. Here, we refer to the previous study of Burgstaller and Pillichshammer (2009) to derive the expected length of the initial tour.

Theorem 22:

Based on the single step improvements, for Euclidean instances and for RLS, the expected approximation ratio after generations is at most , for the hidden constant in the expected fitness gain in Theorem 17, if the current solution is not already locally optimal.

Proof:
The proof is essentially the same as that of Theorem 15. For the Euclidean metric equals times the expected Euclidean distance between two points chosen uniformly at random from the unit square. The latter is derived as  (Burgstaller and Pillichshammer, 2009, Example 3). This gives . Accordingly, , and using Lemma 14 and Theorem 17 yields an expected approximation ratio of at most
formula

Similarly, we can derive upper bounds on the expected approximation ratio after generations for the considered algorithms and the Euclidean metric as presented in Table 3.

Table 3:
Upper bounds on the expected approximation ratio in generations for RLS and (1+1) EA or (1+1) EA* for Euclidean instances due to single-step and consecutive-steps analysis if the current solution is not locally optimal.
Euclidean metric
AlgorithmSingle StepsConsecutive Steps
RLS  ,  
(1+1) EA /EA*  ,  
Euclidean metric
AlgorithmSingle StepsConsecutive Steps
RLS  ,  
(1+1) EA /EA*  ,  

The term is omitted for clarity. Otherwise, the approximation ratio is . The results are based on the theorems mentioned in the caption of Table 2. The constants , and are the hidden constants in the expected fitness gain from Theorems 17, 20, 18, and 21, respectively.

5  Population-Based Algorithms

We now extend our considerations to search heuristics with offspring populations and investigate the algorithms (1+) RLS and (1+) EA shown in Algorithms 4 and 5, respectively. Both algorithms generate offspring in one generation and replace the current solution by an offspring (individual of best fitness among the individuals) if its fitness is not inferior to . In the context of TSP the best individual represents the shortest tour. Offspring populations are popular as all offspring can be generated and evaluated in parallel. In this sense, the number of generations is considered as the parallel runtime of the algorithm. The total number of function evaluations can be derived by multiplying the number of generations by .

formula
formula

5.1  Fitness Gain Based on Single Step Improvements

In one generation, offspring are created and, out of the individuals, a fittest individual is selected for the next generation. Hence, we can calculate the fitness gain based on the probability that any of these offspring creates a sufficiently large improvement.

We first consider the case for RLS and Manhattan instances. The following lemma estimates the probability that one of independent trials will yield a desirable event that happens with some probability .

Lemma 23:
For any , and any
formula
Proof:
The first inequality follows easily from Lemma 8 in Rowe and Sudholt (2014), which states that . The second one follows from
formula

The following theorem shows that the lower bound on the expected fitness gain of (1+) RLS is by a factor of order larger than the lower bound on the expected fitness gain of RLS, at the expense of making fitness evaluations in one generation.

Theorem 24:

For Manhattan instances in generations, (1+) RLS and (1+) EA achieve an expected fitness gain of unless a local optimum is reached.

The minimum in can be explained as follows. For small offspring population sizes , the probability of making a successful 2-Opt step increases linearly with  (cf. Lemma 23). For this probability is , and this asymptotic growth does not increase, even if increases beyond . Hence, increasing the offspring population size beyond does not lead to a further increase of the lower bound on the fitness gain.

Note that to see the expected fitness gain per fitness evaluation, we may simply divide the expected fitness gain per generation by . Then the above factor becomes ; that is, our bounds on the expected fitness gain per fitness evaluation do not improve when using offspring populations. This makes sense as an elitist (1+1) algorithm mutates the best search point seen so far, whereas offspring populations make mutations of the previous generation’s best search point. The advantage of offspring populations lies in the fact that offspring generation can be parallelized. For that reason and for simplicity, we state the fitness gain with respect to the number of generations.

Proof of Theorem 24::
From Theorem 5 we can derive the expected fitness gain given that a step is an improving step as
formula
For each offspring generated in (1+) RLS, the probability for an improving 2-Opt step is at least if the current solution is not locally optimal. For each offspring in the (1+) EA this probability is at least as in Theorem 6. Since we have offspring, the probability of the best one making an improving 2-Opt step can be derived according to Lemma 23 for and , respectively. In both cases
formula
Therefore, for both algorithms the expected value for the fitness gain in any 2-Opt step can be bounded from below as
formula
Hence, the expected value for the fitness gain in generations if no locally optimal solution has been obtained in between can be derived as .

Similarly for Euclidean instances the fitness gains can be derived following Theorems 24, 17, and 18.

Corollary 25:

For Euclidean instances in generations (1+) RLS and (1+) EA achieve an expected fitness gain of unless a locally optimal solution is reached.

5.2  Fitness Gain Based on Linked Steps

For (1+) RLS the analysis on linked steps follows from the previous analysis of RLS (Theorem 10). Recall that Theorem 10 required . Compared to this, the minimum value for  for (1+) RLS is by a factor of order smaller.

Theorem 26:

For Manhattan instances in generations, constant, (1+) RLS achieves an expected fitness gain of unless a local optimum is reached.

Proof:
Let be the random variable that describes the fitness gain obtained in a pair of linked 2-Opt steps of type 0 or 1. For a , the expected fitness gain can be bounded following Theorem 10.
formula
The probability for an improving 2-Opt step is at least if the current solution is not locally optimal. By substituting this probability to Lemma 23, we obtain the probability of improving 2-Opt steps made in one generation as at least . The expected number of improving 2-Opt steps made in generations is therefore .
Let be the number of improving steps. These steps in fact form a sequence of consecutive 2-Opt steps due to RLS accepting only the improving steps. By Lemma 8 we know there are at least disjoint type 0 or 1 pairs in a sequence of consecutive steps. By substituting the value for expected number of improving steps for we obtain the desired result.
formula
As this is greater than . For , we get .
formula
A lower bound for the expected fitness gain for generations is therefore
formula

For Euclidean instances also the results follows from Theorem 20 in a similar way to Theorem 26.

Corollary 27:

For Euclidean instances in generations, constant, (1+) RLS achieves an expected fitness gain of unless a local optimum is reached.

Extending our analysis of linked steps to the (1+) EA, or the (1+) EA*, turns out to be very challenging. Our analysis of linked steps in the (1+1) EA* (Theorems 12 and 21) were based on improving steps made in singular generations, as only singular generations allowed us to form linked pairs of improving 2-Opt steps.

The proof of Theorem 12 showed that the next improving step where a particular edge is being removed is made in a singular generation with probability at least . This argument does not carry over to the (1+) EA: the (1+) EA accepts the best out of offspring, and each offspring may be created in a singular or multi-step mutation. If improvements are easy to find, there is a good chance that the best offspring will result from a multistep mutation.

To demonstrate this effect, assume that there are constants such that the fraction of 2-Opt steps increasing the fitness by more than is at least , where is the maximum fitness increase by any 2-Opt step in the considered metric (Manhattan or Euclidean distances). Such a situation is particularly likely at the start of a run, when the tour contains many long edges. Note that singular mutations yield to a fitness gain of at most . The probability of one offspring increasing the fitness by more than , through a mutation making 2-Opt steps, is at least
formula
where the factor accounts for the fact that the number of 2-Opt steps increasing the fitness by may decrease during the application of the 2-Opt moves.

The probability that no offspring leads to a fitness increase larger than  is then ; hence, with probability the best offspring has a fitness gain larger than , which can happen only in a multistep mutation. So, in this situation with probability at least the best offspring will result from a multistep mutation.

Unless is very small, this drastically decreases the chances of finding linked pairs of 2-Opt steps as in the proof of Theorem 12, and the analysis breaks down. Note that this is a shortcoming of our theoretical approach; we still conjecture that the expected fitness gain can be bounded from below by times the lower bound for the (1+1) EA* from Theorem 12. However, proving this requires novel ideas and possibly a different approach.

The lower bounds on the expected fitness gain for (1+) RLS and (1+) EA are summarized in Table 4.

Table 4:
Expected fitness gain in generations for (1+) RLS and (1+) EA for Manhattan and Euclidean instances due to single-step and consecutive-steps analysis if the algorithm is not in a local optimum. The former applies for any time span ; the latter requires . Otherwise (cf. in a local optima) the expected approximation ratio is . The results for (1+) RLS single step, consecutive steps and (1+) EA single step are based on Theorems 24 and 26 for Manhattan metric and Corollaries 25 and 27 for Euclidean metric, respectively.
 (1+) RLS (1+) EA 
Metric Single Step (any Con. (Single Step (any 
Man    
Euc    
 (1+