Sliding Window Bi-Objective Evolutionary Algorithms for Optimizing Chance-Constrained Monotone Submodular Functions

Evolutionary algorithms (EAs) have been successfully applied to solve a wide range of complex combinatorial optimization problems. The algorithms have received both theoretical and empirical studies, showing their ability to obtain good solutions within a reasonably expected runtime for problems with deterministic settings [3, 7, 8, 11, 19, 26, 31]. Additionally, it has been observed that some EAs also perform effectively on stochastic and dynamic optimization problems. Researchers are actively investigating the advantages and limitations of EAs in solving such problems, particularly within the field of evolutionary computation theory. A variety of studies [12, 14, 20, 18, 27, 29, 32] have been conducted, presenting both the challenges and the advanced implications of these algorithms.

In a stochastic environment, completely eliminating the negative and uncertain impact of stochastic components is challenging. Therefore, it is crucial to minimize these negative effects to prevent unpredictable disruptions in most complex systems. Chance constraint  [1, 5, 9, 15, 22, 23, 24, 25] is a useful technique for handling the effects of stochastic circumstances. It allows the deterministic bound in the constraint to be violated, but only with a very small probability during optimization. However, directly evaluating chance constraints is complex and time-consuming. A practical approach is to transform the stochastic constraint into its corresponding deterministic equivalent for a given confidence level, rather than statistically calculating the probability of violation when optimizing chance-constrained problems with known distribution elements [30, 32].

Submodular functions [16] represent various problems where the incremental benefit of adding solution elements diminishes with the increasing solution size. The optimization of submodular functions is significantly challenging and has been extensively investigated with different types of constraints in previous work [10, 13, 16, 21, 27, 33, 34]. In the paper, we study a chance-constrained version of the submodular optimization problem subject to a knapsack constraint. The problem seeks to find a subset of stochastic elements that maximizes the submodular function value, while ensuring that the actual weight exceeds a given bound with a very small probability. In the previous research, Doerr et al [4] analyzed the performance of the greedy algorithms on the problem. They constructed the surrogate functions using the tail inequalities (Chernoff bound and one-sided side Chebyshev’s inequality) to handle the chance constraint. Their findings demonstrated the greedy algorithms can achieve $(1-o(1))(1-1/e)$-approximation and $(1/2-o(1)(1-1/e)$-approximation for the problem with the uniform independent and identically distributed (IID) weights and the uniform weights with the same dispersion respectively. Subsequently, Neumann et al., [17] applied the Global Simple Evolutionary Multi-objective Algorithm (GSEMO) to the same problem. They employed a similar surrogate approach for estimating violation probability as done in [4] and considered both the submodular function value and the probability as objectives in their bi-objective fitness function. Their theoretical analysis demonstrated that the GSEMO can achieve similar approximation results as [17] within an expected runtime related to the max population size that it can generate. Additionally, their experimental results indicated that the GSEMO could obtain better solutions than the greedy algorithm and other evolutionary algorithms, in solving the problem involving uniform IID weights.

However, the runtime analysis presented in [17] indicates that the growing population size significantly impacts the efficiency of the GSEMO. This is particularly evident when dealing with uniform weights of the same dispersion, where the GSEMO can quickly maintain an exponential number of trade-off solutions in the population. To address this inefficiency, the sliding-selection approach has been introduced, leading to the development of an improved version of GSEMO, named the Sliding Window GSEMO (SW-GSEMO). Neumann and Witt [21] firstly presented the SW-GSEMO’s enhanced performance in solving submodular problems with deterministic weights. In brief, the sliding-selection method involves defining a weight window of size one, which is determined based on the given bound and the ratio of the current time to the total time. A solution is eligible for selection as a parent if its weight falls within the current window; otherwise, the algorithm uniformly selects one individual from the current population.

Within this paper, the SW-GSEMO is slightly updated in terms of the window selection method for optimizing chance-constrained monotone submodular functions. The individuals that are in the window are still potential parents for the mutation, however, when there is no individual in the window area, the algorithm chooses a current solution with the largest function value as the parent for the next mutation unless the current time exceeds the time budget. Following the approach in [17], the fitness function in our algorithm contains two objectives: the evaluated weight and the function value of the solution. We also employ the same surrogate methods based on tail inequalities for weight evaluation. Our study includes settings that include both IID weights, and uniform weights with the same dispersion. Our analysis reveals that the incorporation of the sliding-selection approach reduces the impact of growing population size on the expected runtime. As a result, the expected runtime of the SW-GSEMO to achieve similar expected approximation results shows improvement. Additionally, we empirically assess the performance of the SW-GSEMO on the maximum coverage problem under various settings, bounds, and violation probabilities. We compare its results with those from the original GSEMO and the Non-Dominated Sorting Genetic Algorithm II (NSGA-II ). The initial solutions of all algorithms are empty sets and the NSGA-II uses population sizes of $20$ and $100$ respectively. Furthermore, we provide a visualization of the sliding window’s operation during optimization. Our investigation also seeks why algorithms using surrogates based on the Chernoff bound perform better than those using the one-sided Chebyshev’s inequality when the probability is smaller.

The paper is structured as follows. Section 2 introduces the chance-constrained monotone submodular problem and the investigated settings. Section 3 describes the adopted multi-objective evolutionary algorithms. Our theoretical runtime analysis and proofs of the SW-GSEMO are presented in Section 4. Then we investigate the performance of the different algorithms and visualize the selection behavior in Section 5. Finally, we end with some conclusions in Section 6.

Given a set $V=\{v_{1},...,v_{n}\}$, we consider the optimization of a monotone submodular function $f:2^{V}\to\mathbb{R}_{+}$. A function is called monotone iff for every $S,T\subseteq V$ with $S\subseteq T$, $f(S)\leq f(T)$ holds. A function $f$ is called submodular iff for every $S,T\subseteq V$ with $S\subseteq T$, $v_{i}\in V$ and $v_{i}\notin T$, $f(S\cup\{v\})-f(S)\geq f(T\cup\{v\})-f(T)$ holds. We consider the optimization of such a function $f$ subjected to the chance constraint where each element $v_{i}$ takes on a random weight $W(v_{i})$. Here, the chance-constrained optimization problem can be formulated as

where $W(S)=\sum_{v_{i}\in S}W(v_{i})$ is the total weight of the subset $S$, and $B$ is the given deterministic bound. The parameter $\alpha$ quantifies the probability of exceeding the bound $B$ that can be tolerated.

Following previous work [4, 17], we consider two different settings, which are (1) Uniform IID Weights: the weight of each element $v_{i}\in V$ is sampled from the uniform distribution with the same expected value $E_{W}(v_{i})=a$ and dispersion $\delta(v_{i})=d$ (i.e., $W(v_{i})\in[a-d,a+d$] and $0<d\leq a$); and (2) Uniform Weights with the Same Dispersion: the weight of element $v_{i}\in V$ is sampled from the uniform distribution with the different expected value $E_{W}(v_{i})=a_{i}$ but the same dispersion $\delta(v_{i})=d$ (i.e., $W(v_{i})\in[a_{i}-d,a_{i}+d]$ and $0<d\leq a_{i}$). In both settings, we assume that the expected weight of each element is a positive integer. For further discussion, the subset $S$ is encoded as a decision vector $x=x_{1},x_{2},...,x_{n}$ with length $n$, where $x_{i}=1$ means that the element $v_{i}\in V$ is selected into the subset $S$. Besides, we denote $|x|_{1}$ the number of elements packed into the subset $S$. Since all the settings are based on the uniform distribution, we have the expected weight and variance of the solution $x$ as

and

To handle the chance constraint, we establish the surrogate function based on One Chebyshev’s inequality and Chernoff bound as described in the previous work [4, 17]. The weight calculated by different surrogate functions are respectively formulated as

and

It had been proved that if the surrogate weight of a solution $x$ is less than the bound $B$, then $x$ is feasible [4].

Within the paper, we primarily investigate multi-objective evolutionary algorithms on the given monotone submodular problem. Each solution is considered to be a two-dimensional search point in the objective space. The bi-objective fitness function of the solution $x$ is expressed as

where $f(x)$ denotes the submodular function value of $x$, $W_{sg}(x,\alpha)$ denotes the surrogate weight of $x$. Let $y\in\{0,1\}^{n}$ be another solution in the search space. We say that $x$ (weakly) dominates $y(x\succeq y)$ iff $g_{1}(x)\geq g_{1}(y)$ and $g_{2}(x,\alpha)\leq g_{2}(y,\alpha)$. Note that the infeasible solution is strongly dominated by the feasible one because of the objective function $g_{1}$. Besides, the objective function $g_{2}$ guides the generated solutions approach to the feasible search space.

In previous work [17], the GSEMO (see Algorithm 1) is studied on the chance-constrained monotone submodular problem. Here the GSEMO starts with an initial solution represented by a $0^{n}$ bitstring, signifying an empty set. During the optimization, it maintains a set of non-dominated solutions, continually updating this set as new solutions are generated. In each iteration, the GSEMO randomly and uniformly selects an individual $x$ from the population to serve as a parent for creating offspring $y$ via a standard bit-flip operator, which flips each bit of $x$ independently with a probability of $1/n$. The offspring $y$ is accepted into the population if it is not strictly dominated by any existing solution. Subsequently, any solution in the population that is weakly dominated by $y$ is removed.

The modified SW-GSEMO algorithm is outlined in Algorithm 2. It also begins with a $0^{n}$ bitstring, maintains a population of non-dominating individuals, and employs the standard mutation to generate offspring. However, it differs from the GSEMO in its parent selection method, which defines a sliding window (see Algorithm 3) to select the potential parents instead of applying the random uniform selection in the whole population directly. Within this method, given the total runtime $t_{max}$ and the current time $t$, a current bound $\hat{c}$ is defined as $\hat{c}=t\cdot B/t_{max}$. A window is established as an interval between $[\lfloor\hat{c}\rfloor,\lceil\hat{c}\rceil]$. Unlike the deterministic case where it selects solutions based on deterministic weight, the SW-GSEMO chooses some eligible individuals whose surrogate weights fall within this window. The parent is then randomly selected from these eligible solutions. As the current time $t$ increases, the SW-GSEMO would select solutions with larger surrogate weights that are still within the window. When no individual lies within the current window, the SW-GSEMO generates a sub-population $P^{\prime}$ where the solution $x$ from the original population has the surrogate weight $g_{2}(x)$ that is lower than $\lfloor\hat{c}\rfloor$. Then the solution $x^{\prime}\in P^{\prime}$ with the largest $g_{1}$-value is assigned for the mutation. Moreover, when $t\geq t_{max}$, the SW-GSEMO reverts to selecting parents from the entire population $P$, similar to the GSEMO.

To illustrate that the SW-GSEMO works more efficiently in optimizing the chance-constrained monotone submodular problem than the GSEMO, the expected runtime of SW-GSEMO with the surrogates to reach the same expected result for the problem is analyzed in this section.

The experimental investigations of the SW-GSEMO based on the different surrogate functions are proposed here. The results are compared with those generated from the GSEMO and the NSGA-II in various instances.

In this paper, we investigated the use of SW-GSEMO on chance-constrained monotone submodular optimization problems with IID weights and uniform weights with the same dispersion. Surrogate functions based on Chebshev’s inequality and Chernoff bound have been applied to evaluate the chance constraint. We showed theoretically that the SW-GSEMO with the surrogate can reach the same approximation result in a more efficient way than the GSEMO that was studied in previous work. Furthermore, the algorithm is applied to the maximum coverage problem in the experiments and its results are compared with other multi-objective algorithms under variable instances constructed by different graphs. The experiments demonstrated that the window defined in SW-GSEMO is sliding in the weight interval during the optimization. Additionally, the obtained results show that the SW-GSEMO with the surrogate based on one-sided Chebyshev’s inequality performs better than the GSEMO and NSGA-II (with population sizes 20 and 100) among most of the instances when $\alpha$ is larger, and the SW-GSEMO using the Chernoff bound works best when $\alpha$ is smaller. For future work, it would be interesting to consider other generalized settings with different distributions and covariances as part of the chance-constrained formulation.