Bi-objective Search Research Articles

Finding a Small, Diverse Subset of the Pareto Solution Set in Bi-Objective Search (Extended Abstract)

Finding a Small, Diverse Subset of the Pareto Solution Set in Bi-Objective Search (Extended Abstract)

Towards Effective Multi-Valued Heuristics for Bi-objective Shortest-Path Algorithms via Differential Heuristics

In bi-objective graph search, each edge is annotated with a cost pair, where each cost corresponds to an objective to optimize. We are interested in finding all undominated paths from a given start state to a given goal state (called the Pareto front). Almost all existing works of bi-objective search use single-valued heuristics, which use one number for each objective, to estimate the cost between any given state and the goal state. However, single-valued heuristics cannot reflect the trade-offs between the two costs. On the other hand, multi-valued heuristics use a set of pairs to estimate the Pareto front between any given state and the goal state and are more informed than single-valued heuristics. However, they are rarely studied and have yet to be investigated in explicit state spaces by any existing work. In this paper, we are interested in using multi-valued heuristics to improve bi-objective search algorithms in explicit state spaces. More specifically, we generalize Differential Heuristics (DHs), a class of memory-based heuristics for single-objective search, to bi-objective search, resulting in Bi-objective Differential Heuristics (BO-DHs). We propose several techniques to reduce the memory usage and computational overhead of BO-DHs significantly. Our experimental results show that, with suggested improvement and tuned parameters, BO-DHs can reduce the node expansion and runtime of a bi-objective search algorithm by up to an order of magnitude, paving the way for more effective multi-valued heuristics.

Efficient Multi-Query Bi-Objective Search via Contraction Hierarchies

Contraction Hierarchies (CHs) have been successfully used as a preprocessing technique in single-objective graph search for finding shortest paths. However, only a few existing works on utilizing CHs for bi-objective search exist, and none of them uses CHs to compute Pareto frontiers. This paper proposes an CH-based approach capable of efficiently computing Pareto frontiers for bi-objective search along with several speedup techniques. Specifically, we propose a new preprocessing approach that computes CHs with fewer edges than the existing preprocessing approach, which reduces both the preprocessing times (up to 3x in our experiments) and the query times. Furthermore, we propose a partial-expansion technique, which dramatically speeds up the query times. We demonstrate the advantages of our approach on road networks with 1 to 14 million states. The longest preprocessing time is less than 6 hours, and the average speedup in query times is roughly two orders of magnitude compared to BOA*, a state-of-the-art single-query bi-objective search algorithm.

Simple and efficient bi-objective search algorithms via fast dominance checks

Many interesting search problems can be formulated as bi-objective search problems, that is, search problems where two kinds of costs have to be minimized, for example, travel distance and time for transportation problems. Instead of looking for a single optimal path, we compute a Pareto-optimal frontier in bi-objective search, which is a set of paths in which no two paths dominate each other. Bi-objective search algorithms perform dominance checks each time a new path is discovered. Thus, the efficiency of these checks is key to performance. In this article, we propose algorithms for two kinds of bi-objective search problems. First, we consider the problem of computing the Pareto-optimal frontier of the paths that connect a given start state with a given goal state. We propose Bi-Objective A* (BOA*), a heuristic search algorithm based on A*, for this problem. Second, we consider the problem of computing one Pareto-optimal frontier for each state s of the search graph, which contains the paths that connect a given start state with s. We propose Bi-Objective Dijkstra (BOD), which is based on BOA*, for this problem. A common feature of BOA* and BOD is that all dominance checks are performed in constant time, unlike the dominance checks of previous algorithms. We show in our experimental evaluation that both BOA* and BOD are substantially faster than state-of-the-art bi-objective search algorithms.

Subset Approximation of Pareto Regions with Bi-Objective A* (Extended Abstract)

In bi-objective search, we are given a graph in which each directed arc is associated with a pair of non-negative weights, and the objective is to find the Pareto-optimal solution set. Unfortunately, in many practical settings, this set is too large, and therefore its computation is very time-consuming. In addition, even though bi-objective search algorithms generate the Pareto set incrementally, they do so exhaustively. This means that early during search the solution set covered is not diverse, being concentrated in a small region. To address this issue, we present a new approach to subset approximation of the solution set, that can be used as the basis for an anytime bi-objective search algorithm. Our approach transforms the given task into a target bi-objective search task using two real parameters. For each particular parameter setting, the solutions to the target task is a subset of the solution set of the original task. Depending on the parameters used, the solution set of the target task may be computed very quickly. This allows us to obtain, in challenging road map benchmarks, a rich variety of solutions in times that may be orders of magnitude smaller than the time needed to compute the solution set. We show that by running the algorithm with an appropriate sequence of parameters, we obtain a growing sequence of solutions that converges to the full solution set. We prove that our approach is correct and that Bi-Objective A* prunes at least as many nodes when run over the target task.

Weight Constrained Path Finding with Bidirectional A*

Weight constrained path finding, known as a challenging variant of the classic shortest path problem, aims to plan cost optimum paths whose weight/resource usage is limited by a side constraint. Given the bi-criteria nature of the problem (i.e., the presence of cost and weight), solutions to the Weight Constrained Shortest Path Problem (WCSPP) have some properties in common with bi-objective search. This paper leverages the state-of-the-art bi-objective search algorithm BOBA* and presents WC-BA*, an exact A*-based WCSPP method that explores the search space in different objective orderings bidirectionally. We also enrich WC-BA* with two novel heuristic tuning approaches that can significantly reduce the number of node expansions in the exhaustive search of A*. The results of our experiments on a large set of realistic problem instances show that our new algorithm solves all instances and outperforms the state-of-the-art WCSPP algorithms in various scenarios.

Bounded-Cost Bi-Objective Heuristic Search

There are many settings that extend the basic shortest path search problem. In Bounded-Cost Search, we are given a constant bound and the task is to find a solution within the bound. In Bi-Objective Search, each edge is associated with two costs (objectives) and the task is to minimize both objectives. In this paper, we combine both these settings into a new setting of Bounded-Cost Bi-Objective Search. We are given two bounds, one for each objective and the task is to find a solution within these bounds. We provide a scheme for normalizing the two objectives. We then introduce several algorithms for this new setting and compare them experimentally.

Anytime Approximate Bi-Objective Search

The Pareto-optimal frontier for a bi-objective search problem instance consists of all solutions that are not worse than any other solution in both objectives. The size of the Pareto-optimal frontier can be exponential in the size of the input graph, and hence finding it can be hard. Some existing works leverage a user-specified approximation factor epsilon to compute an approximate Pareto-optimal frontier that can be significantly smaller than the Pareto-optimal frontier. In this paper, we propose an anytime approximate bi-objective search algorithm, called Anytime Bi-Objective A*-epsilon (A-BOA*). A-BOA* is useful when deliberation time is limited. It first finds an approximate Pareto-optimal frontier quickly, iteratively improves it while time allows, and eventually finds the Pareto-optimal frontier. It efficiently reuses the search effort from previous iterations and makes use of a novel pruning technique. Our experimental results show that A-BOA* substantially outperforms baseline algorithms that do not reuse previous search effort, both in terms of runtime and number of node expansions. In fact, the most advanced variant of A-BOA* even slightly outperforms BOA*, a state-of-the-art bi-objective search algorithm, for finding the Pareto-optimal frontier. Moreover, given only a limited amount of deliberation time, A-BOA* finds solutions that collectively approximate the Pareto-optimal frontier much better than the solutions found by BOA*.

Subset Approximation of Pareto Regions with Bi-objective A*

In bi-objective search, we are given a graph in which each directed arc is associated with a pair of non-negative weights, and the objective is to find the Pareto-optimal solution set. Unfortunately, in many practical settings, this set is too large, and therefore its computation is very time-consuming. In addition, even though bi-objective search algorithms generate the Pareto set incrementally, they do so exhaustively. This means that early during search the solution set covers is not diverse, being concentrated in a small region of the solution set. To address this issue, we present a new approach to subset approximation of the solution set, that can be used as the basis for an anytime bi-objective search algorithm. Our approach transforms the given task into a target bi-objective search task using two real parameters. For each particular parameter setting, the solutions to the target task is a subset of the solution set of the original task. Depending on the parameters used, the solution set of the target task may be computed very quickly. This allows us to obtain, in challenging road map benchmarks, a rich variety of solutions in times that may be orders of magnitude smaller than the time needed to compute the solution set. We show that by running the algorithm with an appropriate sequence of parameters, we obtain a growing sequence of solutions that converges to the full solution set. We prove that our approach is correct and that Bi-Objective A* prunes at least as many nodes when run over the target task.

A*pex: Efficient Approximate Multi-Objective Search on Graphs

In multi-objective search, edges are annotated with cost vectors consisting of multiple cost components. A path dominates another path with the same start and goal vertices iff the component-wise sum of the cost vectors of the edges of the former path is ``less than'' the component-wise sum of the cost vectors of the edges of the latter path. The Pareto-optimal solution set is the set of all undominated paths from a given start vertex to a given goal vertex. Its size can be exponential in the size of the graph being searched, which makes multi-objective search time-consuming. In this paper, we therefore study how to find an approximate Pareto-optimal solution set for a user-provided vector of approximation factors. The size of such a solution set can be significantly smaller than the size of the Pareto-optimal solution set, which enables the design of approximate multi-objective search algorithms that are efficient and produce small solution sets. We present such an algorithm in this paper which we call A*pex and which builds on PP-A*, a state-of-the-art approximate bi-objective search algorithm (where there are only two cost components) but (1) makes PP-A* more efficient for bi-objective search and (2) generalizes it to multi-objective search for any number of cost components. We first analyze the correctness of A*pex and then experimentally demonstrate its efficiency advantage over existing approximate algorithms for bi- and tri-objective search.

HARNAS: Human Activity Recognition Based on Automatic Neural Architecture Search Using Evolutionary Algorithms.

Human activity recognition (HAR) based on wearable sensors is a promising research direction. The resources of handheld terminals and wearable devices limit the performance of recognition and require lightweight architectures. With the development of deep learning, the neural architecture search (NAS) has emerged in an attempt to minimize human intervention. We propose an approach for using NAS to search for models suitable for HAR tasks, namely, HARNAS. The multi-objective search algorithm NSGA-II is used as the search strategy of HARNAS. To make a trade-off between the performance and computation speed of a model, the F1 score and the number of floating-point operations (FLOPs) are selected, resulting in a bi-objective problem. However, the computation speed of a model not only depends on the complexity, but is also related to the memory access cost (MAC). Therefore, we expand the bi-objective search to a tri-objective strategy. We use the Opportunity dataset as the basis for most experiments and also evaluate the portability of the model on the UniMiB-SHAR dataset. The experimental results show that HARNAS designed without manual adjustments can achieve better performance than the best model tweaked by humans. HARNAS obtained an F1 score of 92.16% and parameters of 0.32 MB on the Opportunity dataset.

A Simple and Fast Bi-Objective Search Algorithm

Many interesting search problems can be formulated as bi-objective search problems; for example, transportation problems where both travel distance and time need to be minimized. Multi-objective best-first search algorithms need to maintain the set of undominated paths from the start state to each state to compute a set of paths from a given start state to a given goal state (the Pareto-optimal solutions) such that no path in the set is dominated by another path in the set. Each time they find a new path to a state n, they perform a dominance check to determine whether such a path dominates any of the previously found paths to n. Existing algorithms do not perform these checks efficiently, requiring at least a full iteration over the Open list per check. In this paper, we present the first multi-objective algorithm that performs these checks efficiently. Indeed, Bi-Objective A* (BOA*)—our algorithm—requires constant time to check for dominance. Our experimental evaluation shows that BOA*is orders-of-magnitude faster than state-of-the-art search algorithms, such as NAMOA*, Bi-Objective Dijkstra, and Bidirectional Bi-Objective Dijkstra.

Bi-Objective Search with Bi-directional A* (Extended Abstract)

Bi-objective search is a problem of finding a set of optimal solutions in a two-dimensional domain. This study proposes several enhancements to the state-of-the-art bi-objective search with A* and develops its bi-directional variant. Our experimental results on benchmark instances show that our enhanced algorithm is on average five times faster than the state of the art bi-objective search algorithms.

Diversifying Focused Testing for Unit Testing

Software changes constantly, because developers add new features or modifications. This directly affects the effectiveness of the test suite associated with that software, especially when these new modifications are in a specific area that no test case covers. This article tackles the problem of generating a high-quality test suite to cover repeatedly a given point in a program, with the ultimate goal of exposing faults possibly affecting the given program point. Both search-based software testing and constraint solving offer ready, but low-quality, solutions to this: Ideally, a maximally diverse covering test set is required, whereas search and constraint solving tend to generate test sets with biased distributions. Our approach, Diversified Focused Testing (DFT), uses a search strategy inspired by GödelTest. We artificially inject parameters into the code branching conditions and use a bi-objective search algorithm to find diverse inputs by perturbing the injected parameters, while keeping the path conditions still satisfiable. Our results demonstrate that our technique, DFT, is able to cover a desired point in the code at least 90% of the time. Moreover, adding diversity improves the bug detection and the mutation killing abilities of the test suites. We show that DFT achieves better results than focused testing, symbolic execution, and random testing by achieving from 3% to 70% improvement in mutation score and up to 100% improvement in fault detection across 105 software subjects.

A Simple and Fast Bi-Objective Search Algorithm

Many interesting search problems can be formulated as bi-objective search problems, that is, search problems where two kinds of costs have to be minimized, for example, travel distance and time for transportation problems. Bi-objective search algorithms have to maintain the set of undominated paths from the start state to each state to compute the set of paths from the start state to the goal state that are not dominated by some other path from the start state to the goal state (called the Pareto-optimal solution set). Each time they find a new path to a state s, they perform a dominance check to determine whether this path dominates any of the previously found paths to s or whether any of the previously found paths to s dominates this path. Existing algorithms do not perform these checks efficiently. On the other hand, our Bi-Objective A* (BOA*) algorithm requires only constant time per check. In our experimental evaluation, we show that BOA* can run an order of magnitude (or more) faster than state-of-the-art bi-objective search algorithms, such as NAMOA*, NAMOA*dr, Bi-Objective Dijkstra, and Bidirectional Bi-Objective Dijkstra.

Distributed localized bi-objective search

We propose a new distributed heuristic for approximating the Pareto set of bi-objective optimization problems. Our approach is at the crossroads of parallel cooperative computation, objective space decomposition, and adaptive search. Given a number of computing nodes, we self-coordinate them locally, in order to cooperatively search different regions of the Pareto front. This offers a trade-off between a fully independent approach, where each node would operate independently of the others, and a fully centralized approach, where a global knowledge of the entire population is required at every step. More specifically, the population of solutions is structured and mapped into computing nodes. As local information, every node uses only the positions of its neighbors in the objective space and evolves its local solution based on what we term a ‘localized fitness function’. This has the effect of making the distributed search evolve, over all nodes, to a high quality approximation set, with minimum communications. We deploy our distributed algorithm using a computer cluster of hundreds of cores and study its properties and performance on ρMNK-landscapes. Through extensive large-scale experiments, our approach is shown to be very effective in terms of approximation quality, computational time and scalability.

Algorithms for Task Consolidation Problem in a Cloud Computing Environment

Cloud computing has recently emerged as a new paradigm for hosting and delivering services over the Internet. Task consolidation problem in cloud computing systems became an important approach to streamline resource usage which improves energy efficiency. The task consolidation is also known as workload consolidation problem which is the process of assigning set of tasks to set of resources without violating time constraints. Three existing energy conscious heuristics such as ECTC (Energy-Conscious Task Consolidation) Task Consolidation Algorithm and MaxUtil (Maximum rate Utilization) Task Consolidation Algorithm and Bi-objective Task Consolidation algorithm offering different energy saving possibilities were analyzed in this study. The cost functions incorporated effectively capture energy saving possibilities and their capability has been verified by evaluation study. The Bi-objective Task Consolidation algorithm combines the two heuristics to construct the corresponding bi-objective search space. The efficiency of proposed algorithm was proved thought evaluation study consisting of different simulations carried out.

On a biobjective search problem in a line: Formulations and algorithms

In this article we describe three formulations of a multiobjective combinatorial optimization problem, as well as several complexity results and structural properties of these formulations. A multiobjective dynamic programming algorithm is proposed for each of the three formulations. Based on our theoretical and computational results we argue that a clever definition of the recursion, allowing for strong dominance criteria, is crucial in the design of a multiobjective dynamic programming algorithm.