Virtual power plants (VPPs), as a novel integrated energy aggregation model, can effectively realize peak shaving and valley filling in the power system. This paper mainly investigates the model of a VPP consisting of distributed energy resources (DERs) as energy producers and converters, and an electric vehicle parking lot (EVPL) serving as both a battery storage system and an energy consumer. To mitigate the risks associated with uncertainties, a downside risk constraint method is concerned, which has not yet been applied to this model with multiple uncertainties. We concentrate on the risk-embedded scheduling of the VPP with EVPL, taking into account key uncertain factors, such as, the arrival and departure times of the EVs. Numerical results reveal that the primary factor affecting VPP profits is the instability of renewable energy. In our model, compared to risk-neutral strategies, risk-averse strategies lead to a reduction in average profits by 5.66% , highlighting the trade-off between minimizing risk and maximizing profits.
Due to its numerous applications in social marketing and crowdsourcing, the classical topic of designing a budget-feasible mechanism for a submodular valuation function has been well-studied. In this paper, we consider a generalization of this topic: budget-feasible mechanism design for a k-submodular function in the clock auction model. In our problem, each agent has a private cost, and the auctioneer, composed of k departments, attempts to maximize his k-submodular valuation subject to a budget constraint. For the monotone objective, we propose a randomized mechanism with an approximation ratio of 1/5+√(13) . Additionally, the randomized mechanism can also achieve an approximation ratio of 2/15+3√(13) for the non-monotone objective. Our mechanism only requires O(kn) value oracle queries, making it more practical.
An acyclic edge coloring of a graph G colors the edges in G such that adjacent edges receive distinct colors and the subgraph induced by the edges of any two colors is acyclic. It is conjectured that every simple graph with maximum degree can be acyclically edge ( + 2) -colored. The conjecture has been proven true on triangle-free planar graphs, but is unknown true or not for general planar graphs. We make a step forward to acyclically edge color triangle-free toroidal graphs in + 2 colors. This improves the latest ( + 3) -coloring scheme by Chen and Hou, and completely resolves affirmatively the conjecture on such a super class of triangle-free planar graphs. We achieve the result by first showing that every triangle-free toroidal graph has one of the five specified groups of local structures, and then inductively coloring the edges at the presence of each local structure.
We consider m identical machines scheduling problems with fully parallel jobs. Each job J_j requires processing time p_j and can be executed on any machine at any time unit. In this paper, four scheduling problems are considered: (1) minimize the maximum cost, (2) minimize the total completion time, (3) minimize the weighted number of tardy jobs, and (4) minimize the total weighted tardiness. For the first two problems, we develop optimal polynomial-time algorithms to solve them, respectively. For the third problem, we design a polynomial-time algorithm if all weights are equal to one; we propose a pseudo-polynomial-time algorithm for the general case. For the last problem, we design polynomial-time algorithms for solving some special cases of this problem, respectively.
We consider two scheduling problems with rejection and resource matching on m identical parallel-batch machines. A job is either rejected with a rejection cost or accepted for processing. Accepted jobs are processed on parallel-batch machines with a batch capacity of b. There are multiple kinds of resources, one of which could be matched with an accepted job. Each accepted job must consume exactly one kind of resource and each kind of resource can be consumed by at most one accepted job. Job’s processing time may be different when it consumes a different kind of resource. The objective is to minimize the sum of the makespan of the accepted jobs and the total rejection cost of the rejected jobs. When the batch capacity is bounded, we give two approximation algorithms with worst case ratios mb and 2-1/mb , respectively. When the batch capacity is unbounded, we give a polynomial time optimal algorithm.
The quality and safety issues of product have received widespread attention in the fresh agricultural supply chain (FASC). Blockchain technology (BT) provides a new solution for the FASC. In this situation, altruistic behavior also plays an important role. The study incorporates blockchain technology and altruistic preference of supply chain members into a FASC consisting of a supplier and a powerful retailer. The study analyses the optimal strategies of FASC members under the basic scenarios without and with BT and two scenarios based on blockchain system that is the unilateral altruisitc preference of the retailer and the supplier. The study finds that the adoption of BT is related to the cost of building a blockchain platform. Furthermore, the supplier’s altruisitic preference can increase the profit of the retailer and the supplier under certain conditions. The retailer’s altruistic preference can increase the profit of the supplier but reduces himself profit and we design a revenue sharing contract to coordinate the FASC.
Labeled graph is a graph with labels defined on edges. Given a labeled graph G = (V, E) with label set L = {ℓ _1, ℓ _2, … , ℓ _q } , a source s ∈ V and a sink t ∈ V , the Label s-t Cut problem asks to find a minimum size label subset L' ⊆ L such that the removal of all edges from G with labels in L' disconnects s and t. This problem is NP-hard. The current best approximation ratio of Label s-t Cut is O(n^2/3/OPT^1/3) , where OPT is the size of an optimal solution to the problem. This ratio is achieved by a simple purely combinatorial two-phase approximation algorithm. In this paper, we give a new but short (maybe the shortest) proof for the two-phase approximation algorithm. We also perform experiments to evaluate the algorithm. Experimental results show that the two-phase approximation algorithm has actually excellent practical performance in random graphs.
Given an undirected graph G=(V,E) with weighted vertices and edges, a set J of n=|V| independent jobs and m unrelated machines, each vertex v∈ V corresponds to a job J_j∈ J . The combination of prize-collecting vertex cover and scheduling (CPVCS) is to select a subset C⊆ V of vertices and schedule the jobs corresponding to the vertices in C on m unrelated machines, edges that are not covered incur a penalty cost, such that the sum of makespan and the total weight of the uncovered edges by C is minimized, where the makespan is defined as the maximum load of the machines. When the number of machines is an input number, we design a 4-approximation algorithm based on LP-rounding method. When the number of machines is a constant, we design a 3-approximation algorithm by guessing the job with maximum processing time on each machine.
In recent years, the k-submodular maximization problem has garnered significant attention. However, many practical applications cannot be strictly modeled as k-submodular maximization problems. This highlights the necessity of studying the maximization of approximately non-k-submodular. In this paper, we are committed to solving the problem of maximizing ϵ -approximately α -weakly diminishing returns functions under p-system and ℓ knapsack constraints. We first give a greedy algorithm, achieving a 1/(1+ϵ ')(1+1+ϵ/α ^2(1-ϵ )p+2(1+ϵ )^2/α ^2(1-ϵ )^2ℓ ) - approximation with the O(n^2(1+k)log (2n)) time complexity, where ϵ ' is a very small positive number. Then we further introduce an improved algorithm that not only enhances the approximation ratio to min{1,1/α (1+ϵ ')}/1+1+ϵ/α ^2(1-ϵ )[(1+ϵ ')(p+αϵ ')+7/4ℓ ] , but also reduces the time complexity to O(nklogn/ϵ 'log (2n)) .
In two-machine open shop scheduling with exact delays, each job consists of two operations with an arbitrary process order, provided that the second operation must start exactly a certain time after the first operation is completed. It is known that the problem of minimizing makespan cannot be approximated in polynomial time to within a factor of 2-ϵ for any ϵ >0 , unless P=NP . In this paper, we show that similar inapproximability results hold even if all delays are equal. Specifically, it is shown that the problem with equal exact delays cannot be approximated within 4/3-ϵ unless P=NP , and even if each job has equal-length operations, it is NP-hard to approximate the problem to within 5/4-ϵ . When there exist single-operation jobs, we show that the problem cannot be approximated within 7/6-ϵ unless P=NP . On the positive side, we first observe that the O(nlog n) -time exact algorithm for the flow shop problem behaves as a 2-approximation algorithm for the open shop problem, and then we present an improved 5/3 -approximation algorithm for the case with equal-length operations for each job.
This paper introduces the min-max correlation clustering problem with penalties, which is a generalization of the correlation clustering problem. In this problem, each vertex can be clustered or penalized, and the goal of the problem is to minimize the sum of the number of error edges and the penalty cost at the worst vertex. In this paper, we give an integer programming and linear programming relaxation of the problem, and provide a constant approximation algorithm of the problem based on LP-rounding technique.
This paper investigates a two-stage online scheduling problem for parallel machines with the goal of minimizing the makespan. The first stage requires packing a set of n unit jobs into M bags. In the second stage, these bags are allocated to m≤ M machines, with the constraint that jobs within the same bag are processed on the same machine. By introducing an elaborate way of packing, we have developed an online algorithm based on the rule of the LPT and demonstrated that it is the optimal algorithm with a competitive ratio of 4/3 .
Let H(V, E) be a k-regular connected hypergraph with rank R on n vertices and m edges. A set of vertices S⊆ V is an independent set if every two vertices in S are not adjacent. The independent number is the maximum cardinality of an independent set, denoted by α (H) . In this paper, we prove the following inequality: α (H)≥m-(k-2)n-1/R , and the equality holds if and only if H is a hypertree with R-perfect matching. Based on the proofs, some combinatorial algorithms on the independent number are designed.
In this paper, we tackle the multi-modal globally stable matching problem (MGBSM), a challenge crucial for optimizing industry chain partnerships where agents generate preferences based on various criteria. The goal is to achieve a perfect matching where every agent is paired with a partner, and no unmatched pair prefers each other over their current partners across all criteria. Given the NP-hard nature of this problem, the only exact solution method to date requires exhaustive enumeration of all stable matchings. Our key contribution is the development of efficient algorithms to solve MGBSM, beginning with its transformation into a clique problem in graph theory. By harnessing the properties of both stable matchings and cliques, we design an exact algorithm that efficiently determines the existence of a multi-modal globally stable matching and provides a solution when one exists. Recognizing the limitations of perfect matching, especially in practical industry chain scenarios, we introduce the concept of multi-modal group stable matching (MGSM), which relaxes the perfect matching requirement. We focus on identifying the maximum cardinality MGSM (MaxMGSM) and develop a polynomial-time, approximation-preserving reduction from MaxMGSM to the maximum clique problem. This is followed by a fast polynomial-time approximation algorithm with a provable performance guarantee, offering a practical solution for complex industry chain networks.
This paper studies the location-routing problem in the pallet pooling system. Considering the characteristic of simultaneous delivery and pickup, a mixed integer linear programming model is proposed for the location-routing problem with the limitations on carbon emissions and transportation time in the pallet pooling system. Then, a robust optimization model is proposed to deal with the uncertainty of demand. Considering the computational complexity, a heuristic algorithm combining the genetic algorithm and the ant colony optimization is designed to solve the problem efficiently. Numerical experiments are carried out on classical benchmark instances, and the results validate the feasibility and effectiveness of the proposed model.
In this paper, we consider the min-max heterogeneous weighted delivery problem (the MMHWD problem). Specifically, given a weighted graph G=(V,E;w) with length function w:E→R^+ satisfying the triangle inequality, a fixed depot r∈ V , m items, and k vehicles having nonuniform speeds λ _1 , λ _2 , … , λ _k , each item is initially located at its source vertex s_j and it needs to be delivered to its target vertex t_j , j=1,2, … , m , each vehicle can move along some edges of G and only deliver one item at a time and each item only can be continuously delivered by one vehicle, it is asked to find a set 𝒞={C_1,C_2,… , C_k} of k tours for these k vehicles, each starting and ending at the same depot r, and collectively delivering all items, the objective is to minimize the maximum completion time of vehicles, where the completion time of a vehicle is its total length divided by its speed. We obtain the two main results. (1) Given any small constant δ >0 , we design an 134.4434(1+δ ) -approximation algorithm to solve the MMHWD problem, its time complexity is bounded by a polynomial in the input size and 1/δ ; (2) We provide an (φ +9/5-1/k) -approximation algorithm to resolve the MMHWD problem, where φ is the ratio of the largest vehicle speed to the smallest one.
In this paper, we consider the problem of maximizing a monotone approximate k-submodular function under a knapsack constraint. For monotone approximate k-submodular functions, we characterize the approximately k-submodular degree by k-submodularity ratio and present a γ/1+γ·( 1-e^-1) greedy approximation algorithm under a knapsack constraint.
Unmanned Aerial Vehicles have emerged as the optimal solution for target tracking due to their low cost and high maneuverability. We study the problem of target tracking in partially observable adversarial environments. Building on previous research, we develop a data fusion algorithm to optimize target tracking. To enable a single UAV to track a hard-to-observe target with limited sensor capabilities, we construct a target environment tracking model along with its corresponding reward function. In the algorithm design, we incorporate the Exp4-IX algorithm from the framework of an adversarial multi-armed bandit with advice, and we prove that the regret bound of this algorithm exhibits sub-linear growth. In numerical experiments, the Exp4-IX algorithm integrates the Previous Position algorithm, Particle Filtering algorithm, and Trajectory Fitting algorithm, and is benchmarked against the Average Fusion algorithm. The results demonstrate its effectiveness in online fusion for smooth trajectory scenarios. This integration allows the UAV to predict the target’s position with greater accuracy compared to other algorithms.
This study explores heterogeneous two-facility location mixed mechanisms, aiming to develop an approach for positioning two facilities that ensures agent strategyproofness while minimizing social costs. We introduce a mixed mechanism that achieves an approximation ratio of 25/8 , demonstrating a significant improvement over the latest deterministic mechanism, which has an approximation ratio of 17/4 .
We investigate the equally-split bin packing problem (ESBP), which is inspired by a service placement scenario in cloud computing. In this problem, given a set of items, a set of bins already in use, and an infinite number of new bins, one is asked to pack the items using as few bins as possible. Unlike the traditional bin packing problem, ESBP requires that each item must be split equally into at least two pieces, and the pieces of an item must be placed into different bins. We tackle this problem both theoretically and practically. We prove that ESBP is NP-hard and cannot be approximated within a factor better than 4/3 unless P = NP. On the positive side, we propose two algorithms with asymptotic approximation ratios at most e/(e-1)≈ 1.582 . We also propose a heuristic algorithm, which not only minimizes the number of bins used but also reduces the total number of item pieces. We compared these three algorithms and two other methods on a dataset from Alibaba. The experimental result shows that our heuristic algorithm has the best overall performance.
