Journal of Combinatorial Optimization最新文献

Path cover is one of the well-known NP-hard problems that has received much attention. In this paper, we study a variant of path cover, denoted by (hbox {MPC}^{{4}+}_v), to cover as many vertices in a given graph (G = (V, E)) as possible by a collection of vertex-disjoint paths each of order four or above. The problem admits an existing (O(|V|^8))-time 2-approximation algorithm by applying several time-consuming local improvement operations (Gong et al.: Proceedings of MFCS 2022, LIPIcs 241, pp 53:1–53:14, 2022). In contrast, our new algorithm uses a completely different method and it is an improved (O(min {|E|^2|V|^2, |V|^5}))-time 1.874-approximation algorithm, which answers the open question in Gong et al. (2022) in the affirmative. An important observation leading to the improvement is that the number of vertices in a maximum matching M of G is relatively large compared to that in an optimal solution of (hbox {MPC}^{{4}+}_v). Our new algorithm forms a feasible solution of (hbox {MPC}^{{4}+}_v) from a maximum matching M by computing a maximum-weight path-cycle cover in an auxiliary graph to connect as many edges in M as possible.

In this paper, we consider the heterogeneous rooted tree cover (HRTC) problem, which further generalizes the rooted tree cover problem. Specifically, given a complete graph (G=(V,E; w,f; r)) and k construction teams, having nonuniform construction speeds (lambda _{1}), (lambda _{2}), (ldots ), (lambda _{k}), where (rin V) is a fixed common root, (w:Erightarrow {mathbb {R}}^{+}) is an edge-weight function, satisfying the triangle inequality, and (f:Vrightarrow {mathbb {R}}^{+}_{0}) (i.e., ({mathbb {R}}^{+}cup {0})) is a vertex-weight function with (f(r)=0), we are asked to find k trees for these k construction teams, each tree having the same root r, and collectively covering all vertices in V, the objective is to minimize the maximum completion time of k construction teams, where the completion time of each team is the total construction weight of its related tree divided by its construction speed. In addition, substituting k paths for k trees in the HRTC problem, we also consider the heterogeneous rooted path cover (HRPC) problem. Our main contributions are as follows. (1) Given any small constant (delta >0), we first design a (58.3286(1+delta ))-approximation algorithm to solve the HRTC problem, and this algorithm runs in time (O(n^{2}(n+frac{log n}{delta })+log (w(E)+f(V)))). Meanwhile, we present a simple (116.6572(1+delta ))-approximation algorithm to solve the HRPC problem, whose time complexity is the same as the preceding algorithm. (2) We provide a (max {2rho , 2+rho -frac{2}{k}})-approximation algorithm to resolve the HRTC problem, and that algorithm runs in time (O(n^{2})), where (rho ) is the ratio of the largest team speed to the smallest one. At the same time, we can prove that the preceding (max {2rho , 2+rho -frac{2}{k}})-approximation algorithm also resolves the HRPC problem.

In real-life production, the cutting stock problem is often associated with additional constraints and objectives. Among the auxiliary objectives, two of the most relevant are the minimization of the number of different cutting patterns used and the minimization of the maximum number of simultaneously open stacks. The first auxiliary objective arises in manufacturing environments where the adjustment of the cutting tools when changing the cutting patterns incurs increased costs and time spent in production. The second is crucial to face scenarios where the space near the cutting machine or the number of automatic unloading stations is limited. In this paper, we address the one-dimensional cutting stock problem, considering the additional goals of minimizing the number of different cutting patterns used and the maximum number of simultaneously open stacks. We propose two Integer Linear Programming (ILP) formulations and a Constraint Programming (CP) model for the problem. Moreover, we develop new upper bounds on the frequency of the cutting patterns in a solution and address some special cases in which the problem may be simplified. All three approaches are embedded into an iterative exact framework to find efficient solutions. We perform computational experiments using two sets of instances from the literature. The proposed approaches proved effective in determining the entire Pareto front for small problem instances, and several solutions for medium-sized instances with minimum trim loss, a reduced maximum number of simultaneously open stacks, and a small number of different used cutting patterns.

We study the problem of how to fairly and efficiently allocate indivisible items (goods) to agents under budget constraints. Each item has a specific size, and each agent has a budget that limits the total size of the items received. To better explore efficiency, we introduce the concept of tightness, where all agents are tight. An agent is considered as tight if adding any unallocated item to her bundle would exceed her budget. Interestingly, we observe that all individual optimal (IO) allocations, which contain Pareto optimal (PO) allocations, can be extended into a tight allocation while maintaining the values of the agents’ bundles. We achieve an overall negative result for general even identical or binary valuations: there exists no allocation meeting both tightness and envy-freeness up to any item (EFX), and even relaxing it to any desired approximate EFX has been proven to be impossible. However, for single-valued valuations, we illustrate that an EFX and tight (or IO) allocation always exist, and it can be computed using a polynomial algorithm. For single-valued valuations, we establish the existence of 1/2-EFX and PO allocations, with the approximation ratio being the best possible. To further our efforts to study fairness and efficiency, we introduce a relaxed concept of tightness, partial tightness (PT), in which only the unenvied agents are tight. We find that 1/2-EFX and PT allocations are achievable by providing a pseudo-polynomial time algorithm. When agents’ budgets are identical, we can compute a 1/2-EFX and PT allocation in polynomial time.

This paper investigates the superposition problem of two or more individual semi-Markov decision processes (SMDPs). The new sequential decision process superposed by individual SMDPs is no longer an SMDP and cannot be handled by routine iterative algorithms, but we can expand its state spaces to obtain a hybrid-state SMDP. Using this hybrid-state SMDP as an auxiliary and inspired by the Robbins–Monro algorithm underlying the reinforcement learning method, we propose an iteration algorithm based on a combination of dynamic programming and reinforcement learning to numerically solve the superposed sequential decision problem. As an illustration example, we apply our superposition model and algorithm to solve the optimal maintenance problem of a two-component independent parallel system.

Sustainable scheduling is getting more and more attention with economic globalization and sustainable manufacturing. However, fewer studies on the batch scheduling problem consider energy consumption. This paper conducts an investigation into the multi-objective hybrid flow shop batch-scheduling problem with the objectives of minimizing both the makespan and electrical energy consumption. The study aims to select the optimal scheduling solution for the problem by considering batch splitting for all products. In this paper, we propose an improved black widow optimization (IBWO) algorithm to study the problem, which incorporates procreation, cannibalism, and mutation behaviors to maintain the population’s diversity and stability. To achieve our objectives, we use the dynamic entropy weight topsis method to select individual spiders. Finally, we use the nature theorem construction method, which relies on the property theorem, to solve the Pareto solution set and derive the optimization scheme for the hybrid flow shop batch scheduling problem. We verify the effectiveness of the proposed IBWO on instances of varying sizes. When we keep all other factors and cases constant, we compare the IBWO to the NSGA2 algorithm and find that it converges faster for both goals and has lower goals than the NSGA2.

The Euclidean Steiner problem is the problem of finding a set (mathcal{S}mathcal{t}), with the shortest length, such that (mathcal{S}mathcal{t}cup mathcal {A}) is connected, where (mathcal {A}) is a given set in a Euclidean space. The solutions (mathcal{S}mathcal{t}) to the Steiner problem will be called Steiner sets while the set (mathcal {A}) will be called input. Since every Steiner set is acyclic we call it Steiner tree in the case when it is connected. We say that a Steiner tree is indecomposable if it does not contain any Steiner tree for a subset of the input. We are interested in finding the Steiner set when the input consists of infinitely many points distributed on two lines. In particular we would like to find a configuration which gives an indecomposable Steiner tree. It is natural to consider a self-similar input, namely the set (mathcal {A}_{alpha ,lambda }) of points with coordinates ((lambda ^{k-1}cos alpha ,) (pm lambda ^{k-1}sin alpha )), where (lambda >0) and (alpha >0) are small fixed values and (k in mathbb {N}). These points are distributed on the two sides of an angle of size (2alpha ) in such a way that the distances from the points to the vertex of the angle are in a geometric progression. To our surprise, we show that in this case the solutions to the Steiner problem for (mathcal {A}_{alpha ,lambda }), when (alpha ) and (lambda ) are small enough, are always decomposable trees. More precisely, any Steiner tree for (mathcal {A}_{alpha ,lambda }) is a countable union of Steiner trees, each one connecting 5 points from the input. Each component of the decomposition can be mirrored with respect to the angle bisector providing (2^{mathbb N}) different solutions with the same length. By considering only a finite number of components we obtain many solutions to the Steiner problem for finite sets composed of (4k+1) points distributed on the two lines ((2k+1) on a line and 2k on the other line). These solutions are very similar to the ladders of Chung and Graham. We are able to obtain an indecomposable Steiner tree by adding, to the previous input, a single point strategically placed inside the angle. In this case the solution is in fact a self-similar tree (in the sense that it contains a homothetic copy of itself). Finally, we show how the position of the Steiner points in the Steiner tree can be described by a discrete dynamical system which turns out to be equivalent to a 2-interval piecewise linear contraction.

This paper studies the inefficiency of multiplicative approximate Nash Equilibrium for scheduling games. There is a set of machines and a set of jobs. Each job could choose one machine and be processed by the chosen one. A schedule is a (theta )-NE if no player has the incentive to deviate so that it decreases its cost by a factor larger than (1+theta ). The (theta )-NE is a generation of Nash Equilibrium and its inefficiency can be measured by the (theta )-PoA, which is also a generalization of the Price of Anarchy. For the game with the social cost of minimizing the makespan, the exact (theta )-PoA for any number of machines and any (theta ge 0) is obtained. For the game with the social cost of maximizing the minimum machine load, we present upper and lower bounds on the (theta )-PoA. Tight bounds are provided for cases where the number of machines is between 2 and 7 and for any (theta ge 0).

In this paper, we investigate the data mule scheduling with handling time and time span constraints (DMSTC) in which the goal is to minimize the number of data mules dispatched from a depot that are used to serve target sensors located on a wireless sensor network. Each target sensor is associated with a handling time and each dispatched data mule must return to the original depot before time span (D). We also study a variant of the DMSTC, denoted by DMSTC(_l) in which the objective is to minimize the total travel distance of the data mules dispatched. We give exact and approximation algorithms for the DMSTC/DMSTC(_l) on a path and their multi-depot version. For the DMSTC, we show an (O(n^4)) polynomial time algorithm for the uniform 2-depot DMSTC on a path with at least one depot being on the endpoint of the path, where (n) indicates the number of target sensors and an instance of the DMSTC is called uniform if all the handling times are identical. We present a new 2-approximation algorithm for the non-uniform DMSTC on a path and conduct extensive computational experiments on randomly generated instances to show its good practical performance. For the DMSTC(_l), we derive an (O((n+k)^{2}))-time algorithm for the uniform multi-depot DMSTC(_l) on a path, where (k) is the number of depots. For the non-uniform multi-depot DMSTC(_l) on a path or cycle, we give a 2-approximation algorithm.

In the production scheduling of prefabricated components, a scheduling model considering the learning effect of processing time and the deterioration effect of delivery time in this paper is provided. More precisely, it asks for an assignment of a series of independent prefabricated jobs that arrived over time to a single machine for processing, and once the execution of a job is finished, it will be transported to the destination. The information of each prefabricated job including its basic processing time (b_{j}), release time (r_j), and deterioration rate (e_j) of delivery time is unknown in advance and is revealed upon the arrival of this job. Moreover, the actual processing time of prefabricated job (J_j) with learning effect is (p_{j}=b_{j}(a-b t)), where a and b are non-negative parameters and t denotes the starting time of prefabricated job (J_j), respectively. And the delivery time of prefabricated job (J_j) is (q_{j}=e_{j}C_{j}). The goal of scheduling is to minimize the maximum time by which all jobs have been delivered. For the problem, we first analyze offline optimal scheduling and then propose an online algorithm with a competitive ratio of (2-bb_{min }). Furthermore, the effectiveness of the online algorithm is demonstrated by numerical experiments and managerial insights are derived.