Journal of Combinatorial Optimization最新文献

Analysis of higher-order organizations, represented as small connected subgraphs, is a fundamental task on complex networks. This paper studies a new problem of testing higher-order clusterability: given neighbor query access to an undirected graph, can we judge whether this graph can be partitioned into a few clusters of highly-connected cliques? This problem is an extension of the former work proposed by Czumaj et al. (STOC’ 15), who recognized cluster structure on graphs using the framework of property testing. In this paper, the problem of testing whether a well-defined higher-order cluster exists is first defined. Then, an (varOmega (sqrt{n})) query lower bound of this problem is given. After that, a baseline algorithm is provided by an edge-cluster tester on k-clique dual graph. Finally, an optimized (tilde{O}(sqrt{n}))-time algorithm is developed for testing clusterability based on triangles.

In this paper, we consider dynamic facility location problem with unit demand (DFLPUD). We propose a 1.52-approximation algorithm that skillfully integrates dual-fitting and greedy augmentation schemes. Our algorithmic framework begins by formulating DFLPUD as a set covering linear integer programming problem. Then we scale the opening cost of all facilities and use the solution of dual-fitting algorithm to seed a local search to yield an improved performance guarantee 1.52. To the best of our knowledge, this is the best known approximation ratio for DFLPUD.

In a hypergraph H(V, E), a subset of edges (Asubseteq E) forms an edge dominating set if each edge (ein Esetminus A) is adjacent to at least one edge in A. The edge dominating number (gamma '(H)) represents the smallest size of an edge dominating set in H. In this paper, we establish upper bounds on the edge dominating number for hypergraphs with minimum degree (delta ): (1) For (delta le 4), (gamma '(H)le frac{m}{delta }); (2) For (delta ge 5), (gamma '(H)le frac{m}{delta }) holds for hypertrees and uniform hypergraphs; (3) For a random hypergraph model (mathcal H(n,m)), for any positive number (varepsilon >0), (gamma ' (H)le (1+varepsilon )frac{m}{delta }) holds with high probability when m is bounded by some polynomial function of n. Based on the proofs, some combinatorial algorithms on the edge dominating number of hypergraphs with minimum degree are designed.

High-performance computing extensively depends on parallel and distributed systems, necessitating the establishment of quantitative parameters to evaluate the fault tolerability of interconnection networks. The topological structures of interconnection networks in some parallel and distributed systems are designed as n-dimensional ((K_{9}-C_{9})^{n}), obtained through the repeatedly application of the n-th Cartesian product operation. Since the (mathcal {P})-conditional edge-connectivity is proposed by Harary, as a parameter for evaluating the link fault tolerability of the underlying topology graph of the interconnection network system, it has been widely studied in many interconnection networks. The (mathcal {P})-conditional edge-connectivity of a connected graph G, denoted by (lambda (mathcal {P};G)), if any, describes the minimum cardinality of the fault edge-cut of the graph G, whose malfunction divides G into multiple components, with each component satisfying a given property (mathcal {P}) of the graph. In this paper, we primarily define (mathcal {P}_{i}^{t}) to be properties of containing at least (9^t) processors, every remaining processor lying in a lower dimensional subnetwork of the ((K_{9}-C_{9})^{n}), ((K_{9}-C_{9})^{t}), having a minimum degree or average degree of at least 6t, existing two components with each component having at least (9^t) processors, and containing at least one cycle, respectively. We use the properties of the optimal solution to the edge isoperimetric problem of ((K_{9}-C_{9})^{n}) and find that the exact values of the (mathcal {P}_{i})-conditional edge-connectivities of the graph ((K_{9}-C_{9})^{n}) share a common value of (6(n-t)9^t) for (1le ile 5) and (0le tle n-1), except for (i=6), the value is (18n - 6).

It’s a promising way to use Unmanned Aerial Vehicles (UAVs) as mobile base stations to collect data from sensor nodes, especially for large-scale wireless sensor networks. There are a lot of works that focus on improving the freshness of the collected data or the data collection efficiency by scheduling UAVs. Given that sensing data in certain applications is time-sensitive, with its value diminishing as time progresses based on Timeliness of Information (ToI), this paper delves into the UAV Trajectory optimization problem for Maximizing the ToI-based data utility (TMT). We give the formal definition of the problem and prove its NP-Hardness. To solve the TMT problem, we propose a deep reinforcement learning-based algorithm that combines the Action Rejection Mechanism and the Deep Q-Network with Priority Experience Replay (ARM-PER-DQN). Where the action rejection mechanism could reduce the action space and PER helps improve the utilization of experiences with high value, thus increasing the training efficiency. To avoid the unbalanced data collection problem, we also investigate a variant problem of TMT (named V-TMT), i.e., each sensor node can be visited by the UAV at most once. We prove that the V-TMT problem is also NP-Hard, and propose a 2-approximation algorithm as the baseline of the ARM-PER-DQN algorithm. We conduct extensive simulations for the two problems to validate the performance of our designs, and the results show that our ARM-PER-DQN algorithm outperforms other baselines, especially in the V-TMT problem, the ARM-PER-DQN algorithm always outperforms the proposed 2-approximation algorithm, which suggests the effectiveness of our algorithm.

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.