Journal of Combinatorial Optimization最新文献

In this paper, we investigate the minimum resolving dominating set problem which is a emerging combinatorial optimization problem in general graphs. We prove that the resolving dominating set problem is NP-hard and propose a greedy algorithm with an approximation ratio of ((1 + 2ln n)) by establishing a submodular potential function, where n is the node number of the input graph.

We study the fashion game, a classical network coordination/anti-coordination game employed to model social dynamics in decision-making processes, especially in fashion choices. In this game, individuals, represented as vertices in a graph, make decisions based on their neighbors’ choices. Some individuals are positively influenced by their neighbors while others are negatively affected. Analyzing the game’s outcome aids in understanding fashion trends and flux within the population. In an instance of the fashion game, an action profile is formed when all individuals have made their choices. The utility of an individual under an action profile is defined according to the choices he and his neighbors made. A pure Nash equilibria is an action profile under which each individual has a nonnegative utility. To further study the existence of pure Nash equilibria, we investigate an associated optimization problem aimed at maximizing the minimal individual utility, referred to as the utility of a fashion game instance. The fashion game with two different but symmetric actions (choices) has been studied extensively in the literature. This paper seeks to extend the fashion game analysis to scenarios with more than two available actions, thereby enhancing comprehension of social dynamics in decision-making processes. We determine the utilities of all instances on paths, cycles and complete graphs. For instances where each individual likes to anti-coordinate, graph is planar and three actions are available, we illustrate the time complexity of determining the utility of such instances. Additionally, for instances containing both coordinating and anti-coordinating individuals, we extend the results on the time complexity of determining the utility of instances with two available actions to cases with more than two actions.

This paper presents theoretical and practical results for the bin packing problem with scenarios, a generalization of the classical bin packing problem which considers the presence of uncertain scenarios, of which only one is realized. For this problem, we propose approximation algorithms whose ratios are bounded by the square root of the number of scenarios times the approximation ratio for an algorithm for the vector bin packing problem. We also show how an asymptotic polynomial-time approximation scheme is derived when the number of scenarios is constant, that is, not a part of the input. As a practical study of the problem, we present a branch-and-price algorithm to solve an exponential set-cover model and a variable neighborhood search heuristic. Experiments show the competence of the branch-and-price in obtaining optimal solutions for about 59% of the instances considered, while the combined heuristic and branch-and-price optimally solved 62% of the instances considered.

The Maximum Bisection Problem (MBP) is a well-known combinatorial optimization problem that has been proven to be NP-hard. The maximum bisection of a graph is the partition of its set of vertices into two subsets with an equal number of vertices, where the weight of the edge cut is maximal. This work introduces a connected multidimensional generalization of the Maximum Bisection Problem. In this NP-hard problem, weights on edges are vectors of non-negative numbers, and subgraphs induced by partitions must be connected. A mixed integer linear programming (MILP) formulation is proposed with proof of its correctness. The MILP formulation of the problem has a polynomial number of variables and constraints

In this study, we aim to optimize hospital bed allocation to enhance medical service efficiency and quality. We developed an optimization model and algorithms considering cross-departmental bed-sharing costs, patient waiting costs, and the impact on medical quality when patients are assigned to non-primary departments. First, we propose an algorithm to calculate departmental similarity and quantify the effect on patients’ length of stay when admitted to non-primary departments. We then formulate a two-stage cost minimization problem: the first stage involves determining bed allocation for each department, and the second stage involves dynamic admission control decisions. For the second stage, we apply a dynamic programming method and approximate the model using deterministic linear programming to ensure practicality and computational efficiency. Numerical studies validate the effectiveness of our approach. Results show that our model and algorithms significantly improve bed resource utilization and medical service quality, supporting hospital management decisions.

An s, t Hamiltonian path P for an (m times n) rectangular grid graph (mathbb {G}) is a Hamiltonian path from the top-left corner s to the bottom-right corner t. We define an operation “square-switch” on s, t Hamiltonian paths P affecting only those edges of P that lie in some small (2 units by 2 units) square subgrid of (mathbb {G}). We prove that when applied to suitable locations, the result of the square-switch is another s, t Hamiltonian path. Then we use square-switch to achieve a reconfiguration result for a subfamily of s, t Hamiltonian paths we call simple paths, that has the minimum number of bends for each maximal internal subpath connecting any two vertices on the boundary of the grid graph. We give an algorithmic proof that the Hamiltonian path graph (mathcal {G}) whose vertices represent simple paths is connected when edges arise from the square-switch operation: our algorithm reconfigures any given initial simple path P to any given target simple path (P') in (mathcal {O})(( |P |)) time and (mathcal {O})(( |P |)) space using at most ({5} |P |/ {4}) square-switches, where ( |P |= m times n) is the number of vertices in the grid graph (mathbb {G}) and hence in any Hamiltonian path P for (mathbb {G}). Thus the diameter of the simple path graph (mathcal {G}) is at most 5mn/ 4 for the square-switch operation, which we show is asymptotically tight for this operation.

Given a directed graph (G=(V,A)), we tackle the Minimum Feedback Arc Set (MFAS) Problem by designing an efficient algorithm to search for minimal and stable Feedback Arc Sets, i.e. such that none of the arcs can be reintroduced in the graph without disrupting acyclicity and such that for each vertex the number of eliminated outgoing (resp. incoming) arcs is not bigger than the number of remaining incoming (resp. outgoing) arcs. Our algorithm has a good polynomial upper bound and can therefore be applied even on large graphs. We also introduce an algorithm to generate strongly connected graphs with a known upper bound on their feedback arc set, and on such graphs we test our algorithm.

A mixed dominating set in a graph (G=(V,E)) is a subset D of vertices and edges of G such that every vertex and edge in ((Vcup E)setminus D) is a neighbor of some elements in D. The mixed domination number of G, denoted by (gamma _{textrm{md}}(G)), is the minimum size among all mixed dominating sets of G. For natural numbers n and k, where (n > 2k), a generalized Petersen graph P(n, k) is a graph with vertices ( {v_0, v_1, ldots , v_{n-1} }cup {u_0, u_1, ldots , u_{n-1}}) and edges (cup _{0 le i le n-1} {v_{i} v_{i+1}, v_iu_i, u_iu_{i+k}}) where subscripts are modulo n. In this paper, we explicitly construct an optimal mixed dominating set for generalized Petersen graphs P(n, k) for (k in {1, 2}). Moreover, we establish some upper bound on mixed domination number for other generalized Petersen graphs.

Given a monotone submodular set function with a knapsack constraint, its maximization problem has two types of approximation algorithms with running time (O(n^2)) and (O(n^5)), respectively. With running time (O(n^5)), the best performance ratio is (1-1/e). With running time (O(n^2)), the well-known performance ratio is ((1-1/e)/2) and an improved one is claimed to be ((1-1/e^2)/2) recently. In this paper, we design an algorithm with running (O(n^2)) and performance ratio (1-1/e^{2/3}), and an algorithm with running time (O(n^3)) and performance ratio 1/2.

In locally convex spaces, we introduce the new notion of approximate weakly efficient solution of the set-valued optimization problem with variable ordering structures (in short, SVOPVOS) and compare it with other kinds of solutions. Under the assumption of near (mathcal {D}(cdot ))-subconvexlikeness, we establish linear scalarization theorems of (SVOPVOS) in the sense of approximate weak efficiency. Finally, without any convexity, we obtain a nonlinear scalarization theorem of (SVOPVOS). We also present some examples to illustrate our results.