Journal of Combinatorial Optimization最新文献

A network for the transportation of supplies can be described as a rooted tree with a weight of a degree of congestion for each edge. We take the sum of root-leaf distance (SRD) on a rooted tree as the whole degree of congestion of the tree. Hence, we consider the SRD interdiction problem on trees with cardinality constraint by upgrading edges, denoted by (SDIPTC). It aims to maximize the SRD by upgrading the weights of N critical edges such that the total upgrade cost under some measurement is upper-bounded by a given value. The relevant minimum cost problem (MCSDIPTC) aims to minimize the total upgrade cost on the premise that the SRD is lower-bounded by a given value. We develop two different norms including weighted (l_infty ) norm and weighted bottleneck Hamming distance to measure the upgrade cost. We propose two binary search algorithms within O((nlog n)) time for the problems (SDIPTC) under the two norms, respectively. For problems (MCSDIPTC), we propose two binary search algorithms within O((N n^2)) and O((n log n)) under weighted (l_infty ) norm and weighted bottleneck Hamming distance, respectively. These problems are solved through their subproblems (SDIPT) and (MCSDIPT), in which we ignore the cardinality constraint on the number of upgraded edges. Finally, we design numerical experiments to show the effectiveness of these algorithms.

Planar hypergraphs are widely used in several applications, including VLSI design, metro maps, information visualisation, and databases. The minimum ( s-t ) hypercut problem in a weighted hypergraph is to find a partition of the vertices into two nonempty sets, S and ( overline{S} ), with (sin S) and (tin overline{S}) that minimizes the total weight of hyperedges that have at least two endpoints in two different sets. In the present study, we propose an approach that effectively solves the minimum ( s-t ) hypercut problem in (s, t)-planar hypergraphs. The method proposed demonstrates polynomial time complexity, providing a significant advancement in solving this problem. The modelling example shows that the proposed strategy is effective at obtaining balanced bipartitions in VLSI circuits.

Social-media platforms have created new ways for individuals to keep in touch with others, share their opinions and join the discussion on different issues. Traditionally studied by social science, opinion dynamic has attracted the attention from scientists in other fields. The formation and evolution of opinions is a complex process affected by the interplay of different elements that incorporate peer interaction in social networks and the diversity of information to which each individual is exposed. In addition, supplementary information can have an important role in driving the opinion formation and evolution. And due to the character of online social platforms, people can easily end an existing follower-followee relationship or stop interacting with a friend at any time. Taking a step in this direction, we propose the OG–IC model which considers the dynamic of both opinion and relationship in this paper. It not only considers the direct influence of friends but also highlights the indirect effect of group when individuals are exposed to new opinions. And it allows nodes which represent users of social networks to slightly adjust their own opinion and sometimes redefine friendships. A novel problem in social network whose purpose is simultaneously maximizing both the diversity of supplementary information that individuals access to and the influence of supplementary information on individual’s existing opinion is formulated. This problem is proved to be NP-hard and its objective function is neither submodular nor supermodular. However, an algorithm with approximate ratio guarantee is designed based on the sandwich framework. And the effectiveness of our algorithm is experimentally demonstrated on both synthetic and real-world data sets.

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.