Journal of Combinatorial Optimization最新文献

We consider the vertex proper coloring problem for highly restricted instances of geometric intersection graphs of line segments embedded in the plane. Provided a graph in the class UNIT-PURE-k-DIR, corresponding to intersection graphs of unit length segments lying in at most k directions with all parallel segments disjoint, and provided explicit coordinates for segments whose intersections induce the graph, we show for (k = 4) that it is NP-complete to decide if a proper 3-coloring exists, and moreover, (#P)-complete under many-one counting reductions to determine the number of such colorings. In addition, under the more relaxed constraint that segments have at most two distinct lengths, we show these same hardness results hold for finding and counting proper (left( k-1right) )-colorings for every (k ge 5). More generally, we establish that the problem of proper 3-coloring an arbitrary graph with m edges can be reduced in ({mathcal {O}}left( m^2right) ) time to the problem of proper 3-coloring a UNIT-PURE-4-DIR graph. This can then be shown to imply that no (2^{oleft( sqrt{n}right) }) time algorithm can exist for proper 3-coloring PURE-4-DIR graphs under the Exponential Time Hypothesis (ETH), and by a slightly more elaborate construction, that no (2^{oleft( sqrt{n}right) }) time algorithm can exist for counting the such colorings under the Counting Exponential Time Hypothesis (#ETH). Finally, we prove an NP-hardness result for the optimization problem of finding a maximum order proper 3-colorable induced subgraph of a UNIT-PURE-4-DIR graph.

Given a connected graph G, two vertices (u,vin V(G)) doubly resolve (x,yin V(G)) if (d_{G}(x,u)-d_{G}(y,u)ne d_{G}(x,v)-d_{G}(y,v)). The doubly metric dimension (psi (G)) of G is the cardinality of a minimum set of vertices that doubly resolves each pair of vertices from V(G). It is well known that deciding the doubly metric dimension of G is NP-hard. In this work we determine the exact values of doubly metric dimensions of unicyclic graphs which completes the known result. Furthermore, we give formulae for doubly metric dimensions of cactus graphs and block graphs.

In this paper, we propose a physical protocol to verify the first nonzero term of a sequence using a deck of cards. The protocol lets a prover show the value of the first nonzero term of a given sequence to a verifier without revealing which term it is. Our protocol uses (varTheta (1)) shuffles, which is asymptotically lower than that of an existing protocol of Fukusawa and Manabe which uses (varTheta (n)) shuffles, where n is the length of the sequence. We also apply our protocol to construct zero-knowledge proof protocols for three well-known logic puzzles: ABC End View, Goishi Hiroi, and Toichika. These protocols enable a prover to physically show that he/she know solutions of the puzzles without revealing them.

An edge coloring of a graph G is to color all its edges such that adjacent edges receive different colors. It is acyclic if the subgraph induced by any two colors does not contain a cycle. Fiamcik (Math Slovaca 28:139-145, 1978) and Alon et al. (J Graph Theory 37:157-167, 2001) conjectured that every simple graph with maximum degree (Delta ) is acyclically edge ((Delta + 2))-colorable — the well-known acyclic edge coloring conjecture. Despite many major breakthroughs and minor improvements, the conjecture remains open even for planar graphs. In this paper, we prove that planar graphs are acyclically edge ((Delta + 5))-colorable. Our proof has two main steps: Using discharging methods, we first show that every non-trivial planar graph contains a local structure in one of the eight characterized groups; we then deal with each local structure to color the edges in the graph acyclically using no more than (Delta + 5) colors by an induction on the number of edges.

In recent years climate prediction has obtained more attention to mitigate the impact of natural disasters caused by climatic variability. Efficient and effective climate prediction helps palliate negative consequences and allows favourable conditions for managing the resources optimally through proper planning. Due to the environmental, geopolitical and economic consequences, forecasting of atmospheric and oceanic parameters still results in a challenging task. An efficient prediction technique named Sea Lion Autoregressive Deer Hunting Optimization-based Deep Recurrent Neural Network (SLArDHO-based Deep RNN) is developed in this research to predict the oceanic and atmospheric parameters. The extraction of technical indicators makes the devised method create optimal and accurate prediction outcomes by employing the deep learning framework. The classifier uses more training samples and this can be generated by augmenting the data samples using the oversampling method. The atmospheric and the oceanic parameters are considered for the prediction strategy using the Deep RNN classifier. Here, the weights of the Deep RNN classifier are optimally tuned by the SLArDHO algorithm to find the best value based on the fitness function. The devised method obtains minimum mean squared error (MSE), root mean square error (RMSE), mean absolute error (MAE) of 0.020, 0.142, and 0.029 for the All India Rainfall Index (AIRI) dataset.

We study deterministic mechanism design for constrained heterogeneous two-facility location games. The constraint here means that the feasible locations of facilities are specified and the number of facilities that can be built at each feasible location is limited. Given that a set of agents can strategically report their locations on the real line, the authority wants to design strategyproof mechanisms (i.e., mechanisms that can incentivize agents to report truthful private information) to construct two heterogeneous facilities under constraint, while optimizing the corresponding social objectives. Assuming that each agent’s individual objective depends on the sum of her distance to facilities, we consider locating desirable and obnoxious facilities respectively. For the former, we give a deterministic group strategyproof mechanism, which guarantees 3-approximation under the objectives of minimizing the sum cost and the maximum cost. We show that no deterministic strategyproof mechanism can have an approximation ratio of less than 2 under the sum/maximum cost objective. For the latter, we give a deterministic group strategyproof mechanism with 2-approximation under the objectives of maximizing the sum utility and the minimum utility. We show that no deterministic strategyproof mechanism can have an approximation ratio of less than 3/2 under the sum utility objective and 2 under the minimum utility objective, respectively.

Given a graph (G=(V,E)) and a function (r:Vmapsto {0,1,2}), a node (vin V) is said to be Roman dominated if (r(v)=1) or there exists a node (uin N_G[v]) such that (r(u)=2), where ( N_G[v]) is the closed neighbor set of v in G. For (iin {0,1,2}), denote (V_r^i) as the set of nodes with value i under function r. The cost of r is defined to be (c(r)=|V_r^1|+2|V_r^2|). Given a positive integer (Qle |V|), the minimum partial connected Roman dominating set (MinPCRDS) problem is to compute a minimum cost function r such that at least Q nodes in G are Roman dominated and the subgraph of G induced by (V_r^1cup V_r^2) is connected. In this paper, we give a ((3ln |V|+9))-approximation algorithm for the MinPCRDS problem.

Let (G=(V, E)) be a graph that represents an underlying network. Let (tau ) (resp. ({textbf{p}})) be an assignment of non-negative integers as thresholds (resp. incentives) to the vertices of G. The discrete time activation process with incentives corresponding to ((G, tau , {textbf{p}})) is the following. First, all vertices u with ({textbf{p}}(u)ge tau (u)) are activated. Then at each time t, every vertex u gets activated if the number of previously activated neighbors of u plus ({textbf{p}}(u)) is at least (tau (v)). The optimal target vector problem (OTV) is to find the minimum total incentives ({sum }_{vin V} {textbf{p}}(v)) that activates the whole network. We extend this model of activation with incentives, for graphs with weighted edges such that the spread of activation in the network depends on the weight of influence between any two participants. The new version is more realistic for the real world networks. We first prove that the new problem OTVW, is (texttt {NP})-complete even for the complete graphs. Two lower bounds for the minimum total incentives are presented. Next, we prove that OTVW has polynomial time solutions for (weighted) path and cycle graphs. Finally, we extend the discussed model and OTV, for bi-directed graphs with weighted edges and prove that to obtain the optimal target vector in weighted bi-directed paths and cycles has polynomial time solutions.

In this paper, we study approximation algorithms for the problem of maximum weighted target cover with distance limitations (MaxWTCDL). Given n targets (T=left{ t_{1},t_{2},ldots ,t_{n}right} ) on the plane and m mobile sensors (S=left{ s_{1},s_{2},ldots ,s_{m}right} ) randomly deployed on the plane, each target (t_i) has a weight (w_{i}) and the sensing radius of the mobile sensors is (r_{s}), suppose there is a movement distance constraint b for each sensor and a total movement distance constraint B, where (B>b), the goal of MaxWTCDL is to move the mobile sensors within the distance constraints b and B to maximize the weight of covered targets. We present two polynomial time approximation algorithms. One is greedy-based, achieving approximation ratio (frac{1}{2v}) in time (O(mn^2)), where . The other is LP-based, achieving approximation ratio (frac{1}{v}(1-e^{-1})) in time (T_{LP}), where (T_{LP}) is the time needed to solve the linear program.

For the issue of carbon emission mitigation within the automotive supply chain, the cooperation between the vehicle manufacturers and the retailers has been proved to be an efficient measure to enhance emission reduction endeavors. This paper aims to evaluate the effectiveness of the cooperations between a vehicle manufacturer and multiple retailers based on the differential game method. By utilizing the Hamilton–Jacobi–Bellman equation, the equilibrium strategies of the participants under two different cooperation models, i.e., the decentralized model and the Stackelberg leader–follower cooperation model, are analyzed. To be specific, in the decentralized model, each participant independently decides its strategies, whereas the manufacturer cooperates with retailers by offering subsidies in the Stackelberg leader–follower model. Unlike previous studies that solely focused on participants’ decision-making in carbon emission reduction efforts, this paper also examines the retail pricing decisions of the retailers. Additionally, carbon trading is introduced to enhance the realism of our model. Through the theoretical analysis and the numerical experiments on the carbon emission reduction efforts of manufacturers and retailers, as well as the low-carbon reputation of vehicles and the overall system profit under both models, we conclude that the cooperative Stackelberg model outperforms the decentralized model in providing benefits to both parties. Furthermore, such a cooperative approach can foster the long-term development of the automotive supply chain, ultimately contributing to a more sustainable low-carbon future.