Journal of Combinatorial Optimization最新文献

A 3-star is a complete bipartite graph (K_{1,3}). A 3-star packing of a graph G is a collection of vertex-disjoint subgraphs of G in which each subgraph is a 3-star. The maximum 3-star packing problem is to find a 3-star packing of a given graph with the maximum number of 3-stars. A 2-independent set of a graph G is a subset S of V(G) such that for each pair of vertices (u,vin S), paths between u and v are all of length at least 3. In cubic graphs, the maximum 3-star packing problem is equivalent to the maximum 2-independent set problem. The maximum 2-independent set problem was proved to be NP-hard on cubic graphs (Kong and Zhao in Congressus Numerantium 143:65–80, 2000), and the best approximation algorithm of maximum 2-independent set problem for cubic graphs has approximation ratio (frac{8}{15}) (Miyano et al. in WALCOM 2017, Proceedings, pp 228–240). In this paper, we first prove that the maximum 3-star packing problem is NP-hard in claw-free cubic graphs and then design a linear-time algorithm which can find a 3-star packing of a connected claw-free cubic graph G covering at least (frac{3v(G)-8}{4}) vertices, where v(G) denotes the number of vertices of G.

In this work, we introduce a game-theoretic model that assesses the cyber-security risk of cloud networks and informs security experts on the optimal security strategies. Our approach combines game theory, combinatorial optimization, and cyber-security and aims to minimize the unexpected network disruptions caused by malicious cyber-attacks under uncertainty. Methodologically, we introduce the the critical node game, a simultaneous and non-cooperative attacker-defender game where each player solves a combinatorial optimization problem parametrized in the variables of the other player. Each player simultaneously commits to a defensive (or attacking) strategy with limited knowledge about the choices of their adversary. We provide a realistic model for the critical node game and propose an algorithm to compute its stable solutions, i.e., its Nash equilibria. Practically, our approach enables security experts to assess the security posture of the cloud network and dynamically adapt the level of cyber-protection deployed on the network. We provide a detailed analysis of a real-world cloud network and demonstrate the efficacy of our approach through extensive computational tests.

This work deals with the classical Job Shop Scheduling Problem (JSSP) of minimizing the makespan. Metaheuristics are often used on the JSSP solution, but a performance comparable to the state-of-the-art depends on an efficient exploration of the solutions space characteristics. Thus, it is proposed an intensification approach based on the concepts of attraction basins and big valley. Suboptimal solutions obtained by the metaheuristic genetic algorithm are selected and subjected to intensification, in which a binary Bidimensional Genetic Algorithm (BGA) is utilized to enlarge the search neighborhood from a current solution, to escape of attraction basins. Then, the best solution found in this neighborhood is used as the final point of the path relinking strategy derived from the initial suboptimal solution, for exploring possible big valleys. Finally, the best solution in the path is inserted into the population. Trials with usual instances of the literature show that the proposed approach yields greater results with regards to local search, based on permutation of operations on critical blocks, either on the makespan reduction or on the number of generations, and competitive results regarding the contemporary literature.

The online optimization model was first introduced in the research of machine learning problems (Zinkevich, Proceedings of ICML, 928–936, 2003). It is a powerful framework that combines the principles of optimization with the challenges of online decision-making. The present research mainly consider the case that the reveal objective functions are convex or submodular. In this paper, we focus on the online maximization problem under a special objective function (varPhi (x):[0,1]^nrightarrow mathbb {R}_{+}) which satisfies the inequality (frac{1}{2}langle u^{T}nabla ^{2}varPhi (x),urangle le sigma cdot frac{Vert uVert _{1}}{Vert xVert _{1}}langle u,nabla varPhi (x)rangle ) for any (x,uin [0,1]^n, xne 0). This objective function is named as one sided (sigma )-smooth (OSS) function. We achieve two conclusions here. Firstly, under the assumption that the gradient function of OSS function is L-smooth, we propose an ((1-exp ((theta -1)(theta /(1+theta ))^{2sigma })))- approximation algorithm with (O(sqrt{T})) regret upper bound, where T is the number of rounds in the online algorithm and (theta , sigma in mathbb {R}_{+}) are parameters. Secondly, if the gradient function of OSS function has no L-smoothness, we provide an (left( 1+((theta +1)/theta )^{4sigma }right) ^{-1})-approximation projected gradient algorithm, and prove that the regret upper bound of the algorithm is (O(sqrt{T})). We think that this research can provide different ideas for online non-convex and non-submodular learning.

The minimum connected dominating set problem is widely studied due to its applicability to mobile ad-hoc networks and sensor grids. Its variant the minimum 2-connected dominating set (M-2CDS) problem has become increasingly important because its critical role in designing fault-tolerant network. This paper presents a connected dominating degree-based local search (CDD-LS) tailored for solving the M-2CDS. The proposed algorithm implements an improved swap-based neighborhood structure as well as the corresponding fast neighborhood evaluation method using connected dominating degree data structure. The diversification techniques including tabu strategy and perturbaistion help the search jump out of the local optima improving the efficiency. This study investigates the performance of the CDD-LS algorithm on 38 publicly available benchmark datasets. The results demonstrate that the CDD-LS algorithm significantly improves the best runtime in 19 instances, while providing the equivalent performance in 8 instances. Furthermore, the CDD-LS is tested on 18 newly generated instances to check its capability on large-scale scenarios. To gain a deeper understanding of the algorithm’s effectiveness, an investigation into the key components of the CDD-LS algorithm is conducted.

An autonomous robot with a limited vision range finds a path to the goal in an unknown environment in 2D avoiding polygonal obstacles. In the process of discovering the environmental map, the robot has to return to some positions marked previously, the regions where the robot traverses to reach that position are defined as sequences of bundles of line segments. This paper presents a novel algorithm for finding approximately shortest paths along the sequences of bundles of line segments based on the method of multiple shooting. Three factors of the approach including bundle partition, collinear condition, and update of shooting points are presented. We then show that if the collinear condition holds, the exact shortest path of the problem is determined, otherwise, the sequence lengths of paths obtained by the update of the method converges. The algorithm is implemented in Python and some numerical examples show that the running time of path-planing for autonomous robots using our method is faster than that using the rubber band technique of Li and Klette in Euclidean Shortest Paths, Springer, 53–89 (2011).

A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at distance at most 2 from each other receive different colors. Suppose that G is a planar graph with girth 5 and maximum degree (Delta ). We prove that G admits a 2-distance (Delta +7) coloring, which improves the result of Dong and Lin (J Comb Optim 32(2):645–655, 2016). Moreover, we prove that G admits a 2-distance (Delta +6) coloring when (Delta ge 10).

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.