Journal of Combinatorial Optimization最新文献

A (p, 1)-total labelling of a graph G is a mapping f: (V(G)cup E(G)) (rightarrow ) ({0, 1, cdots , k}) such that (|f(u)-f(v)|ge 1) if (uvin E(G)), (|f(e_1)-f(e_2)|ge 1) if (e_1) and (e_2) are two adjacent edges in G and (|f(u)-f(e)|ge p) if the vertex u is incident with the edge e. In this paper, we focus on the list version of a (p, 1)-total labelling. Given a family (L={L(u)subseteq mathbb {N}:uin V(G)cup E(G)}), an L-list (p, 1)-total labelling of G is a (p, 1)-total labelling f of G such that (f(u)in L(u)) for every element (uin V(G)cup E(G)). A graph G is said to be (p, 1)-k-total choosable if it admits an L-list (p, 1)-total labelling whenever the family L contains only sets of size at least k. The smallest k for which a graph G is (p, 1)-k-total choosable is the list (p, 1)-total labelling number of G, denoted by (lambda _{lp}^T(G)). In this paper, we firstly use some important theorems related to Combinatorial Nullstellensatz to prove that the upper bound of (lambda _{lp}^T(C_n)) for cycles (C_n) is (2p+1) with (pge 2). Let G be a graph with maximum degree (Delta (G)ge 6p+3). Then we prove that if G is a planar graph or a 1-planar graph without adjacent 3-cycles, then (lambda _{lp}^T(G)le Delta (G)+2p-1) ((pge 2)).

This paper introduces an integer linear program for a variant of the linear ordering problem. This considers, besides the pairwise preferences in the objective function as the linear ordering problem, positional preferences (weighted rank) in the objective. The objective function is mathematically supported, as the full integer linear program is motivated by the instant run-off voting method to aggregate individual preferences. The paper describes two meta-heuristics, iterated local search and Memetic algorithms to deal with large instances which are hard to solve to optimality. These results are compared with the objective value of the linear relaxation. The instances used are the ones available from the LOP library, and new real instances with preferences given by juries.

This work is to discuss the fault-tolerant facility location problem with submodular penalties. We propose an LP-rounding 2.27-approximation algorithm, where every demand point j has a requirement that (t_{j}) distinct facilities serve it. This is the first constant performance guarantee known for this problem. In addition, we give an LP-rounding 2-approximation algorithm for the case where all requirements are the same.

It is well-known that the classical Johnson’s Rule leads to optimal schedules on a two-stage flowshop. However, it is still unclear how Johnson’s Rule would help in approximation algorithms for scheduling an arbitrary number of parallel two-stage flowshops with the objective of minimizing the makespan. Thus within the paper, we study the problem and propose a new efficient algorithm that incorporates Johnson’s Rule applied on each individual flowshop with a carefully designed job assignment process to flowshops. The algorithm is successfully shown to have a runtime (O(n log n)) and an approximation ratio 7/3, where n is the number of jobs. Compared with the recent PTAS result for the problem (Dong et al. in Eur J Oper Res 218(1):16–24, 2020), our algorithm has a larger approximation ratio, but it is more efficient in practice from the perspective of runtime.

Identifying certain conditions that ensure the Hamiltonicity of graphs is highly important and valuable due to the fact that determining whether a graph is Hamiltonian is an NP-complete problem.For a graph G with vertex set V(G) and edge set E(G), the first Zagreb index ((M_{1})) and second Zagreb index ((M_{2})) are defined as (M_{1}(G)=sum limits _{v_{i}v_{j}in E(G)}(d_{G}(v_{i})+d_{G}(v_{j}))) and (M_{2}(G)=sum limits _{v_{i}v_{j}in E(G)}d_{G}(v_{i})d_{G}(v_{j})), where (d_{G}(v_{i})) denotes the degree of vertex (v_{i}in V(G)). The difference of Zagreb indices ((Delta M)) of G is defined as (Delta M(G)=M_{2}(G)-M_{1}(G)).In this paper, we try to look for the relationship between structural graph theory and chemical graph theory. We obtain some sufficient conditions, with regards to (Delta M(G)), for graphs to be k-hamiltonian, traceable, k-edge-hamiltonian, k-connected, Hamilton-connected or k-path-coverable.

We consider the non-preemptive scheduling problem on identical machines where there is a parameter B and each machine in every unit length time interval can process up to B different jobs. The goal function we consider is the makespan minimization and we develop an EPTAS for this problem. Prior to our work a PTAS was known only for the case of one machine and constant values of B, and even the case of non-constant values of B on one machine was not known to admit a PTAS.

Given a graph G with a terminal set (R subseteq V(G)), the Steiner tree problem (STREE) asks for a set (Ssubseteq V(G) {setminus } R) such that the graph induced on (Scup R) is connected. A split graph is a graph which can be partitioned into a clique and an independent set. It is known that STREE is NP-complete on split graphs White et al. (Networks 15(1):109–124, 1985). To strengthen this result, we introduce convex ordering on one of the partitions (clique or independent set), and prove that STREE is polynomial-time solvable for tree-convex split graphs with convexity on clique (K), whereas STREE is NP-complete on tree-convex split graphs with convexity on independent set (I). We further strengthen our NP-complete result by establishing a dichotomy which says that for unary-tree-convex split graphs (path-convex split graphs), STREE is polynomial-time solvable, and NP-complete for binary-tree-convex split graphs (comb-convex split graphs). We also show that STREE is polynomial-time solvable for triad-convex split graphs with convexity on I, and circular-convex split graphs. Further, we show that STREE can be used as a framework for the dominating set problem (DS) on split graphs, and hence the classical complexity (P vs NPC) of STREE and DS is the same for all these subclasses of split graphs. Finally, from the parameterized perspective with solution size being the parameter, we show that the Steiner tree problem on split graphs is W[2]-hard, whereas when the parameter is treewidth, we show that the problem is fixed-parameter tractable, and if the parameter is the solution size and the maximum degree of I (d), then we show that the Steiner tree problem on split graphs has a kernel of size at most ((2d-1)k^{d-1}+k,~k=|S|).

The packing number of a signed graph ((G, sigma )), denoted (rho (G, sigma )), is the maximum number l of signatures (sigma _1, sigma _2,ldots , sigma _l) such that each (sigma _i) is switching equivalent to (sigma ) and the sets of negative edges (E^{-}_{sigma _i}) of ((G,sigma _i)) are pairwise disjoint. A signed graph packs if its packing number is equal to its negative girth. A reformulation of some well-known conjecture in extension of the 4-color theorem is that every antibalanced signed planar graph and every signed bipartite planar graph packs. On this class of signed planar graph the case when negative girth is 3 is equivalent to the 4-color theorem. For negative girth 4 and 5, based on the dual language of packing T-joins, a proof is claimed by B. Guenin in 2002, but never published. Based on this unpublished work, and using the language of packing T-joins, proofs for girth 6, 7, and 8 are published. We have recently provided a direct proof for girth 4 and in this work extend the technique to prove the case of girth 5.

We study a generalization of the classical Hamiltonian path problem, where multiple salesmen are positioned at the same depot, of which no more than k can be selected to service n destinations, with the objective to minimize the total travel distance. Distances between destinations (and the single depot) are assumed to satisfy the triangle inequality. We develop a non-trivial extension of the well-known Christofides heuristic for this problem, which achieves an approximation ratio of (2-1/(2+k)) with (O(n^3)) running time for arbitrary (kge 1).

Shuffled Frog leaping algorithm (SFLA) is a multi population swarm intelligence algorithm which employs population partitioning techniques during the evolutionary stage. Methods adopted by SFLA for partitioning the population into memeplexes play a critical role in determining its ability to solve complex optimization problems. However, limited research is done in this direction. This work presents supervised machine learning based methods Spectral Partitioning (SCP), Agglomerative Partitioning (AGP) and Ward Hierarchical Partitioning (WHP) for distributing the solutions into memeplexes. The efficacy of variants of SFLA with these methods is assessed over CEC2015 Bound Constrained Single-Objective Computationally Expensive Numerical Optimisation problems. Analysis of results establishes that proposed SCP, AGP and WHP methods outperform Shuffled complex evolution (SCE) partitioning technique; Seed and distance based partitioning technique (SEED), Random partitioning (RAND) and Dynamic sub-swarm partitioning (DNS) for more than 10 functions. Time complexity of all the algorithms is comparable with each other. Statistical analysis using Wilcoxon signed rank sum test indicates that SCP, AGP and WHP perform significantly better than existing approaches for small dimensions.