Journal of Combinatorial Optimization最新文献

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.

Budget games were introduced by Drees, Riechers, and Skopalik (2014) as a model of noncooperative games arising from resource allocation problems. Budget games have several similarities to congestion games, one of which is that the matroid structure of the strategy space is essential for the existence of a pure Nash equilibrium (PNE). Despite these similarities, however, the theoretical relation between budget games and congestion games has been unclear. In this paper, we provide a common generalization of budget games and congestion games, called generalized budget games (g-budget games, for short), to establish a large class of noncooperative games retaining the nice property of the matroid structure. We show that the model of g-budget games includes weighted congestion games and player-specific congestion games under certain assumptions. We further show that g-budget games also include offset budget games, a generalized model of budget games by Drees, Feldotto, Riechers, and Skopalik (2019). We then prove that every matroid g-budget game has a PNE, which extends the result for budget games. We finally a PNE in a certain class of singleton g-budget games can be computed in a greedy manner.

Let (G=(V,E)) be a graph. For an edge (e=xyin E), the closed neighbourhood of e, denoted by (N_G[e]) or (N_G[xy]), is the set (N_G[x]cup N_G[y]). A vertex set (Lsubseteq V) is liar’s vertex-edge dominating set of a graph (G=(V,E)) if for every (e_iin E), (|N_G[e_i]cap L|ge 2) and for every pair of distinct edges (e_i) and (e_j), (|(N_G[e_i]cup N_G[e_j])cap L|ge 3). This paper introduces the notion of liar’s vertex-edge domination which arises naturally from some applications in communication networks. Given a graph G, the Minimum Liar’s Vertex-Edge Domination Problem (MinLVEDP) asks to find a liar’s vertex-edge dominating set of G of minimum cardinality. In this paper, we study this problem from an algorithmic point of view. We show that MinLVEDP can be solved in linear time for trees, whereas the decision version of this problem is NP-complete for general graphs, chordal graphs, and bipartite graphs. We further study approximation algorithms for this problem. We propose two approximation algorithms for MinLVEDP in general graphs and p-claw free graphs. On the negative side, we show that the MinLVEDP cannot be approximated within (frac{1}{2}(frac{1}{8}-epsilon )ln |V|) for any (epsilon >0), unless (NPsubseteq DTIME(|V|^{O(log (log |V|)})). Finally, we prove that the MinLVEDP is APX-complete for bounded degree graphs and p-claw-free graphs for (pge 6).

In this paper, we consider the W-prize-collecting scheduling problem on parallel machines. In this problem, we are given a set of n jobs, a set of m identical parallel machines and a value W. Each job (J_j) has a processing time, a profit and a rejection penalty. Each job is either accepted and processed on one of the machines without preemption, or rejected and paid a rejection penalty. The objective is to minimize the sum of the makespan of accepted jobs and the penalties of rejected jobs, and at the same time the total profit brought by accepted jobs is not less than W. We design a 2-approximation algorithm for the problem based on the greedy method and the list scheduling algorithm.

We study scheduling problems with release times and rejection costs with the objective function of minimizing the maximum lateness. Our main result is a PTAS for the single machine problem with an upper bound on the rejection costs. This result is extended to parallel, identical machines. The corresponding problem of minimizing the rejection costs with an upper bound on the lateness is also examined. We show how to compute a PTAS for determining an approximation of the Pareto frontier on both objective functions on parallel, identical machines. Moreover, we present an FPTAS with strongly polynomial time for the maximum lateness problem without release times on identical machines when the number of machines is constant. Finally, we extend this FPTAS to the case of unrelated machines.

Considering fairness has become increasingly important in recent research. This paper proposes the prize-collecting vertex cover problem with fairness constraints (FPCVC). In a prize-collecting vertex cover problem, those edges that are not covered incur penalties. By adding fairness concerns into the problem, the vertex set is divided into l groups, the goal is to find a vertex set to minimize the cost-plus-penalty value under the constraints that the profit of edges collected by each group exceeds a coverage requirement. In this paper, we propose a hybrid algorithm (combining deterministic rounding and randomized rounding) for the FPCVC problem which, with probability at least (1-1/l^{alpha }), returns a feasible solution with an objective value at most (left( frac{9(alpha +1)}{2}ln l+3right) ) times that of an optimal solution, where (alpha ) is a constant. We also show a lower bound of (Omega (ln l)) for the approximability of FPCVC. Thus, our approximation ratio is asymptotically best possible. Experiments show that our algorithm performs fairly well empirically.

A floorplan (F) is a partition of a polygonal boundary (P) into n-regions satisfying the adjacencies given by an n-vertex graph. Here, it is assumed that the sides of the polygonal boundary are either parallel to the x-axis or y-axis or have slopes (-1) or 1. For a given polygonal boundary P (having m line segments) and a plane triangulated graph G, this paper presents a linear-time algorithm for constructing a floorplan with the required polygonal boundary satisfying all given adjacencies. Further, it has been proved that the number of sides of each region in the obtained floorplan (F) is at most m + 1 (except one region, which can have at most m + 5 sides) for the given polygonal boundary P of length m.

The problem of matroid-based packing of arborescences was introduced and solved in Durand de Gevigney et al. (SIAM J Discret Math 27(1):567-574) . Frank (In personal communication) reformulated the problem in an extended framework. We proved in Fortier et al. (J Graph Theory 93(2):230-252) that the problem of matroid-based packing of spanning arborescences is NP-complete in the extended framework. Here we show a characterization of the existence of a matroid-based packing of spanning arborescences in the original framework. This leads us to the introduction of a new problem on packing of arborescences with a new matroid constraint. We characterize mixed graphs having a matroid-rooted, k-regular, (f, g)-bounded packing of mixed arborescences, that is, a packing of mixed arborescences such that their roots form a basis in a given matroid, each vertex belongs to exactly k of them and each vertex v is the root of least f(v) and at most g(v) of them. We also characterize dypergraphs having a matroid-rooted, k-regular, (f, g)-bounded packing of hyperarborescences.