text: Let c denote a non-negative constant. Suppose that we are given an edge-weighted bipartite graph G=(V,E) with its 2-layered drawing and a family X of intersecting edge pairs. We consider the problem of finding a maximum weighted matching M* such that each edge in M* intersects with at most c other edges in M*, and that all edge crossings in M* are contained in X. In the present paper, we propose polynomial-time algorithms for the problem for c=1 and 2. The time complexities of the algorithms are O((k+m)log n) for c=1 and O(m^4log n+k^3+n(k^2+log n)) for c=2, respectively, where n=|V|, m=|E| and k=|X|.
  @cite_14: A new heuristic procedure for the transportation problem with exclusionary side constraints is developed and implemented. Tabu search, a meta-heuristic method, is used to guide the search to follow a path selectively to prevent from being trapped at local optimal solutions in order to find a global optimal or near global optimal solution. The simplex method on a graph is employed to lead the search from one solution to an adjacent solution in order to take advantage of the underlying network structure of the problem. In the procedure, net changes in cost and in infeasibility are used to measure the attractiveness of a move, and strategic oscillation is used to implement the intensification and diversification functions. A computational experiment was conducted to test the performance of the heuristic procedure and substantial computational results are reported. These results show that the new heuristic procedure finds very good quality solutions and outperforms an existing heuristic procedure in terms of both solution quality and CPU time.
  @cite_8: Abstract The transportation problem with exclusionary side constraints, a practical distribution and logistics problem, is formulated as a 0–1 mixed integer programming model. Two branch-and-bound (B&B) algorithms are developed and implemented in this study to solve this problem. Both algorithms use the Driebeek penalties to strengthen the lower bounds so as to fathom some of the subproblems, to peg variables, and to guide the selection of separation variables. One algorithm also strongly exploits the problem structure in selecting separation variables in order to find feasible solutions sooner. To take advantage of the underlying network structure of the problem, the algorithms employ the primal network simplex method to solve network relaxations of the problem. A computational experiment was conducted to test the performance of the algorithms and to characterize the problem difficulty. The commercial mixed integer programming software CPLEX and an existing special purpose algorithm specifically designed for this problem were used as benchmarks to measure the performance of the algorithms. Computational results show that the new algorithms completely dominate the existing special purpose algorithm and run from two to three orders of magnitude faster than CPLEX.
  @cite_6: 
  @cite_24: In this paper we address the minimum cost perfect matching problem with conflict pair constraints (MCPMPC). Given an undirected graph G with a cost associated with each edge and a conflict set of pairs of edges, the MCPMPC is to find a perfect matching with the lowest total cost such that no more than one edge is selected from each pair in the conflict set. MCPMPC is known to be strongly NP-hard. We present additional complexity results and identify new polynomially solvable cases for the general MCPMPC. Several heuristic algorithms and lower bounding schemes are presented. The proposed algorithms are tested on randomly generated instances. Encouraging experimental results are also reported.
  @cite_19: We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly @math -hard, even if the conflict graph consists only of unconnected edges. This result still holds if the network consists only of disjoint paths of length three. For forcing graphs we also provide a sharp line between polynomially solvable and strongly @math -hard instances for the case where the flow values are required to be integral. Moreover, our hardness results imply that no polynomial time approximation algorithm can exist for both problems. In contrast to this we show that the maximum flow problem with a forcing graph can be solved efficiently if fractional flow values are allowed.
  @cite_13: We study approaches for the exact solution of the NP-hard minimum spanning tree problem under conflict constraints. Given a graph (G(V,E) ) and a set (C E E ) of conflicting edge pairs, the problem consists of finding a conflict-free minimum spanning tree, i.e. feasible solutions are allowed to include at most one of the edges from each pair in (C ). The problem was introduced recently in the literature, with several results on its complexity and approximability. Some formulations and both exact and heuristic algorithms were also discussed, but computational results indicate considerably large duality gaps and a lack of optimality certificates for benchmark instances. In this paper, we build on the representation of conflict constraints using an auxiliary conflict graph ( G (E,C) ), where stable sets correspond to conflict-free subsets of (E ). We introduce a general preprocessing method and a branch and cut algorithm using an IP formulation with exponentially sized classes of valid inequalities for both the spanning tree and the stable set polytopes. Encouraging computational results indicate that the dual bounds of our approach are significantly stronger than those previously available, already in the initial LP relaxation, and we are able to provide new feasibility and optimality certificates.
  @cite_20: For the classical minimum spanning tree problem we introduce disjunctive constraints for pairs of edges which can not be both included in the spanning tree at the same time. These constraints are represented by a conflict graph whose vertices correspond to the edges of the original graph. Edges in the conflict graph connect conflicting edges of the original graph. It is shown that the problem becomes strongly @math -hard even if the connected components of the conflict graph consist only of paths of length two. On the other hand, for conflict graphs consisting of disjoint edges (i.e. paths of length one) the problem remains polynomially solvable.
  @cite_11: We consider the minimum spanning tree problem with conflict constraints (MSTC). The problem is known to be strongly NP-hard and computing even a feasible solution is NP-hard. When the underlying graph is a cactus, we show that the feasibility problem is polynomially bounded whereas the optimization version is still NP-hard. When the conflict graph is a collection of disjoint cliques, (equivalently, when the conflict relation is transitive) we observe that MSTC can be solved in polynomial time. We also identify other special cases of MSTC that can be solved in polynomial time. Exploiting these polynomially solvable special cases we derive strong lower bounds. Also, various heuristic algorithms and feasibility tests are discussed along with preliminary experimental results. As a byproduct of this investigation, we show that if an @e-optimal solution to the maximum clique problem can be obtained in polynomial time, then a ([email protected])-optimal solution to the maximum edge clique partitioning (Max-ECP) problem can be obtained in polynomial time. As a consequence, we have a polynomial time approximation algorithm for the Max-ECP with performance ratio O(n(loglogn)^2log^3n), improving the best previously known bound of O(n).
summary: The conflict pair constraint (or negative disjunctive constraint ) is a generalization of the non-crossing constraint. Represented by a conflict graph @math , this constraint prohibits a solution from including two edges @math such that @math . The minimum cost perfect matching problem with conflict pair constraints is strong NP-hard for a general graph @math even if the conflict graph @math is a collection of single edges @cite_6 and is NP-hard even for @math that consists of 4-cycles @cite_24 ; it turns out that the problem is NP-hard for a bipartite graph. For this type of constraint, there are also studies on the transportation problem @cite_14 @cite_8 , the minimum spanning tree problem @cite_20 @cite_11 @cite_13 , and the max-flow problem @cite_19 .