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_7: Abstract In this work we focus on an extension of the minimum cost flow problem in layered networks. Feasible arc flows must satisfy a specific compatibility restriction in addition to flow balance and capacity restrictions. Namely, at most one of the crossing arcs is allowed to have positive flow on it. This variant of the minimum cost flow problem, which we call the minimum cost noncrossing flow problem, can frequently be encountered in real life. The determination of optimal temporal quay crane allocations to berthed vessels in container terminals, and optimal train schedules through the stations on the same railroad line are two examples. We first analyze the complexity of the problem and show that the noncrossing flow problem is in fact N P -complete in a layered network. Then, we introduce mixed-integer linear programming formulations and discuss a polynomially solvable special case. Next we show a sufficient condition for the existence of a crossing in an optimal solution, which can be used for preprocessing the arcs in order to reduce the problem size. Our computational experiments on a large test set show that our preprocessing algorithm can significantly reduce the number of arcs.
  @cite_21: Given a set of n men represented by n points lying on a line, and n women represented by n points lying on another parallel line, with each person having a list that ranks some people of opposite gender as his her acceptable partners in strict order of preference. In this problem, we want to match people of opposite genders to satisfy people’s preferences as well as making the edges not crossing one another geometrically. A noncrossing blocking pair w.r.t. a matching M is a pair (m, w) of a man and a woman such that they are not matched with each other but prefer each other to their own partners in M, and the segment (m, w) does not cross any edge in M. A weakly stable noncrossing matching (WSNM) is a noncrossing matching that does not admit any noncrossing blocking pair. In this paper, we prove the existence of a WSNM in any instance by developing an (O(n^2) ) algorithm to find one in a given instance.
  @cite_0: Motivated by a crane assignment problem, we consider a Euclidean bipartite matching problem with edge-crossing constraints. Specifically, given n red points and n blue points in the plane, we want to construct a perfect matching between red and blue points that minimizes the length of the longest edge, while imposing a constraint that no two edges may cross each other. We show that the problem cannot be approximately solved within a factor less than 1:277 in polynomial time unless P = NP. We give simple dynamic programming algorithms that solve our problem in two special cases, namely (1) the case where the red and blue points form the vertices of a convex polygon and (2) the case where the red points are collinear and the blue points lie to one side of the line through the red points.
  @cite_23: Consider a bipartite graph G = (O, D, E) where O and D are origin and destination node sets respectively (O = D = n), and E is a set of edges (i, j), i ∈ O and j ∈ D(E= m). Suppose we draw the origin nodes and the destination nodes arranged in two columns and the edges as straight line segments between origins and destinations. A noncrossing matching is a subset of edges M ⊑ E such that no two edges of M intersect (including intersections at nodes). The Maximum Non Crossing Matching (MNCM) is the problem of finding the noncrossing matching of maximum cardinality. Let p denote the cardinality of the MNCM. Problems arising in several fields can be modelled as MNCM: for example the 3-Side Switch Box in VLSI design has been presented in [2] where an O(n 2) time algorithm is proposed. The problem can be reduced to the one of finding the longest increasing subsequence in a permutation of size m [1]. An algorithm for this problem has been proposed by Fredman and slightly improved by Widmayer and Wong[7]; in our case the algorithm complexity is O(m + (m - p) log (p). In this paper some labeling algorithms for the MNCM, which work directly on the bipartite graph, will be proposed. The overall complexity is improved to O(m log log n) or to O(m + minnp, (m - p) log p). Finally the weighted case is considered.
  @cite_5: We consider computing a maximum non-crossing matching in convex bipartite graphs. For a convex bipartite graph of n vertices and m edges, we present an O ( n log n ) time algorithm for finding a maximum non-crossing matching in the graph. The previous best algorithm takes O ( m + n log n ) time (Malucelli et?al., 1993). Since m = ? ( n 2 ) in the worst case, our result improves the previous work.
summary: The non-crossing (or crossing-free) constraint has been considered for some problems of finding an optimal'' subgraph. It is @cite_23 who first studied the algorithmic aspect of the MW-0-CPEMP explicitly. For the edge-unweighted case, they provided a polynomial-time algorithm that runs in @math time or in @math time, where @math denotes the cardinality of a maximum 0-CPE matching. They also extended the algorithm to the edge-weighted case, which yields an @math time algorithm. A bipartite graph is convex if, for every @math , @math @math implies @math for all @math . For the MW-0-CPEMP in edge-unweighted convex bipartite graphs, @cite_5 presented an algorithm whose running time is @math . @cite_0 considered the Euclidean non-crossing bipartite matching problem, where each vertex is represented by a 2D point. The objective is to find a non-crossing perfect matching whose longest edge is minimized. They showed that the problem is NP-hard in general, but that it is polynomially-solvable in some special cases. More recently, @cite_7 showed that the minimum cost non-crossing flow problem on a layered network is NP-hard. Ruangwises and Itoh @cite_21 studied the stable marriage problem under the non-crossing constraint, showing that there exists a weakly stable non-crossing matching for any instance.