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|.
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.
@cite_7 @cite_21 @cite_0 @cite_23 @cite_5
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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. 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. 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. In this paper, we study the non-bipartite maximum matching problem in the semi-streaming model. The maximum matching problem in the semi-streaming model has received a significant amount of attention lately. While the problem has been somewhat well solved for bipartite graphs, the known algorithms for non-bipartite graphs use @math passes or @math time to compute a @math approximation. In this paper we provide the first FPTAS (polynomial in @math ) for the problem which is efficient in both the running time and the number of passes. We also show that we can estimate the size of the matching in @math passes using slightly superlinear space. To achieve both results, we use the structural properties of the matching polytope such as the laminarity of the tight sets and total dual integrality. The algorithms are iterative, and are based on the fractional packing and covering framework. However the formulations herein require exponentially many variables or constraints. We use laminarity, metric embeddings and graph sparsification to reduce the space required by the algorithms in between and across the iterations. This is the first use of these ideas in the semi-streaming model to solve a combinatorial optimization problem.