本文给出了均衡二分图有一个2-因子恰含 k 个大圈的度条件。设 G = (V1,V2;E) 是一个二分图,满足 |V1| = |V2| = n ≥ sk,其中 s ≥ 3 和 k ≥ 1 是两个整数。如果图 G 的最小度至少为 (1 ? 1/s)n + 1,那么 G 有一个2-因子恰含 k 个圈...本文给出了均衡二分图有一个2-因子恰含 k 个大圈的度条件。设 G = (V1,V2;E) 是一个二分图,满足 |V1| = |V2| = n ≥ sk,其中 s ≥ 3 和 k ≥ 1 是两个整数。如果图 G 的最小度至少为 (1 ? 1/s)n + 1,那么 G 有一个2-因子恰含 k 个圈使得每个圈长至少为 2s。展开更多
光谱聚类(spectral clustering,SC)由于在无监督学习中的有效性而受到越来越多的关注。然而其计算复杂度高,不适用于处理大规模数据。近年来提出了许多基于锚点图方法来加速大规模光谱聚类,然而这些方法选取的锚点通常不能很好地体现原...光谱聚类(spectral clustering,SC)由于在无监督学习中的有效性而受到越来越多的关注。然而其计算复杂度高,不适用于处理大规模数据。近年来提出了许多基于锚点图方法来加速大规模光谱聚类,然而这些方法选取的锚点通常不能很好地体现原始数据的信息,从而导致聚类性能下降。为克服这些缺陷,提出了一种二分k-means锚点提取的快速谱聚类算法(fast spectral clustering algorithm based on anchor point extraction with bisecting kmeans,FCAPBK)。该方法利用二分k-means从原始数据中选取一些具有代表性的锚点,构建基于锚点的多层无核相似图;然后通过锚点与样本间的相似关系构造层次二部图。最后在5个基准数据集上分别进行实验验证,结果表明FCAPBK方法能够在较短的时间内获得良好的聚类性能。展开更多
In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
The bipartite Star<sub>123</sub>-free graphs were introduced by V. Lozin in [1] to generalize some already known classes of bipartite graphs. In this paper, we extend to bipartite Star<sub>123</su...The bipartite Star<sub>123</sub>-free graphs were introduced by V. Lozin in [1] to generalize some already known classes of bipartite graphs. In this paper, we extend to bipartite Star<sub>123</sub>-free graphs a linear time algorithm of J. L. Fouquet, V. Giakoumakis and J. M. Vanherpe for finding a maximum matching in bipartite Star<sub>123</sub>, P<sub>7</sub>-free graphs presented in [2]. Our algorithm is a solution of Lozin’s conjecture.展开更多
The maximum weighted matching problem in bipartite graphs is one of the classic combinatorial optimization problems, and arises in many different applications. Ordered binary decision diagram (OBDD) or algebraic decis...The maximum weighted matching problem in bipartite graphs is one of the classic combinatorial optimization problems, and arises in many different applications. Ordered binary decision diagram (OBDD) or algebraic decision diagram (ADD) or variants thereof provides canonical forms to represent and manipulate Boolean functions and pseudo-Boolean functions efficiently. ADD and OBDD-based symbolic algorithms give improved results for large-scale combinatorial optimization problems by searching nodes and edges implicitly. We present novel symbolic ADD formulation and algorithm for maximum weighted matching in bipartite graphs. The symbolic algorithm implements the Hungarian algorithm in the context of ADD and OBDD formulation and manipulations. It begins by setting feasible labelings of nodes and then iterates through a sequence of phases. Each phase is divided into two stages. The first stage is building equality bipartite graphs, and the second one is finding maximum cardinality matching in equality bipartite graph. The second stage iterates through the following steps: greedily searching initial matching, building layered network, backward traversing node-disjoint augmenting paths, updating cardinality matching and building residual network. The symbolic algorithm does not require explicit enumeration of the nodes and edges, and therefore can handle many complex executions in each step. Simulation experiments indicate that symbolic algorithm is competitive with traditional algorithms.展开更多
The optimal semi-matching problem is one relaxing form of the maximum cardinality matching problems in bipartite graphs, and finds its applications in load balancing. Ordered binary decision diagram (OBDD) is a canoni...The optimal semi-matching problem is one relaxing form of the maximum cardinality matching problems in bipartite graphs, and finds its applications in load balancing. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate Boolean functions efficiently. OBDD-based symbolic algorithms appear to give improved results for large-scale combinatorial optimization problems by searching nodes and edges implicitly. We present novel symbolic OBDD formulation and algorithm for the optimal semi-matching problem in bipartite graphs. The symbolic algorithm is initialized by heuristic searching initial matching and then iterates through generating residual network, building layered network, backward traversing node-disjoint augmenting paths, and updating semi-matching. It does not require explicit enumeration of the nodes and edges, and therefore can handle many complex executions in each step. Our simulations show that symbolic algorithm has better performance, especially on dense and large graphs.展开更多
For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s andt such that 2≤ s ≤ t, 0≤ m-s ≤ n-t, andre＋n≤ 2s＋t-1, we prove that if G has at least mn- （2（m - s） ＋...For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s andt such that 2≤ s ≤ t, 0≤ m-s ≤ n-t, andre＋n≤ 2s＋t-1, we prove that if G has at least mn- （2（m - s） ＋ n - t） edges then it contains a subdivision of the complete bipartite K（s,t） with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn- （2（m - s） ＋ n - t ＋ 1） edges for this topological Turan type problem.展开更多
Let G be a properly colored bipartite graph. A rainbow matching of G is such a matching in which no two edges have the same color. Let G be a properly colored bipartite graph with bipartition (X,Y) and . We show that ...Let G be a properly colored bipartite graph. A rainbow matching of G is such a matching in which no two edges have the same color. Let G be a properly colored bipartite graph with bipartition (X,Y) and . We show that if , then G has a rainbow coloring of size at least .展开更多
基金This work is supported by NNSF of China (60172003) and NSF of Shandong Province (Z2000A02).
文摘本文给出了均衡二分图有一个2-因子恰含 k 个大圈的度条件。设 G = (V1,V2;E) 是一个二分图,满足 |V1| = |V2| = n ≥ sk,其中 s ≥ 3 和 k ≥ 1 是两个整数。如果图 G 的最小度至少为 (1 ? 1/s)n + 1,那么 G 有一个2-因子恰含 k 个圈使得每个圈长至少为 2s。
文摘光谱聚类(spectral clustering,SC)由于在无监督学习中的有效性而受到越来越多的关注。然而其计算复杂度高,不适用于处理大规模数据。近年来提出了许多基于锚点图方法来加速大规模光谱聚类,然而这些方法选取的锚点通常不能很好地体现原始数据的信息,从而导致聚类性能下降。为克服这些缺陷,提出了一种二分k-means锚点提取的快速谱聚类算法(fast spectral clustering algorithm based on anchor point extraction with bisecting kmeans,FCAPBK)。该方法利用二分k-means从原始数据中选取一些具有代表性的锚点,构建基于锚点的多层无核相似图;然后通过锚点与样本间的相似关系构造层次二部图。最后在5个基准数据集上分别进行实验验证,结果表明FCAPBK方法能够在较短的时间内获得良好的聚类性能。
基金Supported by the National Natural Science Foundation of China(No.11271300)
文摘In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
文摘The bipartite Star<sub>123</sub>-free graphs were introduced by V. Lozin in [1] to generalize some already known classes of bipartite graphs. In this paper, we extend to bipartite Star<sub>123</sub>-free graphs a linear time algorithm of J. L. Fouquet, V. Giakoumakis and J. M. Vanherpe for finding a maximum matching in bipartite Star<sub>123</sub>, P<sub>7</sub>-free graphs presented in [2]. Our algorithm is a solution of Lozin’s conjecture.
文摘The maximum weighted matching problem in bipartite graphs is one of the classic combinatorial optimization problems, and arises in many different applications. Ordered binary decision diagram (OBDD) or algebraic decision diagram (ADD) or variants thereof provides canonical forms to represent and manipulate Boolean functions and pseudo-Boolean functions efficiently. ADD and OBDD-based symbolic algorithms give improved results for large-scale combinatorial optimization problems by searching nodes and edges implicitly. We present novel symbolic ADD formulation and algorithm for maximum weighted matching in bipartite graphs. The symbolic algorithm implements the Hungarian algorithm in the context of ADD and OBDD formulation and manipulations. It begins by setting feasible labelings of nodes and then iterates through a sequence of phases. Each phase is divided into two stages. The first stage is building equality bipartite graphs, and the second one is finding maximum cardinality matching in equality bipartite graph. The second stage iterates through the following steps: greedily searching initial matching, building layered network, backward traversing node-disjoint augmenting paths, updating cardinality matching and building residual network. The symbolic algorithm does not require explicit enumeration of the nodes and edges, and therefore can handle many complex executions in each step. Simulation experiments indicate that symbolic algorithm is competitive with traditional algorithms.
基金partially supported by NSFC (No.12001296)Fundamental Research Funds for the Central Universities+3 种基金Nankai University (No.63201163)Shi is partially supported by NSFC (No.11922112)Natural Science Foundation of TianjinNankai Universitv (No.63206034)。
文摘The optimal semi-matching problem is one relaxing form of the maximum cardinality matching problems in bipartite graphs, and finds its applications in load balancing. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate Boolean functions efficiently. OBDD-based symbolic algorithms appear to give improved results for large-scale combinatorial optimization problems by searching nodes and edges implicitly. We present novel symbolic OBDD formulation and algorithm for the optimal semi-matching problem in bipartite graphs. The symbolic algorithm is initialized by heuristic searching initial matching and then iterates through generating residual network, building layered network, backward traversing node-disjoint augmenting paths, and updating semi-matching. It does not require explicit enumeration of the nodes and edges, and therefore can handle many complex executions in each step. Our simulations show that symbolic algorithm has better performance, especially on dense and large graphs.
文摘For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s andt such that 2≤ s ≤ t, 0≤ m-s ≤ n-t, andre＋n≤ 2s＋t-1, we prove that if G has at least mn- （2（m - s） ＋ n - t） edges then it contains a subdivision of the complete bipartite K（s,t） with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn- （2（m - s） ＋ n - t ＋ 1） edges for this topological Turan type problem.
文摘Let G be a properly colored bipartite graph. A rainbow matching of G is such a matching in which no two edges have the same color. Let G be a properly colored bipartite graph with bipartition (X,Y) and . We show that if , then G has a rainbow coloring of size at least .
