Cluster analysis plays a very important role in data analysis. In these years, cluster ensemble, as a cluster analysis tool, has drawn much attention for its robustness, stability, and accuracy. Many efforts have been done to combine different initial clustering results into a single clustering solution with better performance. However, they neglect the structure information of the raw data in performing the cluster ensemble. In this paper, we propose a Structural Cluster Ensemble (SCE) algorithm for data partitioning formulated as a set-covering problem. In particular, we construct a Laplacian regularized objective function to capture the structure information among clusters. Moreover, considering the importance of the discriminative information underlying in the initial clustering results, we add a discriminative constraint into our proposed objective function. Finally, we verify the performance of the SCE algorithm on both synthetic and real data sets. The experimental results show the effectiveness of our proposed method SCE algorithm.
@cite_18 defined category utility function to measure the similarity between two partitions. For two partitions @math and @math , the category utility function is given below: @math where @math , @math and @math . Then the corresponding consensus function is obtained by solving the following optimization problem. @math
@cite_18
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Utility functions satisfying gross substitutability have been studied extensively in the economics literature [1,11,12] and recently, the importance of this property has been recognized in the design of combinatorial polynomial time market equilibrium algorithms [8]. This naturally raises the following question: is it possible to design a combinatorial polynomial time algorithm for this general class of utility functions? We partially answer this question by giving an algorithm for separable, differentiable, concave utility functions satisfying gross substitutes. Our algorithm uses the auction based approach of [10]. This paper gives a dichotomy theorem for the complexity of computing the partition function of an instance of a weighted Boolean constraint satisfaction problem. The problem is parameterized by a finite set @math of nonnegative functions that may be used to assign weights to the configurations (feasible solutions) of a problem instance. Classical constraint satisfaction problems correspond to the special case of 0,1-valued functions. We show that computing the partition function, i.e., the sum of the weights of all configurations, is @math -complete unless either (1) every function in @math is of “product type,” or (2) every function in @math is “pure affine.” In the remaining cases, computing the partition function is in P. Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of @math -colorings or the number of independent sets of a graph and also the partition functions of certain “spin glass” models of statistical physics such as the Ising model. Building on earlier work by Dyer and Greenhill [Random Structures Algorithms, 17 (2000), pp. 260-289] and Bulatov and Grohe [Theoret. Comput. Sci., 348 (2005), pp. 148-186], we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or #P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or #P-complete. While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by Hadamard matrices (these turn out to be central in our proofs) we obtain a simple algebraic tractability criterion, which says that the tractable cases are those “representable” by a quadratic polynomial over the field @math . Power allocation across users in two adjacent cells is studied for a code-division multiple access (CDMA) data service. The forward link is considered and cells are modeled as one-dimensional with uniformly distributed users and orthogonal signatures within each cell. Each user is assumed to have a utility function that describes the user's received utility, or willingness to pay, for a received signal-to-interference-plus-noise ratio (SINR). The objective is to allocate the transmitted power to maximize the total utility summed over all users subject to power constraints in each cell. It is first shown that this optimization can be achieved by a pricing scheme in which each base station announces a price per unit transmitted power to the users, and each user requests power to maximize individual surplus (utility minus cost). Setting prices to maximize total revenue over both cells is also considered, and it is shown that, in general, the solution is different from the one obtained by maximizing total utility. Conditions are given for which independent optimization in each cell, which leads to a Nash equilibrium (NE), is globally optimal. It is shown that, in general, coordination between the two cells is needed to achieve the maximum utility or revenue.