The Asymmetric Matrix Partition Problem
نویسندگان
چکیده
An instance of the asymmetric matrix partition problem consists of a matrix A ∈ Rn×m + and a probability distribution p over its columns. The goal is to find a partition scheme that maximizes the resulting partition value. A partition scheme S = {S1, . . . ,Sn} consists of a partition Si of [m] for each row i of the matrix. The partition Si can be interpreted as a smoothing operator on row i, which replaces the value of each entry in that row with the expected value in the partition subset that contains it. Given a scheme S that induces a smoothed matrix A′, the partition value is the expected maximum column entry of A′. We establish that this problem is already APX-hard for the seemingly simple setting in which A is binary and p is uniform. We then demonstrate that a constant factor approximation can be achieved in most cases of interest. Later on, we discuss the symmetric version of the problem, in which one must employ an identical partition for all rows, and prove that it is essentially trivial. Our matrix partition problem draws its interest from several applications like broad matching in sponsored search advertising and information revelation in market settings. We conclude by discussing the latter application in depth.
منابع مشابه
Continuous-time Hopfield neural network-based optimized solution to 2-channel allocation problem
The channel allocation problem in cellular radio systems is NP-complete, and thus its general solution is not known for even the 2-channel case. It is well known that the link gain system matrix (or received-signal power system matrix) of the radio network is (and may be highly) asymmetric, and that as the Hopfield neural network is applied to optimization problems, its weight matrix should be ...
متن کاملA new sequence space and norm of certain matrix operators on this space
In the present paper, we introduce the sequence space [{l_p}(E,Delta) = left{ x = (x_n)_{n = 1}^infty : sum_{n = 1}^infty left| sum_{j in {E_n}} x_j - sum_{j in E_{n + 1}} x_jright| ^p < infty right},] where $E=(E_n)$ is a partition of finite subsets of the positive integers and $pge 1$. We investigate its topological properties and inclusion relations. Moreover, we consider the problem of fin...
متن کاملAn employee transporting problem
An employee transporting problem is described and a set partitioning model is developed. An investigation of the model leads to a knapsack problem as a surrogate problem. Finding a partition corresponding to the knapsack problem provides a solution to the problem. An exact algorithm is proposed to obtain a partition (subset-vehicle combination) corresponding to the knapsack solution. It require...
متن کاملFrom Virasoro Constraints in Kontsevich ’ s Model to W - constraints in 2 - matrix Models
The Ward identities in Kontsevich-like 1-matrix models are used to prove at the level of discrete matrix models the suggestion of Gava and Narain, which relates the degree of potential in asymmetric 2-matrix model to the form of W-constraints imposed on its partition function.
متن کاملTwo-Species Asymmetric Simple Exclusion Process with Open Boundaries
We study the steady state of the two-species Asymmetric Simple Exclusion Process (ASEP) with open boundary conditions. The matrix product method works for the determination of the stationary probability distribution. Several physical quantities are calculated through an explicit representation for the matrix products. By making full use of the relation with the continuous big q-Hermite polynomi...
متن کامل