17 research outputs found
The bipartite unconstrained 0–1 quadratic programming problem: Polynomially solvable cases
We consider the bipartite unconstrained 0-1 quadratic programming problem (BQP01) which is a generalization of the well studied unconstrained 0-1 quadratic programming problem (QP01). BQP01 has numerous applications and the problem is known to be MAX SNP hard. We show that if the rank of an associated m×n cost matrix Q=(qij) is fixed, then BQP01 can be solved in polynomial time. When Q is of rank one, we provide an O(nlogn) algorithm and this complexity reduces to O(n) with additional assumptions. Further, if qij=ai+bj for some ai and bj, then BQP01 is shown to be solvable in O(mnlogn) time. By restricting m=O(logn), we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if m=O(nk) for a fixed k. Finally, if the minimum number of rows and columns to be deleted from Q to make the remaining matrix non-negative is O(logn), then we show that BQP01 is polynomially solvable but it is NP-hard if this number is O(nk) for any fixed k
Low Rank Approximation of Binary Matrices: Column Subset Selection and Generalizations
Low rank matrix approximation is an important tool in machine learning. Given
a data matrix, low rank approximation helps to find factors, patterns and
provides concise representations for the data. Research on low rank
approximation usually focus on real matrices. However, in many applications
data are binary (categorical) rather than continuous. This leads to the problem
of low rank approximation of binary matrix. Here we are given a
binary matrix and a small integer . The goal is to find two binary
matrices and of sizes and respectively, so
that the Frobenius norm of is minimized. There are two models of this
problem, depending on the definition of the dot product of binary vectors: The
model and the Boolean semiring model. Unlike low rank
approximation of real matrix which can be efficiently solved by Singular Value
Decomposition, approximation of binary matrix is -hard even for .
In this paper, we consider the problem of Column Subset Selection (CSS), in
which one low rank matrix must be formed by columns of the data matrix. We
characterize the approximation ratio of CSS for binary matrices. For
model, we show the approximation ratio of CSS is bounded by
and this bound is asymptotically tight. For
Boolean model, it turns out that CSS is no longer sufficient to obtain a bound.
We then develop a Generalized CSS (GCSS) procedure in which the columns of one
low rank matrix are generated from Boolean formulas operating bitwise on
columns of the data matrix. We show the approximation ratio of GCSS is bounded
by , and the exponential dependency on is inherent.Comment: 38 page
Average value of solutions for the bipartite boolean quadratic programs and rounding algorithms
We consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem (BBQP) with m+n variables. A closed-form formula is developed to compute the average objective function value A of all solutions in O(mn) time. However, computing the median objective function value of the solutions is shown to be NP-hard. Also, we show that any solution with objective function value no worse than A dominates at least 2 m+n-2 solutions and this bound is the best possible. Further, we show that such a solution can be identified in O(mn) time and hence the domination ratio of this algorithm is at least 14. We then show that for any fixed natural numbers a and b such that η=ab > 1, no polynomial time approximation algorithm exists for BBQP with domination ratio larger than 1-2(1-η)η(m+n), unless P = NP. It is shown that some powerful local search algorithms can get trapped at a local maximum with objective function value less than A. One of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value. A new integer programming formulation of BBQP is also given and computational results with our rounding algorithms are reported
A divide-and-conquer algorithm for binary matrix completion
We propose an algorithm for low rank matrix completion for matrices with
binary entries which obtains explicit binary factors. Our algorithm, which we
call TBMC (\emph{Tiling for Binary Matrix Completion}), gives interpretable
output in the form of binary factors which represent a decomposition of the
matrix into tiles. Our approach is inspired by a popular algorithm from the
data mining community called PROXIMUS: it adopts the same recursive
partitioning approach while extending to missing data. The algorithm relies
upon rank-one approximations of incomplete binary matrices, and we propose a
linear programming (LP) approach for solving this subproblem. We also prove a
-approximation result for the LP approach which holds for any level of
subsampling and for any subsampling pattern. Our numerical experiments show
that TBMC outperforms existing methods on recommender systems arising in the
context of real datasets.Comment: 14 pages,4 figure