A special class of quadratic pseudo-boolean functions called "half-products" (HP) has recently been introduced. It has been shown that HIP minimization, while NP-hard, admits a fully polynomial time approxim...
详细信息
A special class of quadratic pseudo-boolean functions called "half-products" (HP) has recently been introduced. It has been shown that HIP minimization, while NP-hard, admits a fully polynomial time approximation scheme (FPTAS). In this note, we provide a more efficient FPTAS. We further show how an FPTAS can also be derived for the general case where the HP function is augmented by a problem-dependent constant and can justifiably be assumed to be nonnegative. This leads to an FPTAS for certain partitioning type problems, including many from the field of scheduling. (C) 2008 Elsevier B.V. All rights reserved.
We present a family of local-search-based heuristics for quadratic Unconstrained Binary Optimization (QUBO), all of which start with a (possibly fractional) initial point, sequentially improving its quality by roundin...
详细信息
We present a family of local-search-based heuristics for quadratic Unconstrained Binary Optimization (QUBO), all of which start with a (possibly fractional) initial point, sequentially improving its quality by rounding or switching the value of one variable, until arriving to a local optimum. The effects of various parameters on the efficiency of these methods are analyzed through computational experiments carried out on thousands of randomly generated problems having 20 to 2500 variables. Tested on numerous benchmark problems, the performance of the most competitive variant (ACSIOM) was shown to compare favorably with that of other published procedures.
Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision, and many others. The general solver has a complexity of O(n(3) log(2) n...
详细信息
Submodular function minimization is a key problem in a wide variety of applications in machine learning, economics, game theory, computer vision, and many others. The general solver has a complexity of O(n(3) log(2) n.E + n(4) log (0(1)) n) where E is the time required to evaluate the function and n is the number of variables (Lee et al., 2015). On the other hand, many computer vision and machine learning problems are defined over special subclasses of submodular functions that can be written as the sum of many submodular cost functions defined over cliques containing only a few variables. In such functions, the pseudo-boolean (or polynomial) representation (Boros and Hammer, 2002) of these subclasses are of degree (or order, or clique size) k where k << n. In this work, we develop efficient algorithms for the minimization of this useful subclass of submodular functions. To do this, we define novel mapping that transform submodular functions of order k into quadratic ones. The underlying idea is to use auxiliary variables to model the higher order terms and the transformation is found using a carefully constructed linear program. In particular, we model the auxiliary variables as monotonic booleanfunctions, allowing us to obtain a compact transformation using as few auxiliary variables as possible. The transformed quadratic function can be efficiently minimized using the standard max-flow algorithm with a time complexity of O((n+m)(3)) where m is the total number of auxiliary variables involved in transforming all the higher order terms to quadratic ones. Specifically, we show that our approach for fourth order function requires only 2 auxiliary variables in contrast to 30 or more variables used in existing approaches. In the general case, we give art upper bound for the number or auxiliary variables required to transform a function of order k using Dedekind number, which is substantially lower than the existing bound of 2(2k). (C) 2016 Elsevier B.V. All rights res
暂无评论