Given a system (V, T, f, k), where V is a finite set, T ⊆ V , f : 2 V → R is a submodular function and k ≥ 2 is an integer, the general multiway partition problem (MPP) asks to find a k-partition P = {V 1 , V 2 ,. . ., V k } of V that satisfies V i ∩ T = ∅ for all i and minimizes f (V 1) + f (V 2) + • • • + f (V k), where P is a k-partition of V if (i) V i = ∅, (ii) V i ∩ V j = ∅, i = j, and (iii) V 1 ∪ V 2 ∪ • • • ∪ V k = V hold. MPP formulation captures a generalization in submodular systems of many NP-hard problems such as k-way cut, multiterminal cut, target split and their generalizations in hypergraphs. This paper presents a simple and unified framework for developing and analyzing approximation algorithms for various MPPs.
The enumeration of chemical graphs satisfying given constraints is one of the fundamental problems in chemoinformatics. In this paper, we consider the problem of enumerating (i.e., listing) all treelike chemical graphs from a given path frequency. We propose an exact algorithm for enumerating all solutions to this problem on the basis of the branch-and-bound method. To further improve the efficiency of the enumeration, we introduce a new variant of the compound enumeration problem by adding a specification on the number of multiple bonds to the input and design another exact enumeration algorithm. The experimental results show that our algorithms can efficiently solve instances with larger sizes that are impossible to solve by the previous methods. In particular, we apply the latter algorithm to the enumeration problem of the special treelike chemical structures-alkane isomers. The theoretical and experimental results show that our algorithm works at least as fast as the state-of-the-art algorithms specially designed for generating alkane isomers, however using much less memory space.
We consider the Waring-Goldbach problem for fourth and sixth powers. In particular, we establish that every sufficiently large positive integer under a natural congruence condition can be represented as a sum of 13 fourth powers of prime numbers. This improves upon the earlier result of Kawada and Wooley ['On the Waring-Goldbach problem for fourth and fifth powers',
The two-dimensional (2D) nonseparable linear canonical transform (NS-LCT) is a unitary, linear integral transform that relates the input and output monochromatic, paraxial scalar wave fields of optical systems characterized by a 4×4 ray tracing matrix. In addition to the obvious generalizations of the 1D LCT (which are referred to as separable), the 2D-NS-LCT can represent a variety of nonaxially symmetric optical systems including the gyrator transform and image rotation. Unlike the 1D LCT, the numerical approximation of the 2D-NS-LCT has not yet received extensive attention in the literature. In this paper, (1) we develop a sampling theorem for the general 2D-NS-LCT which generalizes previously published sampling theorems for the 1D case and (2) we determine which sampling rates may be chosen to ensure that the obvious discrete transform is unitary.
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