A partial Latin square (PLS) is a partial assignment of n symbols to an n x n grid such that, in each row and in each column, each symbol appears at most once. The partial Latin square extension problem is an NP-hard problem that asks for a largest extension of a given PLS. We consider the local search such that the neighborhood is defined by (p, q)-swap, i.e., the operation of dropping exactly p symbols and then assigning symbols to at most q empty cells. As a fundamental result, we provide an efficient (p, oo)-neighborhood search algorithm that finds an improved solution or concludes that no such solution exists for p E {1, 2, 3}. The running time of the algorithm is O(nP+ 1). We then propose a novel swap operation, Trellisswap, which is a generalization of (p, q)-swap with p ~ 2. The proposed Trellis-neighborhood search algorithm runs in O(n 3 • 5) time. The iterated local search (ILS) algorithm with Trellisneighborhood is more likely to deliver a high-quality solution than not only ILSs with (p, oo)neighborhood but also state-of-the-art optimization solvers such as IBM ILOG CPLEX and LOCALSOLVER.
A novel framework for inverse quantitative structure–activity relationships (inverse QSAR) has recently been proposed and developed using both artificial neural networks and mixed integer linear programming. However, classes of chemical graphs treated by the framework are limited. In order to deal with an arbitrary graph in the framework, we introduce a new model, called a two-layered model, and develop a corresponding method. In this model, each chemical graph is regarded as two parts: the exterior and the interior. The exterior consists of maximal acyclic induced subgraphs with bounded height, the interior is the connected subgraph obtained by ignoring the exterior, and the feature vector consists of the frequency of adjacent atom pairs in the interior and the frequency of chemical acyclic graphs in the exterior. Our method is more flexible than the existing method in the sense that any type of graphs can be inferred. We compared the proposed method with an existing method using several data sets obtained from PubChem database. The new method could infer more general chemical graphs with up to 50 non-hydrogen atoms. The proposed inverse QSAR method can be applied to the inference of more general chemical graphs than before.
Recently a novel framework has been proposed for designing the molecular structure of chemical compounds using both artificial neural networks (ANNs) and mixed integer linear programming (MILP). In the framework, we first define a feature vector f (C) of a chemical graph C and construct an ANN that maps x = f (C) to a predicted value η(x) of a chemical property π to C. After this, we formulate an MILP that simulates the computation process of f (C) from C and that of η(x) from x. Given a target value y * of the chemical property π , we infer a chemical graph C † such that η(f (C † )) = y * by solving the MILP. In this paper, we use linear regression to construct a prediction function η instead of ANNs. For this, we derive an MILP formulation that simulates the computation process of a prediction function by linear regression. The results of computational experiments suggest our method can infer chemical graphs with around up to 50 non-hydrogen atoms.
CCS CONCEPTS• Computing methodologies → Machine learning; • Mathematics of computing → Integer programming; • Applied computing → Bioinformatics.
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