In this paper, we first study what we call Superset-Subset-Disjoint (SSD) set system. Based on properties of SSD set system, we derive the following (I) to (IV): (I) For a nonnegative integer $k$ and a graph $G=(V,E)$ with $|V|\ge2$, let $X_1,X_2,\dots,X_q\subsetneq V$ denote all maximal proper subsets of $V$ that induce $k$-edge-connected subgraphs. Then at least one of (a) and (b) holds: (a) $\{X_1,X_2,\dots,X_q\}$ is a partition of $V$; and (b) $V\setminus X_1, V\setminus X_2,\dots,V\setminus X_q$ are pairwise disjoint. (II) For $k=1$ and a strongly-connected digraph $G$, whether $V$ is in (a) and/or (b) can be decided in $O(n+m)$ time and we can generate all such $X_1,X_2,\dots,X_q$ in $O(n+m+|X_1|+|X_2|+\dots+|X_q|)$ time, where $n=|V|$ and $m=|E|$. (III) For a digraph $G$, we can enumerate in linear delay all vertex subsets of $V$ that induce strongly-connected subgraphs. (IV) A digraph is Hamiltonian if there is a spanning subgraph that is strongly-connected and in the case (a).

For a set system ( V , C ⊆ 2 V ) , we call each C ∈ C a component. A nonempty subset Y ⊊ C is a removable set (RS) of C if C ∖ Y is a component. We say that a set system has subset-disjoint (SD) property if, for any two components C , C ′ with C ′ ⊊ C , every minimal RS Y of C satisfies either Y ⊆ C ′ or Y ∩ C ′ = ∅ . Assuming that an SD set system is implicitly given by an oracle that returns a minimal RS of a component, we provide an algorithm that enumerates all components in linear time/space with respect to | V | and oracle running time/space. We then extend this algorithm to linear-delay enumeration of all 2-edge-connected (or 2-vertex-connected) induced subgraphs in an undirected graph and of all strongly connected subgraphs in a digraph.

In this paper, we propose a novel family of descriptors of chemical graphs, named cycle-configuration (CC), that can be used in the standard "two-layered (2L) model" of mol-infer, a molecular inference framework based on mixed integer linear programming (MILP) and machine learning (ML). Proposed descriptors capture the notion of ortho/meta/para patterns that appear in aromatic rings, which has been impossible in the framework so far. Computational experiments show that, when the new descriptors are supplied, we can construct prediction functions of similar or better performance for all of the 27 tested chemical properties. We also provide an MILP formulation that asks for a chemical graph with desired properties under the 2L model with CC descriptors (2L+CC model). We show that a chemical graph with up to 50 non-hydrogen vertices can be inferred in a practical time.

A novel framework has recently been proposed for designing the molecular structure of chemical compounds with a desired chemical property using both artificial neural networks and mixed integer linear programming. In this paper, we design a new method for inferring a polymer based on the framework. For this, we introduce a new way of representing a polymer as a form of monomer and define new descriptors that feature the structure of polymers. We also use linear regression as a building block of constructing a prediction function in the framework. The results of our computational experiments reveal a set of chemical properties on polymers to which a prediction function constructed with linear regression performs well. We also observe that the proposed method can infer polymers with up to 50 non-hydrogen atoms in a monomer form.

A novel framework for designing the molecular structure of chemical compounds with a desired chemical property has recently been proposed. The framework infers a desired chemical graph by solving a mixed integer linear program (MILP) that simulates the computation process of two functions: a feature function defined by a two-layered model on chemical graphs and a prediction function constructed by a machine learning method. To improve the learning performance of prediction functions in the framework, we design a method that splits a given data set$\mathcal {C}$into two subsets$\mathcal {C}^{(i)},i=1,2$by a hyperplane in a chemical space so that most compounds in the first (resp., second) subset have observed values lower (resp., higher) than a threshold$\theta$. We construct a prediction function$\psi$to the data set$\mathcal {C}$by combining prediction functions$\psi _{i},i=1,2$each of which is constructed on$\mathcal {C}^{(i)}$independently. The results of our computational experiments suggest that the proposed method improved the learning performance for several chemical properties to which a good prediction function has been difficult to construct.

Drug discovery is one of the major goals of computational biology and bioinformatics. A novel framework has recently been proposed for the design of chemical graphs using both artificial neural networks (ANNs) and mixed integer linear programming (MILP). This method consists of a prediction phase and an inverse prediction phase. In the first phase, an ANN is trained using data on existing chemical compounds. In the second phase, given a target chemical property, a feature vector is inferred by solving an MILP formulated from the trained ANN and then a set of chemical structures is enumerated by a graph enumeration algorithm. Although exact solutions are guaranteed by this framework, the types of chemical graphs have been restricted to such classes as trees, monocyclic graphs, and graphs with a specified polymer topology with cycle index up to 2. To overcome the limitation on the topological structure, we propose a new flexible modeling method to the framework so that we can specify a topological substructure of graphs and a partial assignment of chemical elements and bond-multiplicity to a target graph. The results of computational experiments suggest that the proposed system can infer chemical graphs with around up to 50 non-hydrogen atoms.

A novel framework has recently been proposed for designing the molecular structure of chemical compounds with a desired chemical property using both artificial neural networks and mixed integer linear programming. In the framework, a chemical graph with a target chemical value is inferred as a feasible solution of a mixed integer linear program that represents a prediction function and other requirements on the structure of graphs. In this paper, we propose a procedure for generating other feasible solutions of the mixed integer linear program by searching the neighbor of output chemical graph in a search space. The procedure is combined in the framework as a new building block. The results of our computational experiments suggest that the proposed method can generate an additional number of new chemical graphs with up to 50 non-hydrogen atoms.

We are given an instance (G,I,sigma) with a graph G=(V,E), a set I of items, and a function sigma:V -> 2^I. For a subset X of V, let G[X] denote the subgraph induced from G by X, and I_sigma(X) denote the common item set over X. A subset X of V such that G[X] is connected is called a connector if, for any vertex v in V\X, G[X cup {v}] is not connected or I_sigma(X cup {v}) is a proper subset of I_sigma(X). In this paper, we present the first polynomial-delay algorithm for enumerating all connectors. For this, we first extend the problem of enumerating connectors to a general setting so that the connectivity condition on X in G can be specified in a more flexible way. We next design a new algorithm for enumerating all solutions in the general setting, which leads to a polynomial-delay algorithm for enumerating all connectors for several connectivity conditions on X in G, such as the biconnectivity of G[X] or the k-edge-connectivity among vertices in X in G.

In the present paper, we propose an efficient local search for the minimum independent dominating set problem. We consider a local search that uses k-swap as the neighborhood operation. Given a feasible solution S, it is the operation of obtaining another feasible solution by dropping exactly k vertices from S and then by adding any number of vertices to it. We show that, when k=2, (resp., k=3 and a given solution is minimal with respect to 2-swap), we can find an improved solution in the neighborhood or conclude that no such solution exists in O(n Delta) (resp., O(n Delta^3)) time, where n denotes the number of vertices and Delta denotes the maximum degree. We develop a metaheuristic algorithm that repeats the proposed local search and the plateau search iteratively, where the plateau search examines solutions of the same size as the current solution that are obtainable by exchanging a solution vertex and a non-solution vertex. The algorithm is so effective that, among 80 DIMACS graphs, it updates the best-known solution size for five graphs and performs as well as existing methods for the remaining graphs.