The Graph Coloring Problem (GCP) is a fundamental combinatorial optimization problem that has many practical applications. Degree of SATURation (DSATUR) and Recursive Largest First (RLF) are well known as typical solution construction algorithms for GCP. It is necessary to update the vertex degree in the subgraph induced by uncolored vertices when selecting vertices to be colored in both DSATUR and RLF. There is an issue that the higher the edge density of a given graph, the longer the processing time. The purposes of this paper are to propose a degree updating method called Adaptive Degree Updating (ADU for short) that improves the issue, and to evaluate the effectiveness of ADU for DSATUR and RLF on DIMACS benchmark graphs as well as random graphs having a wide range of sizes and densities. Experimental results show that the construction algorithms with ADU are faster than the conventional algorithms for many graphs and that the ADU method yields significant speed-ups relative to the conventional algorithms, especially in the case of large graphs with higher edge density.

A deterministic pushdown automaton (dpda) having just one stack symbol is called a deterministic restricted one-counter automaton (droca). When it accepts by empty stack, it is called strict. A deterministic one-counter automaton (doca) is a dpda having only one stack symbol, with the exception of a bottom-of-stack marker. The class of languages accepted by strict droca's is a subclass of the class of languages accepted by doca's. Valiant has proved the decidability of the equivalence problem for doca's and the undecidability of the inclusion problem for doca's. Hence the decidablity of the equivalence problem for strict droca's is obvious. In this paper, we present a new direct branching algorithm for checking the inclusion for a pair of languages accepted by strict droca's. Then we show that the worst-case time complexity of our algorithm is polynomial with respect to these automata.

A deterministic pushdown automaton (dpda) having just one stack symbol is called a deterministic restricted one-counter automaton (droca). A deterministic one-counter automaton (doca) is a dpda having only one stack symbol, with the exception of a bottom-of-stack marker. The class of languages accepted by droca's which accept by final state is a proper subclass of the class of languages accepted by doca's. Valiant has proved the decidability of the equivalence problem for doca's and the undecidability of the inclusion problem for doca's. Hence the decidability of the equivalence problem for droca's is obvious. In this paper, we evaluate the upper bound of the length of the shortest input string (witness) that disproves the inclusion for a pair of real-time droca's which accept by final state, and present a new direct branching algorithm for checking the inclusion for a pair of languages accepted by these droca's. Then we show that the worst-case time complexity of our algorithm is polynomial in the size of these droca's.

We propose a learning algorithm for simple deterministic languages from queries and a priori knowledge. To the learner, a special finite subset of the target language, called a representative sample, is provided at the beginning and two types of queries, equivalence queries and membership queries, are available. This learning algorithm constructs nonterminals of a hypothesis grammar based on Ishizaka(1990)'s idea. In Ishizaka(1990)'s algorithm, the learner makes rules as many as possible from positive counterexamples, and diagnoses wrong rules from negative counterexamples. In contrast, our algorithm guesses a simple deterministic grammar and diagnoses them using positive and negative counterexamples based on Angluin(1987)'s algorithm.

A deterministic pushdown automaton (dpda) having just one stack symbol is called a deterministic restricted one-counter automaton (droca). A deterministic one-counter automaton (doca) is a dpda having only one stack symbol, with the exception of a bottom-of-stack marker. The class of languages accepted by droca's which accept by final state is a proper subclass of the class of languages accepted by doca's. Valiant has proved the decidability of the equivalence problem for doca's and the undecidability of the inclusion problem for doca's. Thus the decidability of the equivalence problem for droca's is obvious. In this paper, we evaluate the upper bound of the length of the shortest input string (shortest witness) that disproves the inclusion for a pair of real-time droca's which accept by accept mode, and present a direct branching algorithm for checking the inclusion for a pair of languages accepted by these droca's. Then we show that the worst-case time complexity of our algorithm is polynomial in the size of these droca's.

We are concerned with a subclass of deterministic pushdown automata (dpda) called very simple dpda's, and present a new direct branching algorithm for checking the inclusion for a pair of languages accepted by these dpda's. As usual, we take the maximal thickness (i.e., the length of the shortest input strings that make each stack symbol go to empry) of all stack symbols into account as one parameter of the given dpda's. Then the worst-case time complexity of our algorithm is polynomial with respect to these parameters. Without considering the thickness, the complexity is single exponential in the description length of the given dpda's. As far as we are concerned with very simple dpda's, our algorithm is very simple and direct, and is faster and much better than the previously given algorithms for the inclusion problem of dpda's.

A deterministic pushdown automaton (dpda) having just one stack symbol is called a deterministic restricted one-counter automaton (droca). A deterministic one-counter automaton (doca) is a dpda having only one stack symbol, with the exception of a bottom-of-stack market. The class of languages accepted by droca's is a proper subclass of the class of languages accepted by doca's. Valiant has shown that the regularity problem for doca's is decidable in a single exponential worst-case time complexity. In this paper, we prove that the class of languages accepted by droca's which accept by final state is incomparable with the class of languages accepted by droca's which accept by empty stack (strict droca's), and that the intersection of them is equal to the class of strict regular languages. In addition, we present a new direct branching algorithm for checking the regularity for not only a strict droca but also a real-time droca which accepts by final state. Then we show that the worst-case time complexity of our algorithm is polynomial in the size of each droca.

Many problems can be formulated as maximum clique problems. Hence, it is highly important to develop algorithms that can find a maximum clique very fast in practice. We propose new approximate coloring and other related techniques which markedly improve the run time of the branch-and-bound algorithm MCR (J. Global Optim., 37, pp.95–111, 2007), previously shown to be the fastest maximum-clique-finding algorithm for a large number of graphs. The algorithm obtained by introducing these new techniques in MCR is named MCS. It is shown that MCS is successful in reducing the search space quite efficiently with low overhead. Extensive computational experiments confirm the superiority of MCS over MCR and other existing algorithms. It is faster than the other algorithms by orders of magnitude for several graphs. In particular, it is faster than MCR for difficult graphs of very high density and for very large and sparse graphs, even though MCS is not designed for any particular type of graph. MCS can be faster than MCR by a factor of more than 100,000 for some extremely dense random graphs. This paper demonstrates in detail the effectiveness of each new techniques in MCS, as well as the overall contribution.

We present here a new algorithm for checking equivalence of finite-turn dpda's. It is very direct and much simpler than the previous one by Valiant or by Beeri.

The protein threading problem with profiles is known to be efficiently solvable using dynamic programming. In this paper, we consider a variant of the protein threading problem with profiles in which constraints on distances between residues are given. We prove that protein threading with profiles and constraints is NP-hard. Moreover, we show a strong hardness result on the approximation of an optimal threading satisfying all the constraints. On the other hand, we develop two practical algorithms: CLIQUETHREAD and BBDPTHREAD. CLIQUETHREAD reduces the threading problem to the maximum edge-weight clique problem, whereas BBDPTHREAD combines dynamic programming and branch-and-bound techniques. We perform computational experiments using protein structure data in PDB (Protein Data Bank) using simulated distance constraints. The results show that constraints are useful to improve the alignment accuracy of the target sequence and the template structure. Moreover, these results also show that BBDPTHREAD is in general faster than CLIQUETHREAD for larger size proteins whereas CLIQUETHREAD is useful if there does not exist a feasible threading.

A deterministic pushdown automaton (dpda) having just one stack symbol is called a deterministic restricted one-counter automaton (droca). When it accepts an input by empty stack, it is called strict. This paper is concerned with a subclass of real-time strict droca's, called Szilard strict droca's, and studies the problem of identifying the subclass in the limit from positive data. The class of languages accepted by Szilard strict droca's coincides with the class of Szilard languages (or, associated languages) of strict droca's and is incomparable to each of the class of regular languages and that of simple languages. After providing some properties of languages accepted by Szilard strict droca's, we show that the class of Szilard strict droca's is polynomial time identifiable in the limit from positive data in the sense of Yokomori. This identifiability is proved by giving an exact characteristic sample of polynomial size for a language accepted by a Szilard strict droca. The class of very simple languages, which is a proper subclass of simple languages, is also proved to be polynomial time identifiable in the limit from positive data by Yokomori, but it is yet unknown whether there exists a characteristic sample of polynomial size for any very simple language.