Ryuhei Uehara

On the complexity of reconfiguration problems

Reconfiguration problems arise when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible. We demonstrate that a host of reconfiguration problems derived from NP-complete problems are PSPACE-complete, while some are also NP-hard to approximate. In contrast, several reconfiguration versions of problems in P are solvable in polynomial time.

Efficient algorithms for the longest path problem

The longest path problem is to find a longest path in a given graph While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, very few graph classes are known where the longest path problem can be solved efficiently For a tree, a simple linear time algorithm for the longest path problem is known We first generalize the algorithm, and it then solves the longest path problem efficiently for weighted trees, block graphs, ptolemaic graphs, and cacti We next propose three new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on those classes As a corollary, it is also shown that the problem can be solved efficiently on threshold graphs.

ON COMPUTING LONGEST PATHS IN SMALL GRAPH CLASSES

The longest path problem is the one that finds a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. Among those, for trees, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for some tree-like graph classes by this approach. We next propose two new graph classes that have natural interval representations, and show that the longest path problem can be solved efficiently on these classes.

Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs

This paper deals with the graph isomorphism (GI) problem for two graph classes: chordal bipartite graphs and strongly chordal graphs. It is known that GI problem is GI complete even for some special graph classes including regular graphs, bipartite graphs, chordal graphs, comparability graphs, split graphs, and k-trees with unbounded k. On the other hand, the relative complexity of the GI problem for the above classes was unknown. We prove that deciding isomorphism of the classes are GI complete.

A double classification tree search algorithm for index SNP selection.

AbstractBackgroundIn population-based studies, it is generally recognized that single nucleotide polymorphism (SNP) markers are not independent. Rather, they are carried by haplotypes, groups of SNPs that tend to be coinherited. It is thus possible to choose a much smaller number of SNPs to use as indices for identifying haplotypes or haplotype blocks in genetic association studies. We refer to these characteristic SNPs as index SNPs. In order to reduce costs and work, a minimum number of index SNPs that can distinguish all SNP and haplotype patterns should be chosen. Unfortunately, this is an NP-complete problem, requiring brute force algorithms that are not feasible for large data sets.ResultsWe have developed a double classification tree search algorithm to generate index SNPs that can distinguish all SNP and haplotype patterns. This algorithm runs very rapidly and generates very good, though not necessarily minimum, sets of index SNPs, as is to be expected for such NP-complete problems.ConclusionsA new algorithm for index SNP selection has been developed. A webserver for index SNP selection is available at http://cognia.cu-genome.org/cgi-bin/genome/snpIndex.cgi/

Optimal Attribute-Efficient Learning of Disjunction, Parity and Threshold Functions

Decision trees are a very general computation model. Here the problem is to identify a Boolean function f out of a given set of Boolean functions F by asking for the value of f at adaptively chosen inputs. For classes F consisting of functions which may be obtained from one function g on n inputs by replacing arbitrary n−k inputs by given constants this problem is known as attribute-efficient learning with k essential attributes. Results on general classes of functions are known. More precise and often optimal results are presented for the cases where g is one of the functions disjunction, parity or threshold.

Counting the number of independent sets in chordal graphs

We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant time per output. On the other hand, we prove that the following problems for a chordal graph are #P-complete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is NP-hard, and even hard to approximate.

Parallel approximation algorithms for maximum weighted matching in general graphs

The problem of computing a matching of maximum weight in a given edge-weighted graph is not known to be P-hard or in RNC. This paper presents four parallel approximation algorithms for this problem. The first is an RNC-approximation scheme, i.e., an RNC algorithm that computes a matching of weight at least 1 − e times the maximum for any given constant e > 0. The second one is an NC approximation algorithm achieving an approximation ratio of\( \frac{1} {{2 + \varepsilon }} \) for any fixed e > 0. The third and fourth algorithms only need to know the total order of weights, so they are useful when the edge weights require a large amount of memories to represent. The third one is an NC approximation algorithm that finds a matching of weight at least \( \frac{2} {{3\Delta + 2}} \) times the maximum, where Δ is the maximum degree of the graph. The fourth one is an RNC algorithm that finds a matching of weight at least \( \frac{1} {{2\Delta + 4}} \) times the maximum on average, and runs in Ο(logΔ) time, not depending on the size of the graph.

Linear-time algorithm for sliding tokens on trees

Suppose that we are given two independent sets I b and I r of a graph such that | I b | = | I r | , and imagine that a token is placed on each vertex in I b . Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I b into I r so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between I b and I r whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

A New Approach to Graph Recognition and Applications to Distance-Hereditary Graphs

Algorithms used in data mining and bioinformatics have to deal with huge amount of data efficiently. In many applications, the data are supposed to have explicit or implicit structures. To develop efficient algorithms for such data, we have to propose possible structure models and test if the models are feasible. Hence, it is important to make a compact model for structured data, and enumerate all instances efficiently. There are few graph classes besides trees that can be used for a model. In this paper, we investigate distance-hereditary graphs. This class of graphs consists of isometric graphs and hence contains trees and cographs. First, a canonical and compact tree representation of the class is proposed. The tree representation can be constructed in linear time by using prefix trees. Usually, prefix trees are used to maintain a set of strings. In our algorithm, the prefix trees are used to maintain the neighborhood of vertices, which is a new approach unlike the lexicographically breadth-first search used in other studies. Based on the canonical tree representation, efficient algorithms for the distance-hereditary graphs are proposed, including linear time algorithms for graph recognition and graph isomorphism and an efficient enumeration algorithm. An efficient coding for the tree representation is also presented; it requires ⌈3.59n⌉ bits for a distance-hereditary graph of n vertices and 3n bits for a cograph. The results of coding improve previously known upper bounds (both are 2O(n log n)) of the number of distance-hereditary graphs and cographs to 2⌈3.59n⌉ and 23n, respectively.