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How do we most quickly fold a paper strip (modeled as a line) to obtain a desired mountain-valley pattern of equidistant creases (viewed as a binary string)? Define the folding complexity of a mountain-valley string as the minimum number of simple folds required to construct it. We show that the folding complexity of a length-n uniform string (all mountains or all valleys), and hence of a length-n pleat (alternating mountain/valley), is polylogarithmic in n. We also show that the maximum possible folding complexity of any string of length n is O(n/lgn), meeting a previously known lower bound.

URL to paper listed on conference site: http://www.cs.umanitoba.ca/~cccg2010/accepted.html, A planar point set S is an (i, t) set of ghost chimneys if there exist lines H[subscript 0],H[subscript 1],...,H[subscript t-1] such that the orthogonal projection of S onto H[subscript j] consists of exactly i + j distinct points. We give upper and lower bounds on the maximum value of t in an (i, t) set of ghost chimneys, showing that it is linear in i.

We show how to construct interlocked collections of simple polygons in the plane that fall apart upon removing certain combinations of pieces. Precisely, interior-disjoint simple planar polygons are interlocked if no subset can be separated arbitrarily far from the rest, moving each polygon as a rigid object as in a sliding-block puzzle. Removing a subset S of these polygons might keep them interlocked or free the polygons, allowing them to separate. Clearly freeing removal sets satisfy monotonicity: if S S [prime] and removing S frees the polygons, then so does S [prime]. In this paper, we show that any monotone Boolean function f on n variables can be described by m > n interlocked polygons: n of the m polygons represent the n variables, and removing a subset of these n polygons frees the remaining polygons if and only if f is 1 when the corresponding variables are 1.

For a fixed integer k [greater than or equal to] 0, a k-transmitter is an omnidirectional wireless transmitter with an in nite broadcast range that is able to penetrate up to k "walls", represented as line segments in the plane. We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons., National Science Foundation (U.S.) (NSF grant CCF-0728909)

We give a configuration of 53 points that cannot be covered by disjoint unit disks. This improves the previously known configuration of 55 points., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/10309

The Voronoi game is a two-person game which is a model for a competitive facility location. The game is played on a continuous domain, and only two special cases (one-dimensional case and one-round case) are well investigated. We introduce the discrete Voronoi game in which the game arena is given as a graph. We first analyze the game when the arena is a large complete k-ary tree, and give an optimal strategy. When both players play optimally, the first player wins when k is odd, and the game ends in a tie for even k. Next we show that the discrete Voronoi game isintractable in general. Even for the one-round case in which the strategy adopted by the first player consist of a fixed single node, deciding whether the second player can win is NP-complete. We also show that deciding whether the second player can win is PSPACE-complete in general., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/10302

We investigate the problem of finding common developments that fold to plural incongruent orthogonal boxes. It was shown that there are infinitely many orthogonal polygons that fold to two incongruent orthogonal boxes in 2008. In this paper, we first show that there is an orthogonal polygon that fold to three boxes of size 1×1×5, 1×2×3, and 0×1×11. Although we have to admit a box to have volume 0, this solves the open problem mentioned in literature. Moreover, once we admit that a box can be of volume 0, a long rectangular strip can be folded to an arbitrary number of boxes of volume 0. We next consider for finding common non-orthogonal developments that fold to plural incongruent orthogonal boxes. In literature, only orthogonal folding lines orwith 45 degree lines were considered. In this paper, we show some polygons that can fold to two incongruent orthogonal boxes in more general directions., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/10308

Spanning tree congestion is a relatively new graph parameter, whichhas been studied intensively. This paper studies the complexity of the problemto determine the spanning tree congestion for non-sparse graph classes, while itwas investigated for some sparse graph classes before. We prove that the problemis NP-hard even for chain graphs and split graphs. To cope with the hardnessof the problem, we present a fast (exponential-time) exact algorithm that runs inO ^∗(2^n) time, where n denotes the number of vertices. Additionally, we providea constant-factor approximation algorithm for cographs, and a linear-time algorithmfor chordal cographs., 8th Annual Conference on Theory and Applications of Medels of Computation, TAMC 2011, Tokyo, Japan, May 23-25, 2011., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/9861

Connected bipartite permutation graphs without vertex labels are investigated. First, the number of connected bipartite permutation graphs of n vertices is given. Based on the number, a simple algorithm that generates a connected bipartite permutation graph uniformly at random up to isomorphism is presented. Finally an enumeration algorithm of connected bipartite permutation graphs is proposed. The algorithm is based on reverse search, and it outputs each connected bipartite permutation graph in O(1) time., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/10720

Kaboozle is a puzzle consisting of several square cards, each annotated with colored paths and dots drawn on both sides and holes drilled. The goal is to join two colored dots with paths of the same color (and fill all holes) by stacking the cards suitably. The freedoms here are to reflect, rotate, and order the cards arbitrarily, so it is not surprising that the problem is NP-complete (as we show). More surprising is that any one of these freedoms—reflection, rotation, and order—is alone enough to make the puzzle NP-complete. Furthermore, we show NP-completeness of a particularly constrained form of Kaboozle related to 1D paper folding. Specifically, we suppose that the cards are glued together into a strip, where each glued edge has a specified folding direction (mountain or valley). This variation removes the ability to rotate and reflect cards, and restricts the order to be a valid folded state of a given 1D mountain-valley pattern.

UNO® is one of the world-wide well-known and popular card games. We investigate UNO from the viewpoint of combinatorial algorithmic game theory by giving some simple and concise mathematical models for it. They include cooperative and uncooperative versions of UNO, for example. As a result of analyzing their computational complexities, we prove that even a single-player version of UNO is NP-complete, while it becomes in P in some restricted cases. We also show that uncooperative two-player’s version is PSPACE-complete.

Reconfiguration problems arise when we wish to find a step-by-step transformationbetween two feasible solutions of a problem such that all intermediate results arealso feasible. We demonstrate that a host of reconfiguration problems derived fromNP-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., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/9858

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 amortized 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., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/7872

The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application of the decomposition, we show an O(n) time and space algorithm for finding a longest path in a bipartite permutation graph., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/7873

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., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/7891

This paper deal with the graph isomorphism (GI) problem for two graph classes: chordal bipartite graphs and strongly chrdal graphs. It is known that GI problem is GI complete for some special graph classes including regular graphs, bipartite graphs, chordal graphs, comparability graphs, split graphs, and k-trees for unbounded k. On the other side, the relative complexity of the GI problem for the above classes was unknown. We prove that deciding isomorphism of the classes are GI complete., identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/4896

Background: In 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. Results: We 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 NPcomplete problems. Conclusions: A 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, identifier:https://dspace.jaist.ac.jp/dspace/handle/10119/8529