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Papers by Kenji Kashiwabara

Proceedings of the 6th on ASIA Public-Key Cryptography Workshop - APKC '19

IACR Cryptol. ePrint Arch., 2018

The lattice basis reduction algorithm is a method for solving the Shortest Vector Problem (SVP) o... more The lattice basis reduction algorithm is a method for solving the Shortest Vector Problem (SVP) on lattices. There are many variants of the lattice basis reduction algorithm such as LLL, BKZ, and RSR. Though BKZ has been used most widely, it is shown recently that some variants of RSR are quite efficient for solving a high-dimensional SVP (they achieved many best scores in TU Darmstadt SVP challenge). RSR repeats alternately the generation of new very short lattice vectors from the current basis (we call this procedure “random sampling”) and the improvement of the current basis by utilizing the generated very short lattice vectors. Therefore, it is important for investigating and ameliorating RSR to estimate the success probability of finding very short lattice vectors by combining the current basis. In this paper, we propose a new method for estimating the success probability by the Gram-Charlier approximation, which is a basic asymptotic expansion of any probability distribution b...

Fundamenta Informaticae, 2020

Journal of Graphic Science of Japan, 2006

We consider the following sports scheduling problem. Consider 2n teams in a sport league. Each pa... more We consider the following sports scheduling problem. Consider 2n teams in a sport league. Each pair of teams must play exactly one match in 2n − 1 days. That is, n games are held simultaneously in a day. We want to make a schedule which has n(2n − 1) games for 2n − 1 days. When we make a schedule, the schedule must satisfy a constraint according to the HAP table, which designates a home game or an away game for each team and each date. Two teams cannot play against each other unless one team is assigned to a home game and the other team is assigned to an away game. Recently, D. Briskorn proposed a necessary condition for a HAP table to have a proper schedule. And he proposed a conjecture that such a condition is also sufficient. That is, if a solution to the linear inequalities exists, they must have an integral solution. In this paper, we rewrite his conjecture by using perfect matchings. We consider a monoid in the affine space generated by perfect matchings. In terms of the Hilbe...

arXiv: Geometric Topology, 2017

The reductivity of a spherical curve is the minimal number of a local transformation called an in... more The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configurations for a spherical curve with reductivity four is given by focusing on 5-gons. It has also been unknown if there exists a reduced spherical curve which has no 2-gons and 3-gons of type A, B and C. This paper gives the answer to the question by constructing such a spherical curve.

Convex geometries are closure spaces which satisfy anti-exchange property, and they are known as ... more Convex geometries are closure spaces which satisfy anti-exchange property, and they are known as dual of antimatroids. We consider functions defined on the sets of the extreme points of a convex geometry. Faigle-Kern (Math. Programming 72 (1996) 195-206) presented a greedy algorithm to linear programming problems for shellings of posets, and Kruger (Discrete Appl. Math. 99 (2002) 125-148) introduced b-submodular functions and proved that Faigle-Kern's algorithm works for shellings of posets if and only if the given set function is b-submodular. We extend their results to all classes of convex geometries, that is, we prove that the same algorithm works for all convex geometries if and only if the given set function on the extreme sets is submodular in our sense.

White conjectured that the toric ideal associated with the basis of a matroid is generated by qua... more White conjectured that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges. We present a combinatorial proof of White's conjecture for matroids of rank 3 by using a lemma proposed by Blasiak.

The weak-map order on the matroid base polytopes is the partial order defined by inclusion. Lucas... more The weak-map order on the matroid base polytopes is the partial order defined by inclusion. Lucas proved that the base polytope of no binary matroid includes the base polytope of a connected matroid. A matroid base polytope is said to be decomposable when it has a polytopal decomposition which consists of at least two matroid base polytopes. We shed light on the relation between the decomposability and the weak-map order of matroid base polytopes. We classify matroids into five types with respect to the weak-map order and decomposability. We give an example of a matroid in each class. Moreover, we give a counterexample to a conjecture proposed by Lucas, which says that, when one matroid base polytope covers another matroid base polytope with respect to inclusion, the latter matroid base polytope should be a facet of the former matroid base polytope.

The Electronic Journal of Combinatorics

A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for... more A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for closure systems or equivalently for closure operators. A rooted circuit is a specific type of a pair (X,e)(X,e)(X,e) of a subset XXX, called a stem, and an element enotinXe\not\in XenotinX, called a root. We introduce a notion called a 'prime stem', which plays the key role in this article. Every prime stem is shown to be a pseudo-closed set of an implicational system. If the sizes of stems are all the same, the stems are all pseudo-closed sets, and they give rise to a canonical minimum implicational basis. For an affine convex geometry, the prime stems determine a canonical minimum basis, and furthermore gives rise to an optimal basis. A 'critical rooted circuit' is a special case of a rooted circuit defined for an antimatroid. As a precedence structure, 'critical rooted circuits' are necessary and sufficient to fix an antimatroid whereas critical rooted circuits are not necessari...

The Electronic Journal of Combinatorics

We consider the following sports scheduling problem. Consider 2n2n2n teams in a sport league. Each ... more We consider the following sports scheduling problem. Consider 2n2n2n teams in a sport league. Each pair of teams must play exactly one match in 2n−12n-12n−1 days. That is, nnn games are held simultaneously in a day. We want to make a schedule which has n(2n−1)n(2n-1)n(2n−1) games for 2n−12n-12n−1 days. When we make a schedule, the schedule must satisfy a constraint according to the HAP set, which designates a home game or an away game for each team and each date. Two teams cannot play against each other unless one team is assigned to a home game and the other team is assigned to an away game. Recently, D. Briskorn proposed a necessary condition for an HAP set to have a proper schedule. And he proposed a conjecture that such a condition is also sufficient. That is, if a solution to the linear inequalities exists, they must have an integral solution. In this paper, we rewrite his conjecture by using perfect matchings. We consider a monoid in the affine space generated by perfect matchings. In terms of the Hilb...

The Electronic Journal of Combinatorics

White conjectured that the toric ideal associated with the basis of a matroid is generated by qua... more White conjectured that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges. We present a combinatorial proof of White's conjecture for matroids of rank 3 by using a lemma proposed by Blasiak.

Lecture Notes in Computer Science, 1999

ABSTRACT

Journal of Information Processing, 2015

We shed light on the relation between the decomposability and the weak-map order of matroid base ... more We shed light on the relation between the decomposability and the weak-map order of matroid base polytopes. We classify matroids into five types with respect to the weak-map order and decomposability. We give an example of a matroid in each class. Moreover, we give a counterexample to a conjecture proposed by Lucas, which says that, when one matroid base polytope covers another matroid base polytope with respect to inclusion, the latter matroid base polytope should be a facet of the former matroid base polytope.

2010 9th International Conference on Information Technology Based Higher Education and Training (ITHET), 2010

We propose the concept of a word set for use in a teaching material generation system. The system... more We propose the concept of a word set for use in a teaching material generation system. The system suggests the level of difficulty of a reading material for a student and generates a personalized glossary. We developed a prototype system based on this concept. The reading material is expressed by a word set and is operated by utilizing set operations in the prototype system. Then, we tested and confirmed the system is promising.

Proceedings of the 9th annual ACM international workshop on Web information and data management - WIDM '07, 2007

Recurring patterns of the same link structure can often be observed on Web sites. Patterns are im... more Recurring patterns of the same link structure can often be observed on Web sites. Patterns are important for site administrators in revising Web sites but are difficult to find empirically. We propose a method for detecting such patterns. The first step in the method is viewing a Web site as a directed graph, and identifying pages that have the same substructure by the Anti-Foundation Axiom (AFA). The AFA is a non-standard set theory that allows for a circular structure. The pages identified by AFA are divided into connected components. Then, meaningful sets of pages are selected as patterns by using the Galois lattice of the binary relation between the pages and the connected components. We apply our method to three actual Web sites and succeed in detecting patterns within the target sites.

Proceedings of the sixteenth ACM conference on Hypertext and hypermedia - HYPERTEXT '05, 2005

ABSTRACT

Proceedings of the 6th on ASIA Public-Key Cryptography Workshop - APKC '19

IACR Cryptol. ePrint Arch., 2018

The lattice basis reduction algorithm is a method for solving the Shortest Vector Problem (SVP) o... more The lattice basis reduction algorithm is a method for solving the Shortest Vector Problem (SVP) on lattices. There are many variants of the lattice basis reduction algorithm such as LLL, BKZ, and RSR. Though BKZ has been used most widely, it is shown recently that some variants of RSR are quite efficient for solving a high-dimensional SVP (they achieved many best scores in TU Darmstadt SVP challenge). RSR repeats alternately the generation of new very short lattice vectors from the current basis (we call this procedure “random sampling”) and the improvement of the current basis by utilizing the generated very short lattice vectors. Therefore, it is important for investigating and ameliorating RSR to estimate the success probability of finding very short lattice vectors by combining the current basis. In this paper, we propose a new method for estimating the success probability by the Gram-Charlier approximation, which is a basic asymptotic expansion of any probability distribution b...

Fundamenta Informaticae, 2020

Journal of Graphic Science of Japan, 2006

We consider the following sports scheduling problem. Consider 2n teams in a sport league. Each pa... more We consider the following sports scheduling problem. Consider 2n teams in a sport league. Each pair of teams must play exactly one match in 2n − 1 days. That is, n games are held simultaneously in a day. We want to make a schedule which has n(2n − 1) games for 2n − 1 days. When we make a schedule, the schedule must satisfy a constraint according to the HAP table, which designates a home game or an away game for each team and each date. Two teams cannot play against each other unless one team is assigned to a home game and the other team is assigned to an away game. Recently, D. Briskorn proposed a necessary condition for a HAP table to have a proper schedule. And he proposed a conjecture that such a condition is also sufficient. That is, if a solution to the linear inequalities exists, they must have an integral solution. In this paper, we rewrite his conjecture by using perfect matchings. We consider a monoid in the affine space generated by perfect matchings. In terms of the Hilbe...

arXiv: Geometric Topology, 2017

The reductivity of a spherical curve is the minimal number of a local transformation called an in... more The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configurations for a spherical curve with reductivity four is given by focusing on 5-gons. It has also been unknown if there exists a reduced spherical curve which has no 2-gons and 3-gons of type A, B and C. This paper gives the answer to the question by constructing such a spherical curve.

Convex geometries are closure spaces which satisfy anti-exchange property, and they are known as ... more Convex geometries are closure spaces which satisfy anti-exchange property, and they are known as dual of antimatroids. We consider functions defined on the sets of the extreme points of a convex geometry. Faigle-Kern (Math. Programming 72 (1996) 195-206) presented a greedy algorithm to linear programming problems for shellings of posets, and Kruger (Discrete Appl. Math. 99 (2002) 125-148) introduced b-submodular functions and proved that Faigle-Kern's algorithm works for shellings of posets if and only if the given set function is b-submodular. We extend their results to all classes of convex geometries, that is, we prove that the same algorithm works for all convex geometries if and only if the given set function on the extreme sets is submodular in our sense.

White conjectured that the toric ideal associated with the basis of a matroid is generated by qua... more White conjectured that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges. We present a combinatorial proof of White's conjecture for matroids of rank 3 by using a lemma proposed by Blasiak.

The weak-map order on the matroid base polytopes is the partial order defined by inclusion. Lucas... more The weak-map order on the matroid base polytopes is the partial order defined by inclusion. Lucas proved that the base polytope of no binary matroid includes the base polytope of a connected matroid. A matroid base polytope is said to be decomposable when it has a polytopal decomposition which consists of at least two matroid base polytopes. We shed light on the relation between the decomposability and the weak-map order of matroid base polytopes. We classify matroids into five types with respect to the weak-map order and decomposability. We give an example of a matroid in each class. Moreover, we give a counterexample to a conjecture proposed by Lucas, which says that, when one matroid base polytope covers another matroid base polytope with respect to inclusion, the latter matroid base polytope should be a facet of the former matroid base polytope.

The Electronic Journal of Combinatorics

A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for... more A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for closure systems or equivalently for closure operators. A rooted circuit is a specific type of a pair (X,e)(X,e)(X,e) of a subset XXX, called a stem, and an element enotinXe\not\in XenotinX, called a root. We introduce a notion called a 'prime stem', which plays the key role in this article. Every prime stem is shown to be a pseudo-closed set of an implicational system. If the sizes of stems are all the same, the stems are all pseudo-closed sets, and they give rise to a canonical minimum implicational basis. For an affine convex geometry, the prime stems determine a canonical minimum basis, and furthermore gives rise to an optimal basis. A 'critical rooted circuit' is a special case of a rooted circuit defined for an antimatroid. As a precedence structure, 'critical rooted circuits' are necessary and sufficient to fix an antimatroid whereas critical rooted circuits are not necessari...

The Electronic Journal of Combinatorics

We consider the following sports scheduling problem. Consider 2n2n2n teams in a sport league. Each ... more We consider the following sports scheduling problem. Consider 2n2n2n teams in a sport league. Each pair of teams must play exactly one match in 2n−12n-12n−1 days. That is, nnn games are held simultaneously in a day. We want to make a schedule which has n(2n−1)n(2n-1)n(2n−1) games for 2n−12n-12n−1 days. When we make a schedule, the schedule must satisfy a constraint according to the HAP set, which designates a home game or an away game for each team and each date. Two teams cannot play against each other unless one team is assigned to a home game and the other team is assigned to an away game. Recently, D. Briskorn proposed a necessary condition for an HAP set to have a proper schedule. And he proposed a conjecture that such a condition is also sufficient. That is, if a solution to the linear inequalities exists, they must have an integral solution. In this paper, we rewrite his conjecture by using perfect matchings. We consider a monoid in the affine space generated by perfect matchings. In terms of the Hilb...

The Electronic Journal of Combinatorics

White conjectured that the toric ideal associated with the basis of a matroid is generated by qua... more White conjectured that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges. We present a combinatorial proof of White's conjecture for matroids of rank 3 by using a lemma proposed by Blasiak.

Lecture Notes in Computer Science, 1999

ABSTRACT

Journal of Information Processing, 2015

We shed light on the relation between the decomposability and the weak-map order of matroid base ... more We shed light on the relation between the decomposability and the weak-map order of matroid base polytopes. We classify matroids into five types with respect to the weak-map order and decomposability. We give an example of a matroid in each class. Moreover, we give a counterexample to a conjecture proposed by Lucas, which says that, when one matroid base polytope covers another matroid base polytope with respect to inclusion, the latter matroid base polytope should be a facet of the former matroid base polytope.

2010 9th International Conference on Information Technology Based Higher Education and Training (ITHET), 2010

We propose the concept of a word set for use in a teaching material generation system. The system... more We propose the concept of a word set for use in a teaching material generation system. The system suggests the level of difficulty of a reading material for a student and generates a personalized glossary. We developed a prototype system based on this concept. The reading material is expressed by a word set and is operated by utilizing set operations in the prototype system. Then, we tested and confirmed the system is promising.

Proceedings of the 9th annual ACM international workshop on Web information and data management - WIDM '07, 2007

Recurring patterns of the same link structure can often be observed on Web sites. Patterns are im... more Recurring patterns of the same link structure can often be observed on Web sites. Patterns are important for site administrators in revising Web sites but are difficult to find empirically. We propose a method for detecting such patterns. The first step in the method is viewing a Web site as a directed graph, and identifying pages that have the same substructure by the Anti-Foundation Axiom (AFA). The AFA is a non-standard set theory that allows for a circular structure. The pages identified by AFA are divided into connected components. Then, meaningful sets of pages are selected as patterns by using the Galois lattice of the binary relation between the pages and the connected components. We apply our method to three actual Web sites and succeed in detecting patterns within the target sites.

Proceedings of the sixteenth ACM conference on Hypertext and hypermedia - HYPERTEXT '05, 2005

ABSTRACT