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Combinatorics, Theoretical Computer Science, Algorithm Theory, and Optimization Theory Lab (Hiroshi Hirai)
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A lab studying the theory of optimization and algorithms.
AboutAbout the lab
Algorithms and optimization, together with discrete mathematics.
In the lab of Professor Hiroshi Hirai, research covers optimization, algorithms, and discrete mathematics.
The lab is attempting to develop the theory of optimization and algorithms on metric spaces of nonpositive curvature, which have properties similar to Euclidean space.
It also works on combinatorial optimization that uses computations such as determinants, and on compact representations of lattices and related algorithms.
For undergraduate thesis research, students read texts on optimization, algorithms, and combinatorics in a reading group, and the lab states that programming ability is not required.
Key terms
- Combinatorial Optimization
- A theory of how to select the most favorable choice from a huge but finite set of combinations.
- CAT(0) Space
- A metric space of nonpositive curvature with properties similar to Euclidean space.
- Discrete Convex Analysis
- The theory of convex functions on integer lattice points.
ResearchResearch themes
The lab aims to advance the theory of optimization and algorithms on mathematical stages such as metric spaces and discrete structures. Its research introduction page lists nonpositively curved spaces, algebraic combinatorial optimization, discrete convex analysis, and lattice theory as recent interests.
*Illustrative imageOptimization on Nonpositively Curved Spaces
An attempt to develop the theory of optimization and algorithms on metric spaces of nonpositive curvature, which have properties similar to Euclidean space, in particular on CAT(0) spaces.
*Illustrative imageAlgebraic Combinatorial Optimization
The lab studies combinatorial optimization algorithms that use computations of determinants, Pfaffians, permanents, and hafnians. It also treats linear-algebraic versions of problems in which "subsets" are replaced by "subspaces" and "number of elements" by "dimension."
*Illustrative imageExtensions of Discrete Convex Analysis
An attempt to develop discrete convex analysis not only on integer lattice points but also on more general discrete structures such as graphs, direct products of trees, and Euclidean buildings.
*Illustrative imageCompact Representations of Lattices and Algorithms
The lab studies compact representations of lattices and semilattices, and algorithms on them.
MembersFaculty
- ProfessorHiroshi Hirai
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Imperialist-competitive-algorithm-flowchart-bg” Original author: Icasite Author of the derivative work: Spiritia / CC BY-SA 3.0 (cropped)
- Optimization on Nonpositively Curved Spaces: “Heptagonal tiling honeycomb ideal surface” Roice3 / CC BY-SA 4.0 (cropped)
- Algebraic Combinatorial Optimization: “Determinant as signed volume under multiplication by elementary matrices” Eteri byazrova / CC BY-SA 4.0 (cropped)
- Extensions of Discrete Convex Analysis: “Convex vs. Not-convex” Varagk / CC BY-SA 4.0 (cropped)
- Compact Representations of Lattices and Algorithms: “Finite set-theoretic trees” Jacob Rus / CC BY-SA 4.0 (cropped)
- Related lab (Differential Geometry Lab): “Surfaces of constant positive Gaussian curvature” Salix alba / CC BY-SA 4.0 (cropped)
- Related lab (Graph Theory, Theoretical Computer Science, and Algorithms Lab): “Moreno Sociogram 5th Grade” Martin Grandjean / CC BY-SA 4.0 (cropped)
- Related lab (Ring Theory and Representation Theory Lab): “Combined group axiom commutative diagram” Struoc / CC BY-SA 4.0 (cropped)
- Related lab (Fourier Analysis and Partial Differential Equations Lab): “Ocean waves water” Jon Sullivan / Public domain (cropped)
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