Tillmann Miltzow

Computer Science Department, Université Libre de Bruxelles, Brussels

2018/7/24 Tue 5PM-6PM (Room 3434, Bldg. E6-1)

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem [...]

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]]>Tillmann Miltzow

Computer Science Department, Université Libre de Bruxelles, Brussels

Computer Science Department, Université Libre de Bruxelles, Brussels

2018/7/24 Tue 5PM-6PM (Room 3434, Bldg. E6-1)

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational geometry. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p∈P is seen by at least one guard g∈G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃R. The class ∃R consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP⊆∃R. We prove that the art gallery problem is ∃R-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃R. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many geometric approaches to the problem. This is joint work with Mikkel Abrahamsen and Anna Adamaszek.

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]]>Lectures on topological methods in combinatorics

Ron Aharoni

Department of Mathematics, Technion, Israel

2018/7/17-19 2PM-5PM (Room 3434, Bldg. E6-1)

The lectures will give an introduction to the application of topological methods in matching theory, graph theory, and combinatorics.

Topics that will be covered:

– A topological extension of Hall’s theorem

– combinatorial applications of the nerve theorem

– d-Leray complexes and [...]

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]]>Lectures on topological methods in combinatorics

Ron Aharoni

Department of Mathematics, Technion, Israel

Department of Mathematics, Technion, Israel

2018/7/17-19 2PM-5PM (Room 3434, Bldg. E6-1)

The lectures will give an introduction to the application of topological methods in matching theory, graph theory, and combinatorics.

Topics that will be covered:

Topics that will be covered:

– A topological extension of Hall’s theorem

– combinatorial applications of the nerve theorem

– d-Leray complexes and rainbow matchings

– Matroid complexes and applications

– Open problems

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]]>A course in graph embedding

Jaehoon Kim (김재훈)

School of Mathematics, Birmingham University, Birmingham, UK

2018/6/25-29 10:30AM-12PM, 2:30PM-4PM (Room 3434, Bldg. E6-1)

In this lecture, we aim to learn several techniques to find sufficient conditions on a dense graph G to contain a sparse graph H as a subgraph.

Lecture note (PDF file)

Tentative plan for the course (June 25, [...]

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]]>A course in graph embedding

Jaehoon Kim (김재훈)

School of Mathematics, Birmingham University, Birmingham, UK

School of Mathematics, Birmingham University, Birmingham, UK

2018/6/25-29 10:30AM-12PM, 2:30PM-4PM (Room 3434, Bldg. E6-1)

In this lecture, we aim to learn several techniques to find sufficient conditions on a dense graph G to contain a sparse graph H as a subgraph.

Lecture note (PDF file)

**Tentative plan for the course (June 25, 26, 27, 28, 29)**

Lecture 1 : Basic probabilistic methods

Lecture 2 : Extremal number of graphs

Lecture 3 : Extremal number of even cycles

Lecture 4 : Dependent random choice

Lecture 5 : The regularity lemma and its applications

Lecture 6 : Embedding large graphs

Lecture 7 : The blow-up lemma and its applications I

Lecture 8 : The blow-up lemma and its applications II

Lecture 9 : The absorbing method I

Lecture 10 : The absorbing method II

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]]>Jinyoung Park (박진영)

Department of Mathematics, Rutgers, Piscataway, NJ, USA

2018/06/26 Tuesday 5PM

We discuss the number of proper colorings of hypercubes

given q colors. When q=2, it is easy to see that there are only 2

possible colorings. However, it is already highly nontrivial to figure

out the number of colorings when q=3. Since Galvin (2002) proved the

asymptotics of [...]

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]]>Jinyoung Park (박진영)

Department of Mathematics, Rutgers, Piscataway, NJ, USA

Department of Mathematics, Rutgers, Piscataway, NJ, USA

2018/06/26 Tuesday 5PM

We discuss the number of proper colorings of hypercubes

given q colors. When q=2, it is easy to see that there are only 2

possible colorings. However, it is already highly nontrivial to figure

out the number of colorings when q=3. Since Galvin (2002) proved the

asymptotics of the number of 3-colorings, the rest cases remained open

so far. In this talk, I will introduce a recent work on the number of

4-colorings, mainly focusing on how entropy can be used in counting.

This is joint work with Jeff Kahn.

given q colors. When q=2, it is easy to see that there are only 2

possible colorings. However, it is already highly nontrivial to figure

out the number of colorings when q=3. Since Galvin (2002) proved the

asymptotics of the number of 3-colorings, the rest cases remained open

so far. In this talk, I will introduce a recent work on the number of

4-colorings, mainly focusing on how entropy can be used in counting.

This is joint work with Jeff Kahn.

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]]>Eric Vigoda

College of Computing, Georgia Institute of Technology, Atlanta, GA, USA

2018/6/11 Mon 5PM-6PM

For an unknown graph G on n vertices, given random k-colorings of G, can one learn the edges of G? We present results on identifiability/non-identifiability of the graph G and efficient algorithms for learning G. The results have [...]

The post Eric Vigoda, Learning a graph via random colorings appeared first on KAIST Discrete Math Seminar.

]]>Eric Vigoda

College of Computing, Georgia Institute of Technology, Atlanta, GA, USA

College of Computing, Georgia Institute of Technology, Atlanta, GA, USA

2018/6/11 Mon 5PM-6PM

For an unknown graph G on n vertices, given random k-colorings of G, can one learn the edges of G? We present results on identifiability/non-identifiability of the graph G and efficient algorithms for learning G. The results have interesting connections to statistical physics phase transitions.

This is joint work with Antonio Blanca, Zongchen Chen, and Daniel Stefankovic.

This is joint work with Antonio Blanca, Zongchen Chen, and Daniel Stefankovic.

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]]>Grassmanians and Pseudosphere Arrangements

Michael Dobbins

Department of Mathematics, Binghamton University, Binghamton, NY, USA

2018/5/30 Wednesday 5PM

In this talk I will present a metric space of pseudosphere arrangements, as in the topological representation theorem of oriented matroids, where each pseudosphere is assigned a weight. This gives an extension of the space of [...]

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]]>Grassmanians and Pseudosphere Arrangements

Michael Dobbins

Department of Mathematics, Binghamton University, Binghamton, NY, USA

Department of Mathematics, Binghamton University, Binghamton, NY, USA

2018/5/**30** Wednesday 5PM

In this talk I will present a metric space of pseudosphere arrangements, as in the topological representation theorem of oriented matroids, where each pseudosphere is assigned a weight. This gives an extension of the space of full rank vector configurations of fixed size in a fixed dimension that has nicer combinatorial and topological properties. In rank 3 these spaces, modulo SO(3), are homotopy equivalent to Grassmanians, and the subspaces representing a fixed oriented matroid are contractible. Work on these spaces was partly motivated by combinatorial tools for working with vector bundles.

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]]>Mark Siggers

Department of Mathematics, Kyungpook National University, Daegu

2018/04/03 Tue 5PM

For problems with a discrete set of solutions, a reconfiguration problem defines solutions to be adjacent if they meet some condition of closeness, and then asks for two given solutions it there is a path between them in the set of [...]

The post Mark Siggers, The reconfiguration problem for graph homomorpisms appeared first on KAIST Discrete Math Seminar.

]]>Mark Siggers

Department of Mathematics, Kyungpook National University, Daegu

Department of Mathematics, Kyungpook National University, Daegu

2018/04/03 Tue 5PM

For problems with a discrete set of solutions, a reconfiguration problem defines solutions to be adjacent if they meet some condition of closeness, and then asks for two given solutions it there is a path between them in the set of all solutions.

The reconfiguration problem for graph homomorphisms has seen fair activity in recent years. Fixing a template, the problem Recol(H) for a graph H asks if one can get from one H-colouring of a graph G to another by changing one vertex at a time, always remaining an H-colouring. If the changed vertex has a loop, it must move to an adjecent vertex

Depending on H, the problem seems to be either polynomial time solvable or PSPACE-complete. We discuss many recent results in the area and work towards conjecturing for which H the problem is PSPACE-complete.

This is joint work with Rick Brewster, Jae-baek Lee, Ben Moore and Jon Noel.

The reconfiguration problem for graph homomorphisms has seen fair activity in recent years. Fixing a template, the problem Recol(H) for a graph H asks if one can get from one H-colouring of a graph G to another by changing one vertex at a time, always remaining an H-colouring. If the changed vertex has a loop, it must move to an adjecent vertex

Depending on H, the problem seems to be either polynomial time solvable or PSPACE-complete. We discuss many recent results in the area and work towards conjecturing for which H the problem is PSPACE-complete.

This is joint work with Rick Brewster, Jae-baek Lee, Ben Moore and Jon Noel.

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]]>Hong Liu

Mathematics Institute, University of Warwick, Warwick, UK

2018/4/10 Tue 5PM

Given graphs H1,..., Hk, a graph G is (H1,..., Hk)-free if there is a k-edge-colouring of G with no Hi in colour-i for all i in {1,2,...,k}. Fix a function f(n), the Ramsey-Turán function rt(n,H1,...,Hk,f(n)) is the maximum size of an n-vertex [...]

The post Hong Liu, Two conjectures in Ramsey-Turán theory appeared first on KAIST Discrete Math Seminar.

]]>Hong Liu

Mathematics Institute, University of Warwick, Warwick, UK

Mathematics Institute, University of Warwick, Warwick, UK

2018/4/10 Tue 5PM

Given graphs H_{1},…, H_{k}, a graph G is (H_{1},…, H_{k})-free if there is a k-edge-colouring of G with no H_{i} in colour-i for all i in {1,2,…,k}. Fix a function f(n), the Ramsey-Turán function rt(n,H_{1},…,H_{k},f(n)) is the maximum size of an n-vertex (H_{1},…, H_{k})-free graph with independence number at most f(n). We determine rt(n,K_{3},K_{s},δn) for s in {3,4,5} and sufficiently small δ, confirming a conjecture of Erdős and Sós from 1979. It is known that rt(n,K_{8},f(n)) has a phase transition at f(n)=Θ(√(n\log n)). We prove that rt(n,K_{8},o(√(n\log n)))=n^{2}/4+o(n^{2}), answering a question of Balogh, Hu and Simonovits. The proofs utilise, among others, dependent random choice and results from graph packings. Joint work with Jaehoon Kim and Younjin Kim.

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]]>Antoine Vigneron

School of Electrical and Computer Engineering, UNIST, Ulsan, South Korea

2018/3/27 Tue 5PM

Given a planar subdivision with n vertices, each face having a cone of possible directions of travel, our goal is to decide which vertices of the subdivision can be reached from a given starting point s. [...]

The post Antoine Vigneron, Reachability in a Planar Subdivision with Direction Constraints appeared first on KAIST Discrete Math Seminar.

]]>Antoine Vigneron

School of Electrical and Computer Engineering, UNIST, Ulsan, South Korea

School of Electrical and Computer Engineering, UNIST, Ulsan, South Korea

2018/3/27 Tue 5PM

Given a planar subdivision with n vertices, each face having a cone of possible directions of travel, our goal is to decide which vertices of the subdivision can be reached from a given starting point s. We give an O(n log n)-time algorithm for this problem, as well as an Ω(n log n) lower bound in the algebraic computation tree model. We prove that the generalization where two cones of directions per face are allowed is NP-hard.

The post Antoine Vigneron, Reachability in a Planar Subdivision with Direction Constraints appeared first on KAIST Discrete Math Seminar.

]]>Otfried Cheong

School of Computing, KAIST

2017/3/20 Tue 5PM

We prove a generalization of Pal's 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360 degrees inside Q. We also prove a lower bound [...]

The post Otfried Cheong, The reverse Kakeya problem appeared first on KAIST Discrete Math Seminar.

]]>Otfried Cheong

School of Computing, KAIST

School of Computing, KAIST

2017/3/20 Tue 5PM

We prove a generalization of Pal’s 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360 degrees inside Q. We also prove a lower bound of Ω(m n^{2}) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.

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