The combinatorial properties much-loved Erdős-Rényi random graph G(n,p), which has n vertices and whose edges are present independently with probability p, have been comprehensively studied in the decades since its introduction. In recent years, much research has been devoted to the randomly perturbed graph model, introduced in 2003 by Bohman, Frieze and Martin. In this talk we shall present and motivate this new model of random graphs, and then focus on the Ramsey properties of these randomly perturbed graphs. More precisely, given a pair of graphs (F,H), we ask how many random edges must be added to a dense graph G to ensure that any two-colouring of the edges of the perturbed graph has either a red copy of F or a blue copy of G. This question was first studied in 2006 by Krivelevich, Sudakov and Tetali, who answered it in the case of F being a triangle and H being a clique. We generalise these results, considering pairs of larger cliques, and, should the audience be willing (but even otherwise), shall take a quick look at some of the ideas required in our proofs. This is joint work with Andrew Treglown (Birmingham).

Location: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin Room 005 (Ground Floor)

Dec 17, 2018 | 02:15 PM

Location: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin Room 005 (Ground Floor)

Dec 17, 2018 | 04:00 PM s.t.

In this talk we will look at a new variant of the polynomial method which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like $\mathbb{Z}_4^n$ (Croot, Lev and myself) and $\mathbb{Z}_3^n$ (Ellenberg and Gijswijt) are exponentially small (compared to the size of the group).We will discuss lower and upper bounds for the size of the extremal subsets, including some recent bounds found by Elsholtz and myself. We will also mention some further applications of the method, for instance, the solution of the Erd\H{o}s-Szemer\'edi sunflower conjecture.

Location: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin Room 005 (Ground Floor)

Jan 07, 2019 | 02:15 PM

A major theme in both extremal and probabilistic combinatorics is to find the appearance thresholds for certain spanning structures. Classical examples of such spanning structures include perfect matchings, Hamilton cycles and $H$-tilings, where we look for vertex disjoint copies of $H$ covering all the vertices of some host graph $G$. In this talk we will focus on $H$-tilings in the case that $H$ is a clique, a natural generalisation of a perfect matching. On the one hand there is the extremal question, how large does the minimum degree of an $n$ vertex graph $G$ have to be to guarantee the existence of a clique factor in $G$? On the other hand, there is the probabilistic question. How large does $p$ need to be to almost surely ensure the appearance of a clique factor in the Erd\H{o}s R\'enyi random graph $G(n,p)$? Optimal answers to these questions were given in two famous papers. The extremal question was answered by Hajnal and Szemer\'edi in 1970 and the probabilistic question by Johansson, Kahn and Vu in 2008. In this talk we bridge the gap between these two results by approaching the following question which contains the previous questions as special cases. Given an arbitrary graph of some fixed minimum degree, how many random edges need to be added on the same set of vertices to ensure the existence of a clique tiling? We give optimal answers to this question in all cases. Such results are part of a recent research trend studying properties of what is known as the randomly perturbed graph model, introduced by Bohman, Frieze and Martin in 2003. This is joint work with Jie Han and Andrew Treglown.

Jan 07, 2019 | 04:00 PM s.t.

We present an algorithm to compute all $n$ nondominated points of a multicriteria discrete optimization problem with $d$ objectives using at most $O(n^{\lfloor d/2 \rfloor})$ scalarizations. The method is similar to an algorithm by Klamroth et al. (2015) with the same complexity. As a difference, our method employs a tropical convex hull computation, and it exploits a particular kind of duality which is special for the tropical cones arising. This duality can be seen as a generalization of the Alexander duality of monomial ideals. Joint work with Georg Loho.

Location: Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin Room MA 041 (Ground Floor)

Jan 14, 2019 | 02:15 PM

Location: Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin Room MA 041 (Ground Floor)

Jan 14, 2019 | 04:00 PM s.t.

In a two-colouring of the edges of the complete graph on the natural numbers, what is the densest monochromatic infinite path that we can always find? We measure the density of a path by the upper asymptotic density of its vertex set. This question was first studied by Erdös and Galvin, who proved that the best density is between 2/3 and 8/9. In this talk we settle this question by proving that we can always find a monochromatic path of upper density at least (12+sqrt(8))/17=0.87226…, and constructing a two-colouring in which no denser path exists. This represents joint work with Jan Corsten, Louis DeBiasio and Richard Lang.

Dec 10, 2018 | 04:00 PM s.t.

Progress in satisfiability (SAT) solving has enabled answering long-standing open questions in mathematics completely automatically resulting in clever though potentially gigantic proofs. We illustrate the success of this approach by presenting the solution of the Boolean Pythagorean triples problem. We also produced and validated a proof of the solution, which has been called the ``largest math proof ever''. The enormous size of the proof is not important. In fact a shorter proof would have been preferable. However, the size shows that automated tools combined with super computing facilitate solving bigger problems. Moreover, the proof of 200 terabytes can now be validated using highly trustworthy systems, demonstrating that we can check the correctness of proofs no matter their size.

Dec 10, 2018 | 02:15 PM

Two incidences (u,e) and (v,f) of vertices u, v and edges e, f (respectively) are adjacent if u=v, or e=f, or uv is one of edges e, f. An incidence coloring of a graph G is a coloring of its incidences such that adjacent incidences have distinct colors. We show that every graph of maximal degree 4 has an incidence coloring with 7 colors. Furthermore, we present sufficient conditions for Cartesian product graphs to have incidence colorings with Delta+2 colors where Delta is the maximal degree. In particular, we confirm a conjecture of Pai et al. on incidence colorings of hypercubes. Joint work with B. Lužar and R. Soták.

Dec 03, 2018 | 04:00 PM s.t.

Binary search trees (BSTs) and heaps are perhaps the simplest implementations of the dictionary and the priority queue data types. They are among the most extensively studied structures in computer science, yet, many basic questions about them remain open. What is the best strategy for updating a BST in response to queries? Is there a single strategy that is optimal for every possible scenario? Are self-adjusting trees ("splay trees", Sleator, Tarjan, 1983) optimal? In which cases can we improve upon the logarithmic worst-case cost per operation? Our understanding of heaps is even more limited. Fibonacci heaps (and their relatives) are theoretically optimal in a worst-case sense, but they perform operations outside the "pure" comparison-based heap model (in addition to being complicated in practice). Is there a simple, "self-adjusting" alternative to Fibonacci heaps? Is there, by analogy to BSTs, a heap that can adapt to regularities in the input? Are the two problems related? In my talk I will present some old and new results pertaining to this family of questions.

Dec 03, 2018 | 02:15 PM

The notions of bounded expansion and nowhere denseness capture uniform sparseness of graph classes and render various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce structurally bounded expansion and structurally nowhere dense graph classes, dfined as first-order interpretations of bounded expansion and nowhere dense graph classes. As a first step towards their algorithmic treatment, we provide a characterization of structurally bounded expansion classes via low shrubdepth decompositions, a dense analogue of low treedepth decompositions. We prove that structurally nowhere dense graph classes are vc-minimal.

Location: Humboldt-Universität zu Berlin Institut für Informatik Humboldt-Kabinett (between House 3&4 / 1st Floor [British Reading]) Johann von Neumann-Haus Rudower Chaussee 25 12489 Berlin

Nov 26, 2018 | 04:00 PM s.t.

While most NP-hard graph problems remain NP-hard when restricted to planar graphs, they become somewhat simpler: they often admit algorithms with running time exponential in the square root of the size of the graph (or some other meaningful parameter). This behavior has been dubbed "the square root phenomenon". For many problems, such an algorithm follows from the theory of bidimensionality relying on a linear dependency on the size of the largest grid minor and treewidth of a planar graph. In recent years we have seen a number of other algorithmic techniques, significantly extending the boundary of the applicability of the "square root phenomenon". In my talk I will survey this area and highlight main algorithmic approaches.

Location: Humboldt-Universität zu Berlin Institut für Informatik Humboldt-Kabinett (between House 3&4 / 1st Floor [British Reading]) Johann von Neumann-Haus Rudower Chaussee 25 12489 Berlin

Nov 26, 2018 | 02:15 PM

The associahedron has a natural extension called the multiassociahedron. It is a vertex-decomposable simplicial sphere (in other words, a "combinatorially nice" sphere). Since at least 2003, people tried to construct simplicial polytopes, whose boundary is this simplicial complex, with limited success. In this talk, I will reveal (at least) three combinatorial "known-to-be-challenging" computational problems which should be indispensably faced to solve this problem.

Location: Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin Room MA 041 (Ground Floor)

Nov 19, 2018 | 04:00 PM s.t.

Picture yourself in a committee numerically evaluating a scientific proposal that you find worth funding: A rating of "0" means "easily achievable but not at all innovative", whereas "1" means "very innovative but totally unachievable". In both cases, funding is not recommended. In contrast, "1/2" means "innovative and achievable", in other words: worth funding. All intermediate values are possible. Any rating that is closer to "1/2" than to "1/4" and "3/4" is considered a vote for funding. The proposal passes if 50% of the members support funding, i.e., rate the proposal between "1/2 - 1/8" and "1/2 + 1/8". Now, there are 10 meetings ahead of you. You have an idea how the opinions of the committee members develop. How should your statements look like in the meetings one through ten if you want to have eventually as many supporters of the proposal as possible? This is an instance of the "Optimal Diplomacy Problem" (ODP), introduced by Hegselmann, König, Kurz, Niemann, and Rambau in 2010, published in 2015. How do opinions interact? How is the dynamics of opinions modeled mathematically? What does it mean to "influence others" in this dynamical system? How difficult is it to find optimal diplomacies? How can one compute or at least narrow down optimal diplomacies? What happens if not all informations about committee members are known to the diplomat? In this talk we will discuss our findings, techniques, and open questions based on the arguably most influential model: the Bounded-Confidence model by Hegselmann and Krause. We draw on joint work with Andreas Deuerling, Rainer Hegselmann, Stefan König, Julia Kinkel, Sascha Kurz, and Christoph Niemann.

Nov 19, 2018 | 02:15 PM

Kernelization is a key concept in fixed-parameter algorithmics for polynomial-time preprocessing of NP-hard problems. Herein one seeks to preprocess any instance of a parameterized problem with parameter k to an “equivalent” instance (the so-called (problem) kernel) with size upper-bounded by some function (preferably by some polynomial) in k. Ten years ago, with the development of the so-called composition technique, first NP-hard parameterized problems were proven to presumably admit no polynomial-size problem kernel. Since then the line of Research concerning "limits of kernelization" received a lot of attention. We contribute to this line and present a new technique exploiting triangle-based fractal structures (so called T-fractals) for extending the range of applicability of compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of graph problems dealing with some sort of cuts, hereby answering an open question stated in 2009. Our T-fractals---due to their fractal structure---admit easy-to-prove useful properties exploitable for constructing compositions. They also apply for planar as well as directed variants of the basic problems and also apply to both edge and vertex-deletion problems. This is joint work with Danny Hermelin, André Nichterlein, and Rolf Niedermeier. (Journal version appeared in SIAM Journal on Discrete Mathematics 2018.)

Location: Humboldt-Universität zu Berlin Institut für Informatik Humboldt-Kabinett (between House 3&4 / 1st Floor [British Reading]) Johann von Neumann-Haus Rudower Chaussee 25 12489 Berlin

Nov 12, 2018 | 04:00 PM s.t.

In this lecture I will give an introduction to algebraic and semi-algebraic methods for proving the unsatisfiability of systems of real polynomial equations over the Boolean hypercube. The main goal of this talk is to compare the relative strength of these approaches using methods from proof complexity. On the one hand, I will focus on the static semi-algebraic proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, I will discuss polynomial calculus, which is a dynamic algebraic proof system that models Gröbner basis computations. I am going to present two recent results on the relative strength between algebraic and semi-algebraic proof systems:¹ The first result is that sum-of-squares simulates polynomial calculus: any polynomial calculus refutation of degree d can be transformed into a sum-of-squares refutation of degree 2d and only polynomial increase in size. In contrast, the second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size. ¹) this work was presented at STACS 2018; a preprint is available at https://eccc.weizmann.ac.il/report/2017/154/

Nov 12, 2018 | 02:15 PM

Network analysis defines a number of centrality measures to identify the most central nodes in a network. Fast computation of those measures is a major challenge in algorithmic network analysis. Aside from closeness and betweenness, Katz centrality is one of the established centrality measures. We consider the problem of computing rankings for Katz centrality. In particular, we propose upper and lower bounds on the Katz score of a given node. While previous approaches relied on numerical approximation or heuristics to compute Katz centrality rankings, we construct an algorithm that iteratively improves those upper and lower bounds until a correct Katz ranking is obtained. For a certain class of inputs, this yields an optimal algorithm for Katz ranking computation. Furthermore, Experiments demonstrate that our algorithm outperforms both numerical approaches and heuristics with speedups between 1.5x and 3.5x, depending on the desired quality guarantees. Specifically, we provide efficient parallel CPU and GPU implementations of the algorithms that enable near real-time Katz centrality computation for graphs with hundreds of millions of nodes in fractions of seconds.

Location: Humboldt-Universität zu Berlin Institut für Informatik Room 3.408 (House 3 / 4th Floor [British Reading]) Johann von Neumann-Haus Rudower Chaussee 25 12489 Berlin

Nov 05, 2018 | 04:00 PM s.t.

Network centrality measures indicate the importance of nodes (or edges) in a network. In this talk we will discuss a few popular measures and algorithms for computing complete or partial node rankings based on these measures. These algorithms are implemented in NetworKit, an open-source framework for large-scale network analysis, on which we provide an overview, too. One focus of the talk will be on techniques for speeding up a greedy (1-1/e)-approximation algorithm for the NP-hard group closeness centrality problem. Compared to a straightforward implementation, our approach is orders of magnitude faster and, compared to a heuristic proposed by Chen et al., we always find a solution with better quality in a comparable running time in our experiments. Our method Greedy++ allows us to approximate the group with maximum closeness centrality on networks with up to hundreds of millions of edges in minutes or at most a few hours. In a comparison with the optimum, our experiments show that the solution found by Greedy++ is actually much better than the theoretical guarantee.

Location: Humboldt-Universität zu Berlin Institut für Informatik Room 3.408 (House 3 / 4th Floor [British Reading]) Johann von Neumann-Haus Rudower Chaussee 25 12489 Berlin

Nov 05, 2018 | 02:15 PM

For several graph classes without long induced paths there exists a finite forbidden subgraph characterization for k-colourability. Such a finite set of minimal obstructions allows to provide a “no-certificate" which proves that a graph is not k-colourable. We prove that there are only finitely many 4-critical P6-free graphs, and give the complete list that consists of 24 graphs. In particular, we obtain a certifying algorithm for 3-colouring P6-free graphs, which solves an open problem posed by Golovach et al. (Here, P6 denotes the induced path on six vertices.) Our result leads to the following dichotomy theorem: if H is a connected graph, then there are finitely many 4-critical H-free graphs if and only if H is a subgraph of P6. This answers a question of Seymour. The proof of our main result involves two distinct automatic proofs, and an extensive structural analysis by hand. In this talk we will mainly focus on the algorithmic results by presenting a new algorithm for generating all minimal forbidden subgraphs to k-colourability for given graph classes. This algorithm (combined with new theoretical results) has been successfully applied to fully characterise all forbidden subgraphs for k-colourability for various classes of graphs without long induced paths. (This is joint work with Maria Chudnovsky, Oliver Schaudt and Mingxian Zhong.)

Oct 29, 2018 | 04:00 PM s.t.

Perfect graphs are important objects in graph theory. The perfect graphs include many important families of graphs, and serve to unify results relating colorings and cliques in those families. One of the most famous and most important results is the strong perfect graph theorem conjectured by Claude Berge and proved by Chudnovsky, Robertson and Thomas. This theorem characterizes perfect graphs. Our interest is to give other characterizations of perfect graphs. In this talk, we construct several lattice polytopes arising from a finite simple graph and characterize when the graph is perfect in terms of the lattice polytopes. This talk is based on joint work with Takayuki Hibi and Hidefumi Ohsugi.

Oct 22, 2018 | 04:00 PM s.t.

As we tell our undergraduates, if K is a field, then there are a great number of different ways to describe a linear subspace of K^n. If the base is an algebraic object with less structure than a field, linear algebra becomes more subtle, and some of these descriptions cease to agree. One such setting is tropical geometry. Tropical geometers have reached consensus as to what the "correct" notion of tropical linear subspace is (one way to get it is by a vector of determinants). My subject will be one of the "wrong" descriptions, namely row spaces of matrices, which only produces a subset of the tropical linear spaces. Applications include generalisations of Mason's results from the '70s on presentations of transversal matroids, and a construction in the new area of tropical ideal theory. This work is variously joint with Felipe Rinc\'on, Jorge Alberto Olarte, and Jeffrey and Noah Giansiracusa.

Oct 22, 2018 | 02:15 PM

There has recently been a lot of interplay between extremal graph theory and the theory of graph limits. After a brief survey of the theory of dense graph limits, we focus on exploring a link between Sidorenko's Conjecture, one of the most prominent open problems in extremal graph theory, and weakly norming graphs, one of the concepts studied in the theory of graph limits. Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We are interested in a stronger property, which we call the step Sidorenko property. We relate this property to weakly norming graphs and use our results to construct a bipartite edge-transitive graph that is not weakly norming - this answers a question of Hatami [Israel J. Math. 175 (2010), 125-150]. The talk is based on joint work with Taisa Martins, Peter Pal Pach and Marcin Wrochna.

Location: Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin room MA 041 (ground floor)

Oct 08, 2018 | 02:00 PM s.t.

For positive integers N and r >= 2, an r-monotone coloring of r-tuples from [N] is a 2-coloring by -1 and +1 that is monotone on the lexicographically ordered sequence of r-tuples of every (r+1)-tuple from [N]. Let ORS(n;r) be the minimum N such that every r-monotone coloring of r tuples from [N] contains n elements with all r-tuples of the same color. For every r >= 3, it is known that ORS(n;r) is bounded from above by a tower function of height r-2 with O(n) on the top. The Erdős--Szekeres Lemma and the Erdős--Szekeres Theorem imply ORS(n;2)=(n-1)^2+1 and ORS(n;3) = ((2n-4) choose (n-2))+1, respectively. It follows from a result of Eliáš and Matoušek that ORS (n;4) grows as o tower of height 2. We show that ORS(n;r) grows at least as a tower of height r-2 for every r >= 3. This, in particular, solves an open problem posed by Eliáš and Matoušek and by Moshkovitz and Shapira. Using two geometric interpretations of monotone colorings, we show connections between estimating ORS(n;r) and two Ramsey-type problems that have been recently considered by several researchers. We also prove asymptotically tight estimates on the number of r-monotone colorings.

Location: Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin Room MA 041 (ground floor)

Jul 16, 2018 | 04:00 PM

We consider random simplicial complexes that are generated from the binomial random hypergraphs by taking the downward-closure. We determine when all cohomology groups with coefficients in F_2 vanish. This is joint work with Oliver Cooley, Nicola Del Giudice, and Philipp Spruessel.

Location: Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin room MA 041 (ground floor)

Jul 16, 2018 | 02:15 PM

Most computer scientists and discrete mathematicians are familiar with computational complexity due to the famous P = NP conjecture. Some researchers have a more thorough understanding of the classes P, NP, and PSPACE since problems in logic, graph theory, and combinatorial optimization are often classified by their relationship to one of these classes. More recently, these concepts have found a wider audience through their application to puzzles and games. For example, it is known that the number puzzle Sudoku is NP-complete, whereas the box pushing game Sokoban is PSPACE-complete. In this talk we establish the computational complexity of several puzzles and games. We prove that two new paper and pencil puzzles are NP-complete. These puzzles are called Pencils and Sto-Stone, and were created by the Japanese publisher Nikoli that popularized Sudoku. We then prove that several puzzles that have been implemented as video games are PSPACE-complete. These games include a row shifting puzzle called MazezaM, a turnstile puzzle for the Nintendo Game Boy called Kwirk, and a puzzle called Switches that was invented by undergraduate student Jonathan Gabor while he was attending high school. Our reductions use conventional approaches such as Boolean Satisfiability, as well as some less conventional tools including Gray Codes, and the new Constraint Logic model pioneered by Hearn and Demaine in their influential textbook "Games, Puzzles, and Computation". The talk aims to be accessible to a wide audience, and hopes to encourage researchers to seek similar results for problems (or puzzles!) that are close to them. Most of the new results consist of joint work with undergraduate students at Bard College at Simon's Rock.

Location: Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin room MA 041 (ground floor)

Jul 09, 2018 | 04:00 PM

Matrix rank is well-known to be multiplicative under the Kronecker product, additive under the direct sum, normalised on identity matrices and non-increasing under multiplying from the left and from the right by any matrices. In fact, matrix rank is the only real matrix parameter with these four properties. In 1986 Strassen proposed to study the extension to tensors: find all maps from k-tensors to the reals that are multiplicative under the tensor Kronecker product, additive under the direct sum, normalized on diagonal tensors, and non-increasing under acting with linear maps on the k tensor factors. Strassen called the collection of these maps the "asymptotic spectrum of k-tensors". Strassen proved that understanding the asymptotic spectrum implies understanding the asymptotic relations among tensors, including the asymptotic rank. In particular, knowing the asymptotic spectrum means knowing the arithmetic complexity of matrix multiplication, a central problem in algebraic complexity theory.I will give an overview of known elements in the asymptotic spectrum of tensors, including our recent construction which is based on information theory and moment polytopes. This recent construction is joint work with Matthias Christandl and Peter Vrana.Then I will introduce the analogous object in graph theory: the asymptotic spectrum of graphs. I will explain the relation to Shannon capacity and give an overview of known elements in the asymptotic spectrum of graphs.

Location: Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 10623 Berlin Room MA 041 (ground floor)

Jul 09, 2018 | 02:15 PM

Recently, Balletti and I proved that for the h^*-polynomial h_0^*+h_1^*t+... of a lattice polytope, if we assume h_3^*=0, then (h_1^*, h_2^*) satisfies (i) h_2^*=0; or (ii) h_1^* \leq 3h_2^* + 3; or (iii) (h_1^*,h_2^*)=(7,1). These conditions derive from Scott's theorem (1976), who characterized the possible h^*-polynomials of 2-dimensional lattice polytopes, and Scott's theorem is also essentially valid for lattice polytopes with degree at most 2 (Treutlein (2010)). On the other hand, we proved that it also holds under the assumption h_3^*=0. Since the assumption h_3^*=0 is independent of both dimension and degree of polytopes, we call the conditions (i), (ii), (iii) universal. In this talk, towards finding a new universal condition, we investigate the possibility for the polynomial h_0^*+h_1^*t+... to be the h^*-polynomial of some lattice polytope under the assumption that some of h_i^*'s vanish.

Location: Freie Universität Berlin Institut für Mathematik Arnimallee 2 14195 Berlin Seminar Room

Jul 05, 2018 | 02:15 PM

The study of realization spaces of convex polytopes is one of the oldest subjects in Polytope Theory. Most likely, it goes back to Legendre (1794). A lot of progress took place since that time. However, many questions remained open. In general, computing the dimension of the realization space $\mathcal{R}(P)$ of a d-polytope $P$ is hard, even for $d = 4$, as shown by Mnëv (1988) and Richter-Gebert (1996).In this presentation, we will discuss two criteria to determine the dimension of the realization space, and use them to show that $\dim \mathcal{R}(P) = f_1(P) + 6$ for a 3-polytope $P$, and $\dim \mathcal{R}(P) = df_{d-1}(P)$ (resp. $\dim \mathcal{R}(P) = df_0(P)$ ) for a simple (resp. simplicial) $d$-polytope $P$. We will also discuss the realization spaces of some interesting 2-simple 2-simplicial 4-polytopes. Namely, we will consider the realization space of the 24-cell and of a 2s2s polytope with 12 vertices which was found by Miyata (2011), and give a better bound for/determine its dimension.

Location: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin Room 005 (ground floor)

Jul 02, 2018 | 04:00 PM

We describe some recent developments in treewidth-based algorithms for knots, which exploit the structure of the underlying 4-valent planar graph. In particular, we show how these led to the first general sub-exponential-time algorithm for the HOMFLY-PT polynomial, and we describe some recent progress on parameterised algorithms for unknot recognition.

Location: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin Room 005 (ground floor)

Jul 02, 2018 | 02:15 PM

A famous and wide-open problem, going back to at least the early 1970's, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute one such set of inequalities when a chromatic polynomial $\chi_G(n) = \chi^*_0 \binom {n+d} d + \chi^*_1 \binom {n+d-1} d + \dots + \chi^*_d \binom n d$ is written in terms of a binomial-coefficient basis. More precisely, we prove that $\chi^*_{d-2}+\chi^*_{d-3}+\dots+\chi^*_{d-j-1} \ \ge \ \chi^*_1+\chi^*_2+\dots+\chi^*_j $, for $1 \le j \le \lfloor \frac{ d }{ 2 } \rfloor - 1$. A similar result holds for flow polynomials enumerating either modular or integral nowhere-zero flows of a graph. Our theorems follow from connections among chromatic, flow, order, and Ehrhart polynomials, and the fact that the latter satisfy a decomposition formula into symmetric polynomials due to Stapledon. (This is joint work with Emerson Le\'on.)

Jun 25, 2018 | 04:00 PM

We introduce a family of polytopes, called primitive zonotopes, which can be seen as a generalization of the permutahedron of type $B_d$. We discuss connections to the largest diameter of lattice polytopes and to the computational complexity of multicriteria matroid optimization. Complexity results and open questions are also presented. In particular, we answer a question raised in 1986 by Colbourn, Kocay, and Stinson by showing that deciding whether a given sequence is the degree sequence of a 3-hypergraph is computationally prohibitive. Based on joint works with Asaf Levin (Technion), George Manoussakis (Ben Gurion), Syed Meesum (IMSc Chennai), Shmuel Onn (Technion), and Lionel Pounin (Paris XIII).

Jun 25, 2018 | 02:15 PM

In this talk, I will show how techniques from Combinatorial Optimization and Combinatorics can be leveraged to achieve progress on a well-known question in Integer Programming, namely how to solve integer linear programs (ILPs) with bounded subdeterminants. More precisely, I will present an efficient (even strongly polynomial) algorithm to solve ILPs defined by a constraint matrix whose subdeterminants are all within {-2,-1,0,1,2}. This problem class naturally extends the well-known class of ILPs with a totally unimodular (TU) constraint matrix, i.e., where subdeterminants are within {-1,0,1}, and captures some open problems. We employ a variety of techniques common in Combinatorial Optimization, including polyhedral techniques, matroids, and submodular functions. In particular, using a polyhedral result of Veselov and Chirkov, we reduce the problem to a combinatorial one with a so-called parity constraint. We then show how a seminal, purely combinatorial result of Seymour on decomposing TU matrices, which is tightly related to regular matroids and was originally developed in this context, can be used algorithmically to break the problem into smaller ones. Finally, these smaller problems are amenable to combinatorial optimization techniques like parity-constrained submodular minimization. Additionally, I will highlight some of the many open problems in this field and discuss recent related results.

Jun 18, 2018 | 02:15 PM

The cage problem asks for the smallest number c(k, g) of vertices in a k-regular graph of girth g. The (k, g)-graphs which have c(k, g) vertices are known as cages. While cages are known to exist for all integers k > 1 and g > 2, an explicit construction is known only for some small values of k, g and three infinite families for which g is 6, 8 or 12 and k − 1 is a prime power: corresponding to the generalized g/2-gons of order k − 1. To improve the upper bounds on c(k, 6), c(k, 8) and c(k, 12), when k - 1 is not a prime power, one of the main techniques that has been used so far is to construct small (k, g) graphs by picking a prime power q ≥ k and then finding a small k-regular subgraph of the incidence graph of a generalized g/2-gon of order q. In this talk I will present new constructions in generalized quadrangles and hexagons which improve the known upper bound on c(k, 8) when k = p^{2h} and c(k, 12) when k = p^h, where p is an arbitrary prime. Moreover, we will see a spectral lower bound on the number of vertices in a k-regular induced subgraph of an arbitrary regular graph, which in particular will prove the optimality of a known construction in projective planes. (Joint work with John Bamberg and Gordon Royle)

Location: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin room 005 (ground floor)

Jun 11, 2018 | 04:00 PM s.t.

In 1967, F. Arthur Sherk gave a simple proof that the finite metric planes (of Bachmann and Schmidt) are precisely the affine planes of odd order. Moreover, Sherk’s proof holds for a more general class of incidence structures that do not involve the ‘three-reflection theorem’ whatsoever, and thus yields a beautiful characterisation of the finite affine planes of odd order. By relaxing the first of Sherk’s axioms to ‘every pair of points lies on at most one line’, we can study what we call partial Sherk planes. In this talk, we outline our characterisation of these incidence structures as Bruck nets, in the same vein as Sherk’s result, and what it means for connected combinatorial objects such as mutually orthogonal latin squares. (Joint work with Joanna Fawcett and Jesse Lansdown)

Location: Freie Universität Berlin Institut für Informatik Takustr. 9 14195 Berlin room 005 (ground floor)

Jun 11, 2018 | 02:15 PM

From Kalai's classic paper generalizing Cayley's tree-enumeration formula to simplicial complexes, it is known that simplicial complexes on a small number of vertices can have enormous torsion in homology. Moreover, in a random setting one may find instances of this phenomenon such as, for example, a 3-dimensional simplicial complex on 30 vertices with the torsion subgroup of the second homology group having order larger than 10^82. In this talk I will discuss the problem of explicitly constructing such complexes. In particular, I will discuss my work to use the probabilistic method to construct optimally small (up to a constant factor from a known lower bound) simplicial complexes with prescribed torsion in homology. I will also discuss an application of this work to the problem of counting homotopy types of simplicial complexes.

Jun 04, 2018 | 02:15 PM

May 28, 2018 | 04:01 PM s.t.

In the Steiner Tree problem, a set of terminal vertices in an edge-weighted graph needs to be connected in the cheapest possible way. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on the one hand, Steiner Tree is known to be APX-hard, and on the other hand, it is W[2]-hard if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this, we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals T, and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph H of G of the minimum cost such that there is a directed path from s to t in H for all arcs st of R. It is known that the problem can be solved in time |V(G)|O(|A(R)|) [Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time |V(G)|o(|A(R)|) even if G is planar, unless Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time |V(G)|o(|T|), unless ETH fails. Therefore, there is a significant gap in the complexity with respect to |T| in the exponent. We show that Directed Steiner Network is solvable in time f(|T|) · |V(G)| O(cg · |T|), where cg is a constant depending solely on the genus g of G and f is a computable function.

May 28, 2018 | 04:00 PM s.t.

In the k-Set Packing problem we are given a universe and a family of its subsets, where each of the subsets has size at most k. The goal is to select a maximum number of sets from the family which are pairwise disjoint. It is a well known NP-hard problem, that has been studied from the approximation perspective since the 80's. During the talk I will describe the history of progress on both the weighted and unweighted variants of the problem, with an exposition of methods used to obtain the best known approximation algorithms mostly involving local search based routines.

May 28, 2018 | 02:15 PM