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# 2019 IBS Summer Research Program on Algorithms and Complexity in Discrete Structures

## Sunday, July 21, 2019 - Saturday, August 10, 2019

Room B232, IBS (기초과학연구원)

## Schedule

### July 22 Monday

10:00-11:00 Introduction, 11:00-12:00 Open Problems

### July 23 Tuesday

10:00-10:30 Stefan Kratsch, Humboldt-Universität zu Berlin, Germany
Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

10:45-11:15 Benjamin Bergougnoux, University Clermont Auvergne, France
More applications of the d-neighbor equivalence

11:30-12:00 Yixin Cao, Hong Kong Polytechnic University, China
Enumerating Maximal Induced Subgraphs

13:30-14:30 Open Problems

### July 24 Wednesday

We will go to KAIST for June Huh‘s two talks 15:00-16:00, 16:30-17:30. A IBS shuttle bus @14:30 from the lobby.

### July 25 Thursday

10:00-11:00 Nick Brettell, Durham University, UK
Recent work on characterising matroids representable over finite fields

11:30-12:00 Mamadou M. Kanté, University Clermont Auvergne, France
On recognising k-letter graphs

### July 26 Friday

10:00-11:00 O-joung Kwon (권오정), Incheon National University, Korea
The grid theorem for vertex-minors

11:00-12:00 Progress Report

### July 27 Saturday

8:30 Excursion to Damyang (departure from the accommodation)

### July 29 Monday

10:00-11:00 Archontia Giannopoulou, National and Kapodistrian University of Athens, Greece
The directed flat wall theorem

11:30-12:00 Eunjung Kim (김은정), LAMSADE-CNRS, France
Subcubic even-hole-free graphs have a constant treewidth

### July 30 Tuesday

10:00-11:00 Pierre Aboulker, ENS Ulm, France
Generalizations of the geometric de Bruijn Erdős Theorem

11:30-12:00 Michael Dobbins, Binghamton University, USA
Barycenters of points in polytope skeleta

### July 31 Wednesday

10:00-11:00 Magnus Wahlström, Royal Holloway, University of London, UK
FPT-algorithms via LP-relaxations

11:30-12:00 Édouard Bonnet, ENS Lyon, France
The FPT/W[1]-hard dichotomy of Max Independent Set in H-free graphs

### August 1 Thursday

10:00-11:00 Euiwoong Lee (이의웅), NYU, USA
Losing treewidth by separating subsets

11:30-12:00 Sang-il Oum (엄상일), IBS Discrete Mathematics Group and KAIST, Korea
Branch-depth: Generalizing tree-depth of graphs

### August 2 Friday

10:00-11:00 Dabeen Lee (이다빈), IBS Discrete Mathematics Group, Korea
t-perfect graphs and the stable set problem

11:00-12:00 Progress Report

August 5-9: Free Discussions / Research Collaborations / Progress Report
Tea time: Every weekday 3:30pm

## Abstracts

### July 23 Tuesday

#### Stefan Kratsch, Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets, which have played implicit and explicit roles in many results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class ${\cal C}$, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class ${\cal C}$, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes ${\cal C}$, including the class ${\cal C}_{LP}$ of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or ${\cal C}_{LP}$ graphs.

#### Benjamin Bergougnoux, More applications of the d-neighbor equivalence

In this talk, I will present a framework which gives efficient algorithms for several connectivity problems such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain efficient algorithms parameterized respectively by tree-width, clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width.

Paper available on HAL : https://hal.archives-ouvertes.fr/hal-01799573

#### Yixin Cao, Enumerating Maximal Induced Subgraphs

We study efficient algorithms for the maximal induced subgraphs problem: Given a graph on $n$ vertices, enumerate all its maximal induced subgraphs that belong to a particular hereditary graph class. While its optimization version, known as the vertex deletion problem in literature, has been intensively studied, algorithms for the enumeration version exist only for few simple graph classes, e.g., independent sets and cliques. Enumeration algorithms are mainly categorized in three complexity classes, polynomial total time (polynomial on $n$ and the total number of solutions), incremental polynomial time (for all $s$, the time to output the first $s$ solutions is polynomial on $n$ and $s$), and polynomial delay (for all $s$, the time to output the first $s$ solutions is polynomial on $n$ and linear on $s$). We first show that the maximal induced subgraphs problem can be solved in polynomial total time if and only if it can be solved in incremental polynomial time. We then propose general approaches to develop polynomial-delay algorithms and incremental-polynomial-time algorithms for the problem. They enable us to develop simple algorithms to solve the problem with polynomial delay for a large number of graph classes, and in incremental polynomial time for interval graphs, chordal graphs as well as two of its well-studied subclasses, and all graph classes with finite forbidden induced subgraphs.

### July 25 Thursday

#### Nick Brettell, Recent work on characterising matroids representable over finite fields

Characterising when a matroid is representable over a certain field or set of fields is one of the oldest problems in matroid theory, dating back to Whitney’s 1935 paper. In this talk, I will discuss some of the more recent work in this area, with a focus on work I have been involved with towards characterising matroids representable over all fields of size at least four.

#### Mamadou M. Kanté, On recognising k-letter graphs

$k$-letter graphs are a subclass of graphs of linear clique-width $k$, but are well-quasi-ordered by the induced subgraph ordering. I present a simple FPT algorithm for recognising $k$-letter graphs and also an upper bound on the size of minimal obstructions. The characterisation is based on a simple observation allowing an MSOL-definability. (joint work with V. Lozin)

### July 26 Friday

#### O-joung Kwon (권오정), The grid theorem for vertex-minors

We prove that, for a circle graph $H$, every graph with sufficiently large rank-width contains a vertex-minor isomorphic to $H$. This is joint work with Jim Geelen, Rose McCarty, and Paul Wollan.

### July 29 Monday

#### Archontia Giannopoulou, The directed flat wall theorem

At the core of the Robertson-Seymour theory of Graph Minors lies a powerful structure theorem which captures, for any fixed graph $H$, the common structural features of all the graphs not containing $H$ as a minor [Neil Robertson, Paul D. Seymour: Graph Minors. XVI. Excluding a non-planar graph. J. Comb. Theory, Ser. B 89(1): 43-76 (2003)]. An important step towards this structure theorem is the Flat Wall Theorem [Neil Robertson, Paul D. Seymour: Graph Minors .XIII. The Disjoint Paths Problem. J. Comb. Theory, Ser. B 63(1): 65-110 (1995)], which has a lot of algorithmic applications (for example, the minor-testing and the disjoint paths problem with fixed number terminals).
In this paper, we prove the directed analogue of this Flat Wall Theorem. Our result builds on the recent Directed Grid Theorem by two of the authors (Kawarabayashi and Kreutzer), and we hope that this is an important and significant step toward the directed structure theorem, as with the case for the undirected graph for the graph minor project.
Joint work with Ken-ichi Kawarabayashi, Stephan Kreutzer, and O-joung Kwon.

#### Eunjung Kim (김은정), Subcubic even-hole-free graphs have a constant treewidth

It is known that even-hole-free graphs can have arbitrarily large rankwidth, but all known constructions have many high-degree vertices. It has been believed in the structural graph theory community that the rankwidth shall be bounded if the maximum degree is bounded. We prove that there exists a universal constant $c$ such that every even-hole-free graph of degree at most three has treewidth at most $c$. As a by-product of the proof, we also show that there exists a function $f$ such that for any $r$, $K_r$-minor-free and even-hole-free graph has treewidth at most $f(r)$; this extends the result of Silva et. al. (Discrete Applied Mathematics 2010) addressing the case of planar graphs.
This is a joint work with Pierre Aboulker and Isolde Adler.

### July 30 Tuesday

#### Pierre Aboulker, Generalizations of the geometric de Bruijn Erdős Theorem

A classic Theorem of de Bruijn and Erdős states that every noncollinear set of n points in the plane determines at least n distinct lines. The line $L(u,v)$ determined by two points $u$, $v$ in the plane consists of all points $p$ such that

• $\operatorname{dist}(p, u)+\operatorname{dist}(u, v)=\operatorname{dist}(p, v)$ (i.e. $u$ is between $p$ and $v$) or
• $\operatorname{dist}(u, p)+\operatorname{dist}(p, v)=\operatorname{dist}(u, v)$ (i.e. $p$ is between $u$ and $v$) or
• $\operatorname{dist}(u, v)+\operatorname{dist}(v, p)=\operatorname{dist}(u, p)$ (i.e. $v$ is between $u$ and $p$).

With this definition of line $L(uv)$ in an arbitrary metric space $(V, \operatorname{dist})$, Chen and Chvátal conjectured that every metric space on n points, where n is at least 2, has at least n distinct lines or a line that consists of all n points. The talk will survey results on and around this conjecture.

#### Michael Dobbins, Barycenters of points in polytope skeleta

In this talk I will classify the n-tuples of dimensions such that any target point in any d-polytope is the barycenter of points in faces of the prescribed dimensions. Specifically, for a $(nk+r)$-polytope and target point, we can always find $r$ points in faces of dimension $k+1$, and $n-r$ points in faces of dimension $k$, that have their barycenter at the given target. This is tight in the sense that we cannot require even one of the points to be in a face of dimension less than $k$, and we cannot require more than $n-r$ of the points to be in faces of dimension $k$. This is joint work with Florian Frick.

### July 31 Wednesday

#### Magnus Wahlström, FPT-algorithms via LP-relaxations

LP-relaxations are traditionally (within theoretical computer science) used for computing approximate solutions to NP-hard problems, but across the last few years there have been several examples where LP-relaxations have been used to guide FPT-algorithms — that is, exact (non-approximate) algorithms whose running time is bounded in terms of a “parameter” of the input instance.  This approach has given algorithms that are simultaneously simpler and faster for a range of central problems in parameterized complexity.  At the same time, this is applicable only to specific problems and relaxations; an arbitrary LP-relaxation, even one that has good properties in an approximation sense, will in general give no useful guidance with respect to exact solutions.

In this talk, we give an overview of these FPT applications, and the conditions they impose on the LP-relaxation. Our main focus is on an approach of “discrete relaxation” referred to as Valued CSPs, but we also briefly survey more immediately combinatorial conditions, as well as related algorithms that solve these problems more efficiently (e.g., so-called linear-time FPT algorithms) by bypassing the LP-solver.

#### Édouard Bonnet, The FPT/W[1]-hard dichotomy of Max Independent Set in H-free graphs

Maximum Independent Set (MIS) is in general graphs the paradigmatic W[1]-hard problem. In stark contrast, polynomial-time algorithms are known when the inputs are restricted to structured graph classes such as, for instance, perfect graphs (which includes bipartite graphs, chordal graphs, co-graphs, etc.) or claw-free graphs. In this talk, we introduce some variants of co-graphs with “parameterized noise”, that is, graphs that can be turned into disjoint unions or complete sums by the removal of a certain number of vertices and the edition of a certain number of edges per incident vertex, both controlled by the parameter. We give a series of FPT Turing-reductions on these classes and use them to make some progress on the parameterized complexity of MIS in $H$-free graphs. We present the current state of the (randomized) FPT vs W[1]-hard dichotomy of MIS in $H$-free graphs. We then show that for every fixed $t \geq 1$, MIS is FPT in $P(1,t,t,t)$-free graphs, where $P(1,t,t,t)$ is the graph obtained by substituting all the vertices of a four-vertex path but one end of the path by cliques of size t. In particular, this helped finishing the classification when H has at most five vertices.
This follows joint works with Nicolas Bousquet, Pierre Charbit, Stéphan Thomassé, and Rémi Watrigant.

### August 1 Thursday

#### Euiwoong Lee (이의웅), Losing treewidth by separating subsets

We study the problem of deleting the smallest set S of vertices (resp. edges) from a given graph $G$ such that the induced subgraph (resp. subgraph) $G \setminus S$ belongs to some class of graphs $H$. When graphs in H have treewidth bounded by $t$, we construct a framework for both vertex and edge deletion versions where approximation algorithms for these problems can be obtained from approximation algorithms for natural graph partitioning problems called $k$-Subset Vertex Separator and $k$-Subset Edge Separator.
For the vertex deletion, our framework, combined with the previous best result for $k$-Subset Vertex Separator, yields improved approximation ratios for fundamental problems such as $k$-Treewidth Vertex Deletion and Planar-$F$ Vertex Deletion. For the edge deletion, we give improved approximation algorithms for $k$-Subset Edge Separator and their applications to various problems in bounded-degree graphs.
Joint work with Anupam Gupta, Jason Li, Pasin Manurangsi, and Michal Wlodarczyk.

#### Sang-il Oum (엄상일), Branch-depth: Generalizing tree-depth of graphs

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A$ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs.
Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.
This is a joint work with Matt DeVos and O-joung Kwon.

### August 2 Friday

#### Dabeen Lee (이다빈), t-perfect graphs and the stable set problem

A graph $G$ is called $t$-perfect if the stable set polytope of $G$ can be obtained after adding the odd cycle inequalities to the standard edge formulation for the problem. A motivation for studying $t$-perfect graphs is algorithmic: as the odd cycle inequalities can be separated in polynomial time, one can compute a maximum weight stable set in a $t$-perfect graph by an LP algorithm in polynomial time. There are rich classes of $t$-perfect graphs, but a complete characterization of $t$-perfect graphs is still open. Due to this, no graph-theoretic / combinatorial algorithm is known, although there is an algorithm for the unweighted case based on a combinatorial approximate LP algorithm. In this survey talk, I will (i) review classes of known $t$-perfect graphs, (ii) briefly explain the algorithm for the unweighted case and a possibility of extending it to the weighted case, and (iii) discuss some research directions towards finding a graph-theoretic algorithm for computing a maximum weight stable set in a $t$-perfect graph.

#### Participants (not including IBS researchers)

1. Pierre Aboulker, ENS Ulm, France
2. Rémy Belmonte, University of Electro-Communications, Japan
3. Benjamin Bergougnoux, University of Paris Diderot, France
4. Édouard Bonnet, ENS Lyon, France
5. Nick Brettell, Durham University, UK
6. Yixin Cao, Hong Kong Polytechnic University, China
7. Michael Dobbins, Binghamton University, USA
8. Chris Eppolito, Binghamton University, USA
9. Archontia Giannopoulou, National and Kapodistrian University of Athens, Greece
10. Mamadou M. Kante, University Clermont Auvergne, France
11. Eunjung Kim (김은정), LAMSADE-CNRS, France
12. Stefan Kratsch, Humboldt-Universität zu Berlin, Germany
13. O-joung Kwon (권오정), Incheon National University, Korea
14. Michael Lampis, Université Paris-Dauphine, France
15. Euiwoong Lee (이의웅), NYU, USA
16. Valia Mitsou, Université Paris-Diderot, France
17. Florian Sikora, University Paris Dauphine, France
18. Magnus Wahlström, Royal Holloway, University of London, UK

An invitation-only summer research program will be held in the summer of 2019. There will be 10-20 participants to work together at DIMAG. Talks are open.

#### Accommodation

Everyone will stay at the Daedeok Innopolis Guest House (대덕특구게스트하우스).

• Address: 27-5, Expo-ro 123beon-gil, Yuseong-gu, Daejeon, 34125, Republic of Korea (대전광역시 유성구 엑스포로 123번길 27-5)
• Tel: (+82)-042-865-2500

#### Organizers

• Eunjung Kim (김은정), LAMSADE-CNRS, France
• Sang-il Oum (엄상일), IBS Discrete Mathematics Group and KAIST, Korea

## Details

Start:
Sunday, July 21, 2019
End:
Saturday, August 10, 2019
Event Category:

Room B232
IBS (기초과학연구원)

## Organizers

Sang-il Oum (엄상일)
Eunjung Kim
기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209