Given a graph $G$, the ball centered at vertex $v$ of radius $r$ is the set containing all vertices at distance at most $r$ from $v$. We prove that for every integer $t\ge 1$ there exists a constant $c_t$ such that for every $K_t$-minor-free graph $G$, and every set $S$ of balls in $G$, the minimum size of a set of vertices of $G$ intersecting all the balls of $S$ is at most $c_t$ times the maximum number of vertex-disjoint balls in $S$.

A number of old conjectures related to edge-colourings of 3-regular graphs remain vastly open. Among them, the Petersen colouring conjecture is a very strong statement, which implies most of these, such as the Berge—Fulkerson conjecture, and the 5-Cycle-double-cover conjecture. It states that every bridgeless cubic graph admits an edge-colouring with 5 colours such that for every edge e, the set of colours assigned to the edges adjacent to e has cardinality either 2 or 4, but not 3.

Let A and B be two sets of points in R^d, where |A|=|B|=n and the distance between them is defined by some bipartite measure dist(A,B). We study several problems in which the goal is to translate the set B, so that dist(A, B) is minimized. The main measures that we consider are (i) the diameter in two and three dimensions, that is diam(A,B) = max {d(a,b) : a in A, b in B}, where d(a,b) is the Euclidean distance between $a$ and $b$,

Les 21ièmes Journées Graphes et Algorithmes (JGA 2019) auront lieu à Bruxelles, du 13 au 15 novembre 2019. Il s'agit du rendez-vous annuel de la communauté francophone de la théorie des graphes et de ses applications.

A fundamental step towards solving combinatorial problems using techniques from linear programming theory is to encode the space of feasible solutions for such problems as the set of vertices of polytopes of small extension complexity. In this work, we establish several connections between the extension complexity of polytopes, formal language theory, and group theory. In particular, we introduce the notion of homogeneous context-free grammars and show that polytopes corresponding to languages accepted by polynomial-size non-uniform families of grammars with polynomial homogeneity have polynomial extension complexity.

We prove that every intersection graph of axis-parallel rectangles in the plane with clique number ω has chromatic number O(ω log ω), which is the first improvement of the original O(ω²) bound of Asplund and Grünbaum from 1960. As a consequence, we obtain a polynomial-time O(log log n)-approximation algorithm for Maximum Weight Independent Set in axis-parallel rectangles, improving the previous best approximation ratio of O(log n/log log n). Joint work with Parinya Chalermsook.

Chattopadhyay, Mande and Sherif (to appear in STOC 2019) recently exhibited a total Boolean function, the sink function, that has polynomial approximate rank and polynomial randomized communication complexity. This gives an exponential separation between randomized communication complexity and logarithm of the approximate rank, refuting the log-approximate-rank conjecture. We show that even the quantum communication complexity of the sink function is polynomial, thus also refuting the quantum log-approximate-rank conjecture.

Many computational geometry algorithms require greater accuracy for output than for input. So, to send or use the output, we often need to round it off. If this is done naively, intersections can occur during rounding. An algorithm from Devillers, Lenhart, Lazard was presented last year to solve this problem. The worst case of this algorithm is \(O(n^15)\). An implementation of this algorithm shows that on average, it is \(O(n \sqrt{n})\).

The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of n cities that visits each city c a prescribed number kcof times. Travel costs may not be symmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families. The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984).

Let P be a set of n points in the plane. In the traditional geometric algorithms view, P is given as an unordered sequence of locations (usually pairs of x and y coordinates). There are many interesting and useful structures that one can build on top of P: the Delaunay triangulation, Voronoi diagram, well-separated pair decomposition, quadtree, etc. These structures can all be represented in O(n) space but take Omega(n log n) time to construct on a Real RAM, and, hence, contain information about P that is encoded in P but cannot be directly read from P.

In this talk I will explore the question how much information about the
structure of a set of points can be derived from their locations, especially
when we are *uncertain* about the locations of the points.