Events

Title: Approximation algorithms for stochastic orienteering  

Abstract: This week we revisit the stochastic orienteering problem in which we are given a metric graph where each node has a deterministic reward and a random size. The goal is to adaptively decide on which nodes to visit to maximize the expected reward, while not exceeding the budget B on the distance plus the size of jobs processed.

Title: Approximating Weighted Connectivity Augmentation below Factor 2

Abstract:  The Weighted Connectivity Augmentation Problem (WCAP) asks to increase the edge-connectivity of a graph in the cheapest possible way by adding edges from a given set. It is one of the most elementary network design problems for which no better-than-2 approximation algorithm has been known, whereas 2-approximations can be easily obtained through a variety of well-known techniques.

Title: Exact Zarankiewicz numbers through linear hypergraphs

Abstract: The \emph{Zarankiewicz number} $Z_{2,2}(m,n)$ is usually defined as the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that has no $K_{2,2}$ subgraph. An equivalent definition is that $Z_{2,2}(m,n)$ is the greatest total degree of a linear hypergraph with $m$ vertices and $n$ edges. A hypergraph is \emph{linear} if each pair of vertices appear together in at most one edge. The equivalence of the two definitions can be seen by considering the bipartite incidence graph of the linear hypergraph.

Title: Asymptotics of the Euler characteristic of Kontsevich's commutative graph complex

Abstract: I will present results on the asymptotic growth rate of the Euler characteristic of Kontsevich's commutative graph complex. By a work of Chan-Galatius-Payne, these results imply the same asymptotic growth rate for the top-weight Euler characteristic of M_g, the moduli 
space of curves, and establish the existence of large amounts of unexplained cohomology in this space. This asymptotic growth rate 
follows from new generating functions for the edge-alternating sum of graphs without odd automorphisms. I will give an overview on this 
interaction between topology and combinatorics and illustrate the combinatorial and analytical tools that were needed to obtain these 
generating functions.

Title: Greedy algorithm for stochastic matching is a 2-approximatio

Abstract: We will discuss the greedy algorithm for the stochastic matching problem. In this problem, we are given an undirected graph where each edge is assigned a probability p_e in [0, 1] and each vertex is assigned a patience t_v in Z+. We begin each step by probing an edge e which is not adjacent to any edges in our matching. The probe will succeed with probability p_e, and if it does, we add e to our matching. Otherwise, we may not probe e again. We also may not probe edges adjacent to a vertex v more than t_v times. The goal is to maximize the number of edges we add to our matching. 

Title: Integer programs with bounded subdeterminants and two nonzeros per row

Abstract: Determining the complexity of integer linear programs with integer coefficient matrices whose subdeterminants are bounded by a constant is currently a very actively discussed question in the field. In this talk, I will present a strongly polynomial-time algorithm for such integer programs with the further requirement that every constraint contains at most two variables. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than k vertex-disjoint odd cycles, where k is any constant. Previously, polynomial-time algorithms were only known for k = 0 (bipartite graphs) and for k = 1.

This is joint work with Samuel Fiorini, Gwenaël Joret, and Yelena Yuditsky, which recently appeared at FOCS this year.

Title: A recursive spectral bound for independence

Abstract: We discuss an upper bound for the independence number of a graph, in the spirit of the well-known Hoffman bound. Our bound involves the largest Laplacian eigenvalue of the graph; more surprisingly, it also involves the independence number of a certain induced graph. We illustrate the bound on several examples.

Title: k-Connectedness and k-Factors in the Semi-Random Graph Process

Abstract: The semi-random graph process is a single player graph game where the player is initially presented an edgeless graph with n vertices. In each round, the player is offered a vertex u uniformly at random and subsequently chooses a second vertex v deterministically according to some strategy, and adds edge uv to the graph. The objective for the player is then to ensure that the graph fulfils some specified property as fast as possible.

Title: Algebraicity of solutions of functional equations with one catalytic variable

Abstract: Abstract: Numerous combinatorial enumeration problems reduce to the study of functional equations which can be solved by a uniform method introduced by Bousquet-Mélou and Jehanne in 2006. In my talk, I will first briefly explain this result and its proof. Then I will present a new generalization of it to the case of systems of functional equations with one catalytic variable. The method is constructive and yields an algorithm for computing the minimal polynomials of interest.

Title: Boosted Sampling

Abstract: We will discuss the boosted sampling technique introduced by Gupta et al. which approximates the stochastic version of problems by using nice approximation algorithms for the deterministic version of the problem. We will focus on rooted stochastic Steiner trees as an example, though other problems are covered by this approach (such as vertex cover and facility location). The problem is given to us in two stages: in the first stage we may choose some elements at a cheaper cost, and in the second stage our actual requirements are revealed to us, and we can buy remaining needed elements at a more expensive cost (where costs get scaled by some factor in the second stage). We will see that if our problem is sub-additive, and we have an alpha-approximation algorithm for the deterministic version of our problem with a beta-strict cost-sharing function then we can get an (alpha + beta)-approximation for the stochastic version of our problem. We also discuss related problems, for example the (not sub-additive!) unrooted stochastic Steiner tree problem.