Separating path systems

We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a separating path system of size $O(n)$ and prove this in certain interesting special cases. In particular, we establish this conjecture for random graphs and graphs with linear minimum degree. We also obtain tight bounds on the size of a minimal separating path system in the case of trees.Comment: 21 pages, fixed misprints, Journal of Combinatoric

Two Optimal Strategies for Active Learning of Causal Models from Interventional Data

From observational data alone, a causal DAG is only identifiable up to Markov equivalence. Interventional data generally improves identifiability; however, the gain of an intervention strongly depends on the intervention target, that is, the intervened variables. We present active learning (that is, optimal experimental design) strategies calculating optimal interventions for two different learning goals. The first one is a greedy approach using single-vertex interventions that maximizes the number of edges that can be oriented after each intervention. The second one yields in polynomial time a minimum set of targets of arbitrary size that guarantees full identifiability. This second approach proves a conjecture of Eberhardt (2008) indicating the number of unbounded intervention targets which is sufficient and in the worst case necessary for full identifiability. In a simulation study, we compare our two active learning approaches to random interventions and an existing approach, and analyze the influence of estimation errors on the overall performance of active learning

Experiment Selection for Causal Discovery

Randomized controlled experiments are often described as the most reliable tool available to scientists for discovering causal relationships among quantities of interest. However, it is often unclear how many and which different experiments are needed to identify the full (possibly cyclic) causal structure among some given (possibly causally insufficient) set of variables. Recent results in the causal discovery literature have explored various identifiability criteria that depend on the assumptions one is able to make about the underlying causal process, but these criteria are not directly constructive for selecting the optimal set of experiments. Fortunately, many of the needed constructions already exist in the combinatorics literature, albeit under terminology which is unfamiliar to most of the causal discovery community. In this paper we translate the theoretical results and apply them to the concrete problem of experiment selection. For a variety of settings we give explicit constructions of the optimal set of experiments and adapt some of the general combinatorics results to answer questions relating to the problem of experiment selection

Separating Path Systems for the Complete Graph

For any graph $G$, a separating path system of $G$ is a family of paths in $G$ with the property that for any pair of edges in $E(G)$ there is at least one path in the family that contains one edge but not the other. We investigate the size of the smallest separating path system for $K_n$, denoted $f(K_n)$. Our first main result is a construction that shows $f(K_n) \leq \left(\frac{21}{16}+o(1)\right)n$ for sufficiently large $n$. We also show that $f(K_n) \leq n$ whenever $n=p,p+1$ for prime $p$. It is known by simple argument that $f(K_n) \geq n-1$ for all $n \in \mathbb{N}$. A key idea in our construction is to reduce the problem to finding a single path with some particular properties we call a Generator Path. These are defined in such a way that the $n$ cyclic rotations of a generator path provide a separating path system for $K_n$. Hence existence of a generator path for some $K_n$ gives $f(K_n) \leq n$. We construct such paths for all $K_n$ with $n \leq 20$, and show that generator paths exist whenever $n$ is prime.Comment: 23 pages, 3 figure

Two optimal strategies for active learning of causal models from interventions

Abstract From observational data alone, a causal DAG is in general only identifiable up to Markov equivalence. Interventional data generally improves identifiability; however, the gain of an intervention strongly depends on the intervention target, i.e., the intervened variables. We present active learning strategies calculating optimal interventions for two different learning goals. The first one is a greedy approach using single-vertex interventions that maximizes the number of edges that can be oriented after each intervention. The second one yields in polynomial time a minimum set of targets of arbitrary size that guarantees full identifiability. This second approach proves a conjecture of Eberhard

COMBINATORIAL ALGORITHMS FOR GRAPH DISCOVERY AND EXPERIMENTAL DESIGN

In this thesis, we study the design and analysis of algorithms for discovering the structure and properties of an unknown graph, with applications in two different domains: causal inference and sublinear graph algorithms. In both these domains, graph discovery is possible using restricted forms of experiments, and our objective is to design low-cost experiments. First, we describe efficient experimental approaches to the causal discovery problem, which in its simplest form, asks us to identify the causal relations (edges of the unknown graph) between variables (vertices of the unknown graph) of a given system. For causal discovery, we study algorithms for the problem of learning the causal relationships between a set of observed variables in the presence of hidden or unobserved variables while minimizing a suitable cost of interventions on the observed variables. An intervention on a set of variables helps learn the presence of causal relations adjacent to them. Under various cost models for interventions, we design combinatorial algorithms for causal discovery by identifying new connections between discrete optimization, graph property testing, and efficient intervention design. Next, we investigate query-efficient experimental approaches for estimating various graph properties, such as the number of edges and graph connectivity. The access to the graph, or equivalently performing an experiment, is via a Bipartite Independent Set (BIS) oracle. The BIS oracle is related to the interventional access model used in our work for causal graph discovery, with other applications in group testing and fine-grained complexity. In this setting, we develop non-adaptive algorithms that lead to efficient implementations in highly parallelized and low-memory streaming settings

Covering and Separation for Permutations and Graphs

This is a thesis of two parts, focusing on covering and separation topics of extremal combinatorics and graph theory, two major themes in this area. They entail the existence and properties of collections of combinatorial objects which together either represent all objects (covering) or can be used to distinguish all objects from each other (separation). We will consider a range of problems which come under these areas. The first part will focus on shattering k-sets with permutations. A family of permutations is said to shatter a given k-set if the permutations cover all possible orderings of the k elements. In particular, we investigate the size of permutation families which cover t orders for every possible k-set as well as study the problem of determining the largest number of k-sets that can be shattered by a family with given size. We provide a construction for a small permutation family which shatters every k-set. We also consider constructions of large families which do not shatter any triple. The second part will be concerned with the problem of separating path systems. A separating path system for a graph is a family of paths where, for any two edges, there is a path containing one edge but not the other. The aim is to find the size of the smallest such family. We will study the size of the smallest separating path system for a range of graphs, including complete graphs, complete bipartite graphs, and lattice-type graphs. A key technique we introduce is the use of generator paths - constructed to utilise the symmetric nature of Kn. We continue this symmetric approach for bipartite graphs and study the limitations of the method. We consider lattice-type graphs as an example of the most efficient possible separating systems for any graph