Goal Inference as Inverse Planning

Infants and adults are adept at inferring agents ’ goals from in-complete or ambiguous sequences of behavior. We propose a framework for goal inference based on inverse planning, in which observers invert a probabilistic generative model of goal-dependent plans to infer agents ’ goals. The inverse plan-ning framework encompasses many specific models and rep-resentations; we present several specific models and test them in two behavioral experiments on online and retrospective goal inference

Distortion is Fixed Parameter Tractable

We study low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective. Let M = M(G) be the shortest path metric of an edge weighted graph G, with the vertex set V (G) and the edge set E(G), on n vertices. We give the first fixed parameter tractable algorithm that for an unweighted graph metric M and integer d either constructs an embedding of M into the line with distortion at most d, or concludes that no such embedding exists. Our algorithm requires O(nd 4 (2d + 1) 2d) time which is a significant improvement over the best previous algorithm of Bădoiu et al. that runs in time O(n 4d+2 d O(1)). Because of its apparent similarity to the notoriously hard Bandwidth Minimization problem, we find it surprising that this problem turns out to be fixed parameter tractable. We extend our results on embedding unweighted graph metric into the line in two ways. First, we give an algorithm to construct small distortion embeddings of weighted graph metrics. The running time of our algorithm is O(n(dW) 4 (2d + 1) 2dW) where W is the largest edge weight of the input graph. To complement this result, we show that the exponential dependence on the maximum edge weight is unavoidable. In particular, we show that deciding whether a weighted graph metric M(G) with maximum weight W < |V (G) | can be embedded into the line with distortion at most d is NP-Complete for every fixed rational d ≥ 2. This rules out any possibility of an algorithm with running time O((nW) h(d) ) where h is a function of d alone. Secondly, we consider more general host metrics for which analogous results hold. In particular, we prove that for any tree T with maximum degree ∆, embedding M into a shortest path metric of T is fixed parameter tractable, parameterized by (∆, d)

Bandwidth on AT-free graphs

We study the classical Bandwidth problem from the viewpoint of parameterized algorithms. In the Bandwidth problem we are given a graph G = (V, E) together with a positive integer k, and asked whether there is an bijective function β: {1,..., n} → V such that for every edge uv ∈ E, |β −1 (u) − β −1 (v) | ≤ k. The problem is notoriously hard, and it is known to be NP-complete even on very restricted subclasses of trees. The best known algorithm for Bandwidth for small values of k is the celebrated algorithm by Saxe [SIAM Journal on Algebraic and Discrete Methods, 1980], which runs in time 2 O(k) n k+1. In a seminal paper, Bodlaender, Fellows and Hallet [STOC 1994] ruled out the existence of an algorithm with running time of the form f(k)n O(1) for any function f even for trees, unless the entire W-hierarchy collapses. We initiate the search for classes of graphs where Bandwidth is fixed parameter tractable (FPT), that is, solvable in time f(k)n O(1) for some function f. In this paper we present an algorithm with running time 2 O(k log k) n 2 for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial FPT algorithm for Bandwidth on a graph class where the problem remains NP-complete