Tractable Cost-Optimal Planning over Restricted Polytree Causal Graphs

Abstract

Causal graphs are widely used to analyze the complexity of planning problems. Many tractable classes have been identified with their aid and state-of-the-art heuristics have been derived by exploiting such classes. In particular, Katz and Keyder have studied causal graphs that are hourglasses (which is a generalization of forks and inverted-forks) and shown that the corresponding cost-optimal planning problem is tractable under certain restrictions. We continue this work by studying polytrees (which is a generalization of hourglasses) under similar restrictions. We prove tractability of cost-optimal planning by providing an algorithm based on a novel notion of variable isomorphism. Our algorithm also sheds light on the k-consistency procedure for identifying unsolvable planning instances. We speculate that this may, at least partially, explain why merge-and-shrink heuristics have been successful for recognizing unsolvable instances.

A Refined Understanding of Cost-Optimal Planning with Polytree Causal Graphs

“…Although our algorithm is polynomial-time, its running time is admittedly not impressive. However, previous tractability results like Aghighi, Jonsson, and Ståhlberg (2015) and Ståhlberg (2017), have similar tower functions despite using more restrictions than we do. We believe that our bounds can be considerably improved by an even more careful analysis, combining these proof techniques with others to give an even more refined picture of optimal plans.…”

“…It is easy to verify that (inverted) forks and hourglasses are polytrees. Aghighi, Jonsson, and Ståhlberg (2015) show that cost-optimal planning is tractable for instances with bounded domain size and a polytree causal graph with bounded diameter, the length Inverted fork Fork Hourglass…”

Generating SAS+ Planning Tasks of Specified Causal Structure

“…Figure 2 exemplifies selected graph structures. Probably the most explored causal graph structure in terms of complexity analysis is the polytree (Domshlak and Dinitz 2001;Domshlak and Brafman 2002;Domshlak 2007, 2008a;Giménez and Jonsson 2008;Bäckström and Jonsson 2013;Aghighi, Jonsson, and Ståhlberg 2015;Bäckström, Jonsson, and Ordyniak 2019). Other explored structures include chains (Domshlak and Dinitz 2001;Jonsson 2008, 2009), DAGs and directed-path singly connected graphs (Domshlak and Dinitz 2001;Domshlak and Brafman 2002;Katz and Domshlak 2007, 2008a,b, 2010Bäckström and Jonsson 2013), as well as (inverted) forks and (inverted) trees (Domshlak and Dinitz 2001;Domshlak and Brafman 2002;Katz and Domshlak 2007, 2008a,b, 2010Katz and Keyder 2012).…”

“…Most existing heuristics that work on the multi-valued representation exploit the causal information in one way or another. Further, starting with the seminal work of Bäckström and Nebel (1995), the research on the complexity of planning tasks had a major focus on the characterization of planning fragments by their causal graph structure (Domshlak and Dinitz 2001;Domshlak and Brafman 2002;Katz and Domshlak 2007, 2008a,b, 2010Jonsson 2008, 2009;Katz and Keyder 2012;Bäckström and Jonsson 2013;Aghighi, Jonsson, and Ståhlberg 2015;Bäckström, Jonsson, and Ordyniak 2019), as well as some local structural characteristics, such as kdependence Domshlak 2007, 2008a; Giménez and Jonsson 2012), classifying these fragments into a variety of complexity classes. For these two reasons, various planners' performance heavily relies on structural characteristics of the input planning task.…”