A Combinatorial Search Perspective on Diverse Solution Generation

Finding diverse solutions has become important in many combinatorial search domains, including Automated Planning, Path Planning and Constraint Programming. Much of the work in these directions has however focussed on coming up with appropriate diversity metrics and compiling those metrics in to the solvers/planners. Most approaches use linear-time greedy algorithms for exploring the state space of solution combinations for generating a diverse set of solutions, limiting not only their completeness but also their effectiveness within a time bound. In this paper, we take a combinatorial search perspective on generating diverse solutions. We present a generic bi-level optimization framework for finding cost-sensitive diverse solutions. We propose complete methods under this framework, which guarantee finding a set of cost sensitive diverse solutions satisficing the given criteria whenever there exists such a set. We identify various aspects that affect the performance of these exhaustive algorithms and propose techniques to improve them. Experimental results show the efficacy of the proposed framework compared to an existing greedy approach. In many real-world domains involving combinatorial search such as automated planning, path planning and constraint programming, generating diverse solutions is of much importance. In the case of automated planning, real-world scenario often involves working with unknown or partially known user preferences (Kambhampati 2007), as the user preferences are many times difficult to be articulated and specified completely. Such situations lead to multiple, often, large number of plans that satisfy a given problem instance. In order to facilitate serving the user with a closest plan possible as per her (hidden) preferences, presenting a diverse set of plans to the user is explored (Roberts, Howe, and Ray 2014; Nguyen et al. 2012) so that the user can make a well-informed decision. In the constraint programming domain, diverse (resp. similar) solutions are explored in order to handle unknown user preferences as well as to generate robust solutions (Hebrard et al. 2005). Several methods have been proposed in the literature for finding a diverse set of plans. In the context of constraint programming, (Hebrard et al. 2005) presents a complete method which creates K copies of the Constraint SatisfacCopyright c © 2016, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. tion Problem (CSP), each copy with a different set of variable names, adds ( K 2 ) additional constraints for handling the minimum distance requirement between all pairs, and uses off-the-shelf solvers to generate solutions. As one would expect, they report that this approach generates prohibitively large CSPs and therefore propose a greedy method. The greedy approach which has since been widely adopted (Petit and Trapp 2015; Bloem 2015; Roberts, Howe, and Ray 2014; Nguyen et al. 2012) works as follows: Obtain a candidate solution satisfying any given cost criteria, add this to the solution K-set, provide feedback to the method finding candidate solutions about the current composition of the K-set so that it tries to find the next candidate solution distant to the current K-set. Upon obtaining the next candidate solution, the greedy method adds it to the K-set if it indeed satisfies the distance criteria and provides feedback to find the next solution, otherwise the candidate solution is simply discarded. This process is continued until a set of K diverse solutions is found. (Roberts, Howe, and Ray 2014; Eiter et al. 2013) and (Nguyen et al. 2012) consider the first solution generated to be the starting solution (permanent member) for constructing the K-set through the above greedy approach. (Bloem 2015) considers an optimal solution to be the starting solution of the greedy method. (Petit and Trapp 2015) attempt to address the issue of fixed starting solution by running the greedy approach multiple times with different optimal solutions as the starting solution on each occasion. A pertinent issue with the above approaches is that the first solution (or an optimal solution) is always considered to be part of the solution set, which may often result in not finding a K-set even when there exists one, even with a very good feedback strategy to search for distant solutions after finding the initial solution. Note that an optimal solution (or the first found solution) need not be part of a diverse set of the required size at all. Figure 1 shows an example of an instance where the optimal solution is not part of the most diverse solution set of size 2 (assuming that the distance between two plans is inversely proportional to the number of edges/actions they have in common; p1 and p2 have edge a in common and p2 and p3 have edge b in common). In this paper, we address this problem in depth by proposing complete algorithms which guarantee to find a set of K diverse solutions whenever there exists one. In order to ac-