Inconsistency and Accuracy of Heuristics with A* Search

Many studies in heuristic search suggest that the accuracy of the heuristic used has a positive impact on improving the performance of the search. In another direction, historical research perceives that the performance of heuristic search algorithms, such as A* and IDA*, can be improved by requiring the heuristics to be consistent – a property satisfied by any perfect heuristic. However, a few recent studies show that inconsistent heuristics can also be used to achieve a large improvement in these heuristic search algorithms. These results leave us a natural question: which property of heuristics, accuracy or consistency/inconsistency, should we focus on when building heuristics? While there are studies on the heuristic accuracy with the assumption of consistency, no studies on both the inconsistency and the accuracy of heuristics are known to our knowledge. In this study, we investigate the relationship between the inconsistency and the accuracy of heuristics with A* search. Our analytical result reveals a correlation between these two properties. We then run experiments on the domain for the Knapsack problem with a family of practical heuristics. Our empirical results show that in many cases, the more accurate heuristics also have higher level of inconsistency and result in fewer node expansions by A*. Introduction Heuristic search has been playing a practical role in solving hard problems. One of the most popular heuristic algorithms is A search (Hart, Nilson, and Raphael 1968), which is essentially best-first search with an additive evaluation f(x) = g(x) + h(x), where g(x) is the cost of the current path from the start node to node x, and h(x) is an estimation of the cheapest cost h(x) from x to a solution node. The function h is called a heuristic function, or heuristic for short. An important property of A search is its admissibility: A will always return an optimal solution if the heuristic h it uses is admissible, meaning h(x) never exceeds h(x). Research on A and other similar heuristic search algorithms, such as IDA (Korf 1985), has focused on understanding the impact of properties of the heuristic function on the quality of the search. A well-studied subclass of admissible heuristics is the one with the consistency property. Heuristic h is called consistent if h(x) ≤ c(x, x) + h(x) for all pairs of nodes (x, x), where c(x, x) is the cheapest cost from x to x. Consistency was introduced in the original A paper (Hart, Nilson, and Raphael 1968) and later became a desirable property of admissible heuristics for two perceptions. First, since the perfect heuristic h is consistent, it is expected that a good heuristic should also be consistent. The consistency is believed to enable A to forgo reopening nodes (Pearl 1984, p. 82) and thus can reduce the number of node expansions. Second, inconsistent admissible heuristics seem rare. In fact, it is assumed by many researchers (Korf 2000) that “almost all admissible heuristics are consistent.” The portrait of inconsistent heuristics was usually painted negatively until recently, when Zahavi et al. (2007) discovered that inconsistency is actually not that bad. They demonstrated by empirical results that in many cases, inconsistency can be used to achieve large performance improvements of IDA. They then promoted the use of inconsistent heuristics and showed how to turn a consistent heuristic into an inconsistent heuristic using the bidirectional pathmax (BPMX) method of Felner et al. (2005). Follow-up studies (Felner et al. 2011; Zhang et al. 2009) have also provided positive results of inconsistent heuristics with A search and encouraged researchers to explore inconsistency as a means to further improve the performance of A. In another line of research on heuristics, there have been extensive investigations on the impact of the accuracy of the heuristic on the performance of A (and IDA). While there are a few negative results (Korf and Reid 1998; Korf, Reid, and Edelkamp 2001; Helmert and Röger 2008), most studies (Pohl 1977; Gaschnig 1979; Nam Huyn 1980; Sen, Bagchi, and Zhang 2004; Dinh, Russell, and Su 2007; Dinh et al. 2012) in this line support the intuition that in many search spaces, improving the accuracy of the heuristic can improve the efficiency of A. Some of the negative results (Korf and Reid 1998; Korf, Reid, and Edelkamp 2001) on the benefit of heuristic accuracy were actually obtained under the assumption that the heuristic is consistent. Other negative results only apply to specific planning domains (Helmert and Röger 2008) or contrived search spaces with an overwhelming number of solutions (Dinh et al. 2012). In light of the newly discovered benefit of inconsistent heuristics and the well-established positive results on the accuracy of heuristics, it is natural to ask so which property, consistency/inconsistency or accuracy, of heuristics really matter to the performance of A?. Is there any relationship between these properties of heuristics? The goal of paper is to address these questions. In this work, we first analyze a correlation between inconsistency and accuracy of heuristics. Our analytical result reveals that the level of inconsistency of a heuristic can serve as an upper bound on the level of accuracy of the heuristic (see Theorem 1 for details.) We then investigate the relationship between the inconsistency and accuracy of heuristics as well as their impact on the performance of A, by running experiments on a practical domain for the Knapsack problem taken from (Dinh et al. 2012). Our study differs from the previous works (Felner et al. 2011; Zhang et al. 2009) on inconsistent heuristics with A in both the search space used and the construction of heuristics. While Felner et al. and Zhang et al. use undirected graphs and focus on the reduction in node re-expansions as a benefit of inconsistency, our experiments are done on a directed acyclic graph on which A will never reopen nodes, regardless of the heuristic used. For this search graph, we use a family of heuristics that arise in practice, which allow us to compare the inconsistency level and the accuracy level of many heuristics within this family. Recall that Felner et al. and Zhang et al. incorporated BPMX into A and compared the performance of A with other less well-known heuristic algorithms (B, B’, C). However, as pointed out by Zahavi et al. (2007), BPMX is only applicable for undirected graphs, thus is inapplicable for the search space we consider.