Solving the Snake in the Box Problem with Heuristic Search: First Results

Abstract

Snake in the Box (SIB) is the problem of finding the longest simple path along the edges of an n-dimensional cube, subject to certain constraints. SIB has important applications in coding theory and communications. State of the art algorithms for solving SIB apply uninformed search with symmetry breaking techniques. We formalize this problem as a search problem and propose several admissible heuristics to solve it. Using the proposed heuristics is shown to have a huge impact on the number of nodes expanded and, in some configurations, on runtime. These results encourage further research in using heuristic search to solve SIB, and to solve maximization problems more generally.

Generalized Longest Path Problems

“…Another admissible heuristic for Snake-in-the-box appears in (Palombo et al 2015). The remaining graph is partitioned into disjoint sets of connected components.…”

“…∀x ∈ V, L(x) = {x} ∪ N (x). The star-shaped exclusion set proposed in (Palombo et al 2015), where not all vertices can be on the same snake path, is one such exclusion set. Note that it is better to make such sets as small as possible, thus unlike (Palombo et al 2015) we use a 4-vertex star shape, which we henceforth call a "Y" exclusion.…”

“…They used LSP to demonstrate their findings and proposed an admissible heuristic for LSP. Then, Palombo et al (2015) proposed several admissible heuristics for solving the Snake-in-the-box (SIB) problem (Kautz 1958). SIB is a reminiscent of LSP that is important for a useful type of efficient error correction codes.…”

“…We then introduce methods for generating admissible heuristics for GLSP based on exclusion sets and must-include paths. We also extend the notion of patternbased heuristics for Snake-in-the-box (Palombo et al 2015) taken from a bi-connected projection of the graph to include components that are not necessarily disjoint. Finally, we describe an implementation, including a new incremental implementation of the heuristics.…”

Finding Optimal Longest Paths by Dynamic Programming in Parallel

“…Additionally to the constraint of not allowing repeated vertices in the path it is required that for any two vertices u, v there is no edge between u and v unless u and v are adjacent in the path. Heuristic search LP algorithms can be adapted to efficiently solve SIB (Palombo et al 2015) by designing SIB specific heuristics. However, these techniques cannot be transferred to solving the general LP problem since they rely on SIB specific heuristics and therefore these results are not relevant for our work.…”