Heuristic Contraction Hierarchies with Approximation Guarantee

We present a new heuristic point-to-point routing algorithm based on contraction hierarchies (CH). Given an epsilon >= 0, we can prove that the length of the path computed by our algorithm is at most (1+epsilon) times the length of the optimal (shortest) path. CH is based on node contraction: removing nodes from a network and adding shortcut edges to preserve shortest path distances. Our algorithm tries to avoid shortcuts even when a replacement path is epsilon times longer.

💡 Research Summary

This paper presents a significant extension of the Contraction Hierarchies (CH) speed-up technique for point-to-point shortest path queries by introducing a principled approximation scheme. The core idea is to trade exact optimality for a provable approximation guarantee, thereby reducing preprocessing effort and extending applicability to graph classes where exact CH performs poorly.

The authors propose a “Heuristic Contraction” process (Algorithm 2) that modifies the core node contraction operation of standard CH. In exact CH, when contracting a node u, a shortcut between neighbors v and w is added if the shortest path between them avoiding u is longer than the path via u. The heuristic version relaxes this rule: a shortcut is added only if the avoiding path P is longer than (1+ε)(˜c(v,u)+˜c(u,w)). Here, ε ≥ 0 is a user-defined error parameter, and ˜c is a second weight stored per edge, called the “witness memory.” If a witness path P prevents a shortcut, the algorithm distributes the error ratio γ across all edges of P by reducing their ˜c values. This careful bookkeeping ensures a global invariant (Lemma 1): for every edge, c(e)/(1+ε) ≤ ˜c(e) ≤ c(e). This prevents the accumulation of errors during sequential node contractions and is crucial for the final guarantee.

Surprisingly, the query algorithm (Algorithm 3) remains identical to the bidirectional Dijkstra-like search used in exact CH. It performs a forward search from the source in the “upward” graph (edges to higher-ordered nodes) and a backward search from the target in the “downward” graph. The central theoretical result (Theorem 1) proves that this query, using the graph built by the heuristic contraction, returns a path whose length is at most (1+ε) times the optimal shortest path distance. The proof relies on two key lemmas: Lemma 2 shows that any path in the heuristic CH can be transformed into a canonical “up-then-down” form without increasing its ˜c weight sum. Combining this with the properties of ˜c from Lemma 1 yields the approximation factor.

To enhance practical query performance, the paper adapts the essential “stall-on-demand” optimization for the heuristic setting (Algorithm 5). Stall-on-demand prunes the search by stopping the expansion of a node if a shorter path to it already exists. In exact CH, this shorter path can be any path. However, the authors demonstrate that this condition breaks the approximation guarantee in heuristic CH (see Figure 1). They derive a corrected stalling condition: a node u is stalled only if a shorter path exists that can be split into an upward prefix P′₁ and a (potentially non-upward) suffix P′₂, such that c(P′₁) + (1+ε)c(P′₂) < c(P), where P is the upward path currently used to reach u. Theorem 2 proves that incorporating this modified condition preserves the (1+ε) approximation guarantee.

The proposed heuristic CH framework provides a flexible trade-off via the parameter ε. Setting ε=0 recovers exact CH. Larger ε values allow the algorithm to avoid more shortcuts during preprocessing, leading to smaller graph sizes and faster queries, at the cost of a bounded increase in path length. This opens doors for applying CH-like acceleration to problem domains where exact preprocessing is too costly or where multiple objective functions (e.g., travel time and distance) need to be supported efficiently from a single precomputed data structure. The work thus successfully bridges the gap between highly efficient but inflexible exact speed-up techniques and adaptable approximation algorithms with rigorous guarantees.