DFS-Based Reserialization Methods

• DFS-based reserialization is a framework that applies depth-first search on auxiliary graphs, enabling optimal decompositions and efficient resource-constrained routing.
• The approach employs label-based DFS with dominance pruning to systematically generate and prune candidate solutions for improved decision-making in complex routing problems.
• Succinct graph encodings integrate DFS traversals with constant-time query support, achieving near-optimal space usage and rapid operations across various graph classes.

Depth-First Search-based Reserialization refers to a set of algorithmic frameworks and data structures optimized for generating and outputting depth-first search (DFS) decompositions of combinatorial objects by traversing auxiliary graphs or succinctly encoded separable graph structures. The approach has prominent applications in both resource-constrained combinatorial optimization, particularly routing problems, and space-efficient low-level graph algorithms. The paradigm leverages label-based DFS to enumerate feasible solutions under complex constraints, as well as succinct data-structural methods to facilitate rapid post-DFS queries and compressed serializations.

1. Formal Model: Auxiliary Graphs and DFS-based Label Enumeration

In vehicle routing and related combinatorial optimization settings, DFS-based reserialization constructs and traverses an auxiliary directed graph $G=(V,A)$ built from a “giant tour” customer sequence $T=(v_1,\ldots,v_n)$. Node set $V=\{0,1,\ldots,n\}$ indexes tour prefixes, with $0$ as the depot. Each arc $(i\rightarrow j)\in A$ (with $0\leq i<j\leq n$) models the feasibility of serving subsequence $(v_{i+1},\ldots,v_j)$ as a single trip without violating resource bounds (e.g., capacity, fleet size).

A DFS “split procedure” searches $G$ in label space, where a label $\ell=(i,Q,c,r)$ describes the current prefix endpoint $i$, accumulated load $Q$, cost $c$, and remaining resource(s) $r$ (fleet cardinality or vector for heterogeneous fleets). Labels are maintained per node, and dominance pruning is applied: $\ell_1$ dominates $\ell_2$ at node $i$ iff $c_1\leq c_2$, $Q_1\leq Q_2$, $r_1\geq r_2$, and at least one inequality is strict. Dominated labels are immediately discarded (Duhame et al., 2022).

2. Explicit DFS-Based Split Algorithms and Metaheuristic Integration

A practical algorithm instantiates label-based DFS via an explicit stack and per-node label sets. Starting from $\ell_0=(0,0,0,R_{\max})$, the procedure iteratively pops labels, extends along feasible arcs, applies resource updates (e.g., $c\leftarrow c+d_{i,j}$, $Q\leftarrow Q+q_{i,j}$, $r_k\leftarrow r_k-1$ for fleet modeling), performs dominance pruning, and generates parent pointers for reconstructing optimal splits. Enumeration limits $N_{\max}, L_{\max}$ (total labels, labels per node) provide computational tractability.

These DFS split procedures are used as subroutines within metaheuristics, notably the GRASPxELS framework: GRASP constructs randomized giant tours, which are split via the DFS procedure; then XELS applies perturbations and local searches using Split at each iteration. Empirical results on HVRP and real-life DLP instances demonstrate that DFS-based split is competitive with or superior to greedy split in solution quality and execution time, particularly for medium-sized instances ($n\approx100$–$150$). Average split times for DFS are $56$–$60$ ms (Taillard) and $400$ ms (large DLP), with optimality guaranteed in the unlimited enumeration regime (Duhame et al., 2022).

3. Succinct Graph Encodings for Space-Efficient DFS Reserialization

For separable graphs—those admitting $O(n^\epsilon)$-size balanced separators such as planar, bounded-genus, or $H$-minor-free classes—DFS-based reserialization exploits compact representations that support DFS traversal and output in $o(n)$ time and $o(n)$ extra bits (excluding permanent structure storage).

The succinct encoding recursively divides $G$ into “mini” and “micro” pieces, each with $O(r^\epsilon)$ or $O(\tilde{r}^\epsilon)$ boundary vertices and table-based lookup for micro pieces. The graph is stored in $Z(n)+o(n)$ bits (where $Z(n)$ is the information-theoretic minimum), supporting $O(1)$ translation between vertex indices at each nesting level and constant-time adjacency/degree queries (Elberfeld et al., 28 Apr 2025).

DFS is performed using iterator objects for boundary vertices and table lookups for micro pieces; states per boundary vertex are updated and stored in $o(n)$ bits, with at most $o(n)$ iterators and color bits active. Each step is atomic ($O(1)$), yielding overall $o(n)$ time and space bounds.

4. Augmentation and Constant-Time DFS Query Support

After DFS traversal, the succinct encoding is augmented with the generated DFS tree (“palm tree”) by embedding parent/child pointers and color information directly into the structure. For each mini boundary vertex $v$, a small array $L_v$ records the mini-piece IDs in which $v$ has children, enabling efficient enumeration of descendants. The aggregate augmentation cost is $o(n)$ bits.

Standard DFS-related queries are then supported in $O(1)$ time by exploiting the piecewise representation:

• Lowest-Common Ancestor (LCA) queries reduce to precomputed Harel–Tarjan structures on contracted palm trees of size $o(n)$, with local reference substitutions for non-boundary vertices.
• Lowpoint, s–t numbering, depth, and #descendants queries are resolved by boundary reference plus small offset or direct table lookup, ensuring constant-time response per query (Elberfeld et al., 28 Apr 2025).

5. Planar Graph Specialization and Comparative Analysis

For planar graphs ($\epsilon=1/2$), the complete encoding, micro-table construction, and DFS tree augmentation are all achievable in $O(n)$ bits and expected $O(n)$ time (succinct variant with Raman–Raman–Satti FID) or truly linear time/space (compact variant with Baumann–Hagerup FID). All subsequent DFS-based queries are constant-time, and full DFS output is performed in $o(n)$ time and space. A direct corollary is that planar graphs support DFS reserialization and rich structural queries, using asymptotically optimal storage and runtime parameters (Elberfeld et al., 28 Apr 2025).

Compared to naïve serialization—the classical approach requiring adjacency lists, stack management, and $O(\log n)$ bits per vertex—DFS-based reserialization via succinct encoding uses only $Z(n)+o(n)$ bits, eliminating the $\Theta(n\log n)$ storage and time overheads.

6. Correctness, Optimality, and Algorithmic Significance

Under unlimited enumeration bounds ($N_{\max}\rightarrow\infty$, $L_{\max}\rightarrow\infty$), the DFS split procedure for routing decompositions returns optimal resource-constrained shortest paths and hence optimal splits. The correctness follows from exhaustive depth-first label expansion with non-domination guarantees: every feasible path contributing to an optimum is explicitly generated and pruned only if suboptimal, with parent pointers supporting certificate reconstruction (Duhame et al., 2022).

In the context of succinct DFS reserialization, the space-efficient traversals systematically enumerate the search tree implicitly or explicitly and produce outputs and query support in resource-optimal fashion. The integration into metaheuristics and low-level graph libraries illustrates the versatility and foundational impact of DFS-based reserialization for combinatorial optimization, graph algorithms, and data-structural innovation.