Compact Path Representation in DAGs via Colored Edge Pebbling

Abstract: Compactly representing a variation graph is a core problem in computational pangenomics that is usually attacked with techniques that have been originated on texts and adapted to graphs. In this paper we propose a new framework that takes a topology-centric perspective instead. A variation graph is modeled as a directed acyclic graph (DAG) together with a set of distinguished paths, where each path is assigned a distinct color. Our compact representation is centered on pebbling the graph, i.e. placing colored pebbles on edges so that every predefined path can be univocally reconstructed from the pebbled edges. In particular, a saturated pebbling marks each chosen edge with every path (color) traversing it. We first propose a data structure to represent and query a variation graph with storage space depending on the size of the pebbling. The supported queries are: (i) path query, which recovers a path given its color, and (ii) edge query, which reports the colors of paths traversing a given edge. We then prove that the problem of finding a pebbling of minimum size is solvable in polynomial time. On the contrary, we prove that finding a saturated pebbling of minimum size is NP-hard, but can be reduced to the minimum-weight set cover problem, allowing us to leverage integer linear programming (ILP) solvers. We show how to exploit saturated pebblings to achieve faster queries times than minimum size pebbling. Our framework opens a new algorithmic viewpoint on developing more efficient variation graph representations rooted on the study of the topology of those graphs.

• The paper introduces colored edge pebbling and polynomial-time algorithms for Min Pebbling, with running time O(min{|V||E| + Σᵢ|Vᵢ|, |H||E|}) for constructing minimum guardians.
• The paper proves Min Saturated Pebbling is NP-complete even when every predefined path has at most three edges, via a reduction from Vertex Cover on cubic triangle-free graphs.
• The paper reduces saturated pebbling to weighted set cover and provides succinct data structures using O(|E| + |P| log|H|) extra bits, with linear-time path reconstruction and local edge-query performance under saturation.

Overview and motivation

This paper addresses the compact representation of a set of distinguished paths in a directed acyclic graph (DAG), motivated by computational pangenomics, where a variation graph encodes many genomes as colored paths over a shared graph structure. Rather than adapting sequence-centric indexing techniques, the authors adopt a topology-centric perspective: they ask how few edges must be marked so that every predefined path can be reconstructed unambiguously from the marks alone. The central abstraction is colored edge pebbling: placing pebbles of color $i$ on edges of path $h_i$ such that consecutive pebbled edges are connected by unique paths in the graph — the unique path condition. A subsequence of edges satisfying this condition is called a guardian for the path.

The paper formalizes two optimization problems. Min Pebbling asks for a minimum-cardinality pebbling; because guardians for different colors do not interact, it decomposes into independent per-path minimization. Min Saturated Pebbling adds the constraint that any pebbled edge carries all colors of paths traversing it, coupling the paths into a global optimization problem. The paper's main contributions are polynomial-time algorithms for Min Pebbling, an NP-hardness proof for Min Saturated Pebbling (even when each path has at most three edges), a reduction of Min Saturated Pebbling to minimum-weight set cover with an ILP formulation, and succinct data structures supporting edge queries (report $\col(e)$) and path queries (reconstruct $h_i$) in space proportional to $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$ bits beyond a representation of $G$.

Polynomial-time algorithms for Min Pebbling

Both algorithms exploit a simple characterization: an edge $(u,v)$ on path $h_i$ must be pebbled exactly when there exist multiple paths in $G$ between its tail and the next "checkpoint" vertex. The first algorithm precomputes a matrix $A$ where $h_i$0 counts paths from $h_i$1 to $h_i$2 capped at 2, via dynamic programming in reverse topological order in $h_i$3 time, then scans each path backwards, pebbling edge $h_i$4 whenever $h_i$5. Correctness follows from an exchange argument showing that the greedy backward scan produces a minimum-size guardian per color. The second algorithm avoids the full matrix by computing, on the fly during a reverse traversal, the bounded path count from each vertex to the current checkpoint, achieving $h_i$6 total time. Together these yield:

$h_i$7

The implication is that the local variant of the problem is fully tractable, and the choice between the two algorithms depends only on whether $h_i$8 or $h_i$9 dominates. The authors note that minimum-size pebblings need not be unique, so both algorithms output one optimal solution among possibly several.

NP-hardness of Min Saturated Pebbling

The hardness result is established by reduction from Vertex Cover on cubic triangle-free graphs. A key structural lemma shows that any cubic triangle-free graph can be oriented into a transitive-free DAG in linear time using a Brooks 3-coloring: orienting edges from lower to higher color yields acyclicity, and transitivity would force a triangle. From this DAG $\col(e)$0, the reduction builds a variation graph $\col(e)$1 whose gadget per original edge consists of node edges, an arc edge, a shortcut edge, and $\col(e)$2 fake edges on each side, giving rise to shortcut paths (single edge), arc paths (three edges), and fake paths (three edges). Uniqueness lemmas show that node/arc-edge paths project to the unique paths of the transitive-free $\col(e)$3.

The core correspondence is that the size of a minimum saturated pebbling equals $\col(e)$4, where $\col(e)$5 is the minimum total degree of a vertex cover. Shortcut edges must always be pebbled ($\col(e)$6 pebbles), fake paths admit empty guardians, arc edges are never selected in an optimum (an exchange argument replaces an arc edge carrying $\col(e)$7 colors by a node edge carrying at most $\col(e)$8), and the selected node edges correspond exactly to a degree-weighted vertex cover. Since all vertices have degree three in the cubic case, $\col(e)$9, so deciding whether a saturated pebbling of size at most $h_i$0 exists decides Vertex Cover. This establishes that Min Saturated Pebbling is NP-complete even when every predefined path has at most three edges — a notably strong restriction, since the hardness does not stem from long or complex paths but from the global coupling induced by saturation.

Reduction to set cover and ILP formulation

Despite the hardness, the problem admits an exact formulation as minimum-weight set cover. For each path $h_i$1, a fake edge appended after $h_i$2 forces coverage of the final element. The universe consists of pairs $h_i$3 with $h_i$4; the set associated with edge $h_i$5 contains $h_i$6 whenever $h_i$7 is a cardinal ancestor of $h_i$8 along $h_i$9, weighted by $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$0. Two lemmas show the equivalence: guardian edges cover the universe, and conversely any set cover induces valid guardians via the unique-path condition. Consequently,

$O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$1

and a minimum saturated pebbling can be extracted from an optimal cover by discarding the mandatory fake edges. The resulting ILP has $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$2 binary variables and $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$3 constraints, making the problem amenable to practical ILP solvers. An implicit consequence is that Min Saturated Pebbling inherits the $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$4-approximability of weighted set cover, although the paper does not pursue approximation guarantees explicitly.

Succinct representation and query algorithms

The data structure stores the pebbles grouped by edge rank in an array $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$5, together with a bitvector encoding block boundaries and rank/select structures over both. It uses $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$6 extra bits and supports membership tests, enumeration of $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$7, guardian lengths, random access to the $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$8-th edge of $O(|E| + |\mathcal{P}|\log|\mathcal{H}|)$9, and inverse rank queries, all within polylogarithmic overhead.

Path reconstruction runs in $G$0 time: a reverse scan computes, for every vertex, the nearest guardian anchor reachable from it, then a forward walk emits the unique inter-anchor segments guaranteed by the unique path condition.

Edge queries are more involved. For a general pebbling, the algorithm computes cardinal ancestors/descendants of the query endpoints by DP, determines the closest guardian edges $G$1 and $G$2 for each color, and checks consecutiveness, running in $G$3 time. For minimum-size pebblings, a lemma showing that no color can appear on three guardian edges whose heads are cardinal ancestors of a common vertex bounds the relevant work to $G$4. The strongest result exploits saturation: if the query edge is itself pebbled, $G$5 is read directly in $G$6 time; otherwise, traversals restricted to unpebbled edges identify the component $G$7 and its boundary pebbled edges $G$8, and the query completes in

$G$9

This is a local guarantee: query cost scales with the "unpebbled neighborhood" of the query edge rather than the whole graph, which is precisely the regime saturation is designed to enable. The trade-off is explicit — saturated pebblings may use more pebbles than minimum ones (the paper's example uses nine versus eight), and the worst-case edge-query bound still involves the $(u,v)$0 term.

Limitations and open questions

The framework assumes the underlying graph is acyclic, whereas real pangenome graphs frequently contain cycles and bidirected structures; extending the definitions and algorithms beyond DAGs remains open. Edge queries answer only for single edges; generalizing to arbitrary subpaths — a capability naturally supported by the GBWT — is left unaddressed. Although the set-cover reduction enables exact ILP solving, no tractable fragments of Min Saturated Pebbling (e.g., restricted graph classes or bounded treewidth) are identified. Finally, the unique path condition is one possible reconstruction criterion; alternative criteria based on unique minimum-weight paths or other uniqueness properties are proposed but not analyzed.

Conclusion

This paper introduces colored edge pebbling as a topology-centric framework for compactly representing predefined paths in DAGs, establishing a clean complexity dichotomy: Min Pebbling is solvable in near-linear-per-path polynomial time, while its saturated variant is NP-hard even for paths of length three yet reduces exactly to weighted set cover. The accompanying succinct data structures achieve space proportional to the pebbling size and support path extraction in linear time and edge-color queries whose cost, under saturation, depends on the local unpebbled structure around the query edge. The work connects bubble structure, vertex cover, and set cover within a single combinatorial model and provides a theoretical foundation for topology-aware variation graph indexing complementary to sequence-centric approaches such as the GBWT.