Source-linked AI summary
Constructive Arboricity in Almost-Linear Time for Dense Graphs
Iman Gholami, Mohammad T. Hajiaghayi, Mahdi JafariRaviz, Alireza Kaviani, Soheil Mohammadkhani, AmirMohammad Shahrezaei
TL;DR
The paper tackles finding a graph’s exact arboricity and constructing an optimal forest partition. A randomized algorithm uses an orientation-based approach and achieves a running time of O(min{n^(2+o(1)), m^(4/3+o(1))}), with almost-linear time on dense graphs.
Problem
The task is to determine a graph’s exact arboricity and partition its edges into that many forests.
Source: §1.1 · p. 3Method
The algorithm finds the minimum outgoing bound k, then uses ExactK to test for a k-forest partition and, if needed, a (k+1)-forest partition.
Source: §4.5 · p. 16Results
O(min{n^(2+o(1)), m^(4/3+o(1))}) time computes exact arboricity and constructs an optimal forest partition; for m=Θ(n^2), the time is m^(1+o(1)).
Source: §1 · p. 3, §1.1 · pp. 3–4Takeaways & Limitations
For dense graphs, an optimal forest partition can be constructed in almost-linear time in the number of edges.
Source: §1.1 · p. 4The theorem covers simple, unweighted, undirected graphs; it does not extend to undirected multigraph or weighted inputs.
Source: §4.5 · p. 17
Abstract
from arXiv · showhide
The arboricity of a graph is the minimum number of forests needed to partition its edges. We present a randomized algorithm that computes the exact arboricity of a simple graph and constructs an optimal forest partition in $\min\{n^{2+o(1)},m^{4/3+o(1)}\}$ time both in expectation and with high probability, where $n$ and $m$ denote the numbers of vertices and edges. In particular, it runs in almost-linear time, $O(m^{1+o(1)})$, on dense graphs with $m=Θ(n^2)$. We also give the first known almost-linear-time algorithm that constructs a partition into at most one more forest than optimal.
1 Introduction
The paper presents randomized algorithms for constructing forest partitions, including an exact-arboricity algorithm and an additive-one construction. Its approach uses orientations and star insertion to test whether an optimal partition exists.
1 Introduction
- Recent flow and minimum-cut approaches compute arboricity values, but the cited minimum-cut reduction does not construct forests; earlier constructive bounds include Gabow’s eO(m·a(G)).The paper’s contribution therefore addresses construction as well as arboricity computation.Source: §1 · p. 3
1.1 Main results
- The randomized algorithm returns the exact arboricity and an optimal forest partition in O(min{n^2+o(1), m^4/3+o(1)}) time, in expectation and with high probability.Every returned partition is correct.Source: §1.1 · p. 3, §1 · p. 3
- For dense graphs with m = Θ(n^2), the algorithm constructs an optimal forest partition in m^1+o(1) time.Source: §1.1 · p. 4
- The exact algorithm finds the smallest feasible k-out orientation, then tests whether k forests suffice and otherwise calls ExactK with k+1.The orientation guarantees k ≤ a(G) ≤ k+1.Source: §1.1 · p. 4
- Given a k-out orientation, a randomized algorithm constructs k+1 forests in O~(m + nk) time, in expectation and with high probability.After finding an orientation with k = ρ(G), this yields at most a(G) + 1 forests.Source: §1.1 · p. 4
- Vertex Reduction either extends a k-forest partition after removing a specified vertex or reports that no such partition exists, using at most one minimum-cost-flow computation.Its running time is (nk)^1+o(1); the star-completion procedure chooses between two construction approaches.Source: §1.1 · p. 4
2 A near-linear constructive additive-one algorithm
The additive-one algorithm randomly colors outgoing edges to form pseudoforests, controls their cycle count, then converts them into forests. Its conversion uses a surplus graph and deterministic exchange operations.
2 A near-linear constructive additive-one algorithm
- Independent random assignment of each vertex’s outgoing edges to distinct colors makes each color class a pseudoforest; cycle counting bounds the expected number of monochromatic cycles.The cycle-count argument counts directed cycles in the supplied k-out orientation and sums their monochromatic probabilities.Source: §2 · p. 5, §2.1 · p. 5
2.1 Number of cycles
- With probability at least 1 − n^-10, the coloring passed to conversion has at most 2m log2 n/k cycles.The algorithm tries up to ⌈10 log2 n⌉ independent colorings, accepting one below the threshold.Source: §2.1 · pp. 5–6
2.2 Converting pseudoforests into forests
- Removing one edge per cycle creates matching sets whose union is the surplus graph H; deterministic conversion turns the pseudoforests into k+1 forests in O~(m + nk + k|H|) time.The conversion makes H colorful, then removes its cycles through exchanges that preserve a valid partition.Source: §2.2 · pp. 6–7
2.3 Proof of Theorem 2
- The coloring loop takes O~(m) time, and the cycle threshold bounds the surplus-graph term in the conversion cost.The conversion’s stated bound is O~(m + nk + k|H|).Source: §2.1 · p. 6, §2.2 · p. 6, §2.3 · p. 7
3 Vertex Reduction
This section reduces insertion of one vertex’s star into k existing forests to a minimum-cost-flow problem. The flow either produces a recolored k-forest partition or a density certificate that k forests cannot suffice.
3 Vertex Reduction
- Given a k-forest partition of H−r, the procedure either extends it to H, possibly recoloring edges, or returns a set X proving that k forests cannot suffice.The failure certificate has |E_H(X)| > k(|X|−1).Source: §3 · p. 7
3.1 Build the flow network
- A minimum-cost flow routes units representing star edges through rooted-forest color paths, prioritizing the insertion of as many star edges as possible.Subject to that choice, the flow minimizes costs on edges that encode color changes.Source: §3.1 · p. 8
3.2 Construct the color classes from the flow
- Flow paths retain or reroute original edges and insert selected star edges; each resulting color has outdegree at most one, with r a sink.Flow units are matched through the network to determine each edge’s new color and direction.Source: §3.2 · pp. 8–10
3.3 Why failure means no partition exists
- When not all star edges can be selected, a projected residual cut yields X with |E_H(X)| > k(|X|−1), certifying that no k-forest partition exists.Each forest contains at most |X|−1 edges inside X.Source: §3.3 · p. 10
3.4 Why each new color class is a forest
- A cycle in a new color class would be consistently oriented, contain a color change, and induce a negative-cost residual walk, contradicting flow optimality.A color change has negative residual cost, while a minimum-cost flow cannot have a negative-cost closed walk.Source: §3.4 · pp. 10–11
3.5 Size and running time
- Rooting forests, building the network, decoding the flow, and extracting the cut and output take (nk)^{1+o(1)} deterministic time.The integral minimum-cost-flow computation and remaining work fit within this bound.Source: §3.5 · p. 11
4 The exact algorithm and its two time bounds
The section develops ExactK through core peeling and two alternative construction branches, then wraps it with minimum-outgoing-bound search to obtain an optimal forest partition. Choosing between the branches gives the same algorithm a time bound of O(n) + η(m)O(min{m^{4/3}, n^2}).
4.1 Peeling and the core
- Peeling degree-at-most-k vertices leaves a core with M = Θ(kN); successful core partitions extend to the graph, and density witnesses remain valid.An empty core can be partitioned directly by reverse restoration.Source: §4.1 · p. 12
4.2 First approach: Deletion and star restoration
- A deterministic hitting-set deletion produces a (k−1)-out orientation of Q−R, after which star restoration either builds k forests or rules them out.The constraints require deleting each vertex of outdegree k or one of its outgoing neighbors.Source: §4.2 · p. 13
4.3 Second approach: Rooted tree construction
- The rooted construction either certifies a density violation or returns k arc-disjoint spanning trees whose removal of the added root edges yields a partition of the original edges.The rooted-branch construction takes e O(kM) deterministic time.Source: §4.3 · pp. 13–14
4 The exact algorithm and its two time bounds
- ExactK either constructs a k-forest partition or certifies none exists; the wrapper tests the minimum outgoing bound k and, if needed, retries at k+1.The second call uses the same orientation and succeeds when the first call rules out k forests.Source: §4.4 · p. 15, §4.5 · pp. 16–17
4.4 Choosing between the two approaches
- Choosing the smaller branch multiplier gives the same algorithm O(n) + η(m)O(min{m^{4/3}, n^2}) time.The two branch bounds are O(n) + η(m)[m + kM] and O(n) + η(m)[m + M(1 + N/k)].Source: §4.4 · pp. 15–16
4.5 Proof of Theorem 1
- Binary search finds the minimum outgoing bound k; because k ≤ a(G) ≤ k+1, ExactK at k and, if necessary, k+1 yields an optimal partition.The orientation search uses exact maximum-flow computations.Source: §4.5 · pp. 16–17
AI acknowledgment
The authors acknowledge that GPT Astra helped develop some proofs and contributed to the paper’s writing and exposition.
AI acknowledgment
- GPT Astra assisted with some proofs and the writing and exposition; the authors remain responsible for correctness and final content.Source: AI acknowledgment · p. 17