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arxiv: 2502.02477 · v3 · submitted 2025-02-04 · 💻 cs.DS

Speeding-up Graph Algorithms via Clique Partitioning

Akshar Chavan , Sanaz Rabinia , Daniel Grosu , Marco Brocanelli This is my paper

Pith reviewed 2026-05-23 03:43 UTC · model grok-4.3

classification 💻 cs.DS
keywords graph algorithmsclique partitioningbipartite cliquestripartite graphsedge reductionmatchingall-pairs shortest pathspreprocessing
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The pith

A clique partitioning method replaces bipartite cliques with tripartite graphs to cut edges while keeping all paths intact.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an algorithm that identifies bipartite cliques in a graph and replaces each one with a tripartite graph structure. This change reduces the total number of edges but keeps every path between vertices unchanged. The new method runs in O(m n^δ) time, which improves on the previous O(m n^δ log² n) bound. Experiments show it reduces more edges than prior work and, when used before other algorithms, speeds up matching by up to 2.07 times and all-pairs shortest paths by up to 1.74 times on large graphs.

Core claim

The paper claims that by partitioning the graph into cliques and restructuring bipartite cliques into tripartite graphs, one obtains a sparser graph that retains complete path information, allowing any path-based algorithm to be run directly on it with no loss of correctness, and that this restructuring can be done in O(m n^δ) time for 0 ≤ δ ≤ 1.

What carries the argument

The bipartite clique to tripartite graph replacement, which serves as the mechanism to reduce edges while preserving paths.

If this is right

  • The proposed algorithm achieves up to 21.26% higher edge reduction than the Feder-Motwani algorithm.
  • It runs up to 105.18× faster than the Feder-Motwani algorithm.
  • Preprocessing with this method yields up to 2.07× speedup for matching algorithms on large synthetic graphs.
  • Preprocessing yields up to 1.74× speedup for all-pairs shortest path algorithms on real-world graphs.
  • Edge reduction reaches up to 74.36% on synthetic graphs with over a billion edges and 46.8% on real-world graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This restructuring could be applied as a general preprocessing step for any algorithm that relies on path information in graphs.
  • The approach might generalize to directed graphs more broadly or to weighted graphs if the tripartite replacement is adjusted accordingly.
  • Future work could explore combining this with other graph compression techniques to achieve even greater reductions.
  • If the time complexity holds for δ close to 0, it could make preprocessing feasible even for very large n.

Load-bearing premise

Replacing each bipartite clique with a tripartite graph preserves every path present in the original graph.

What would settle it

Finding a graph and a pair of vertices where a path exists before restructuring but not after, or measuring the running time on a family of graphs to check if it exceeds the claimed O(m n^δ) bound.

Figures

Figures reproduced from arXiv: 2502.02477 by Akshar Chavan, Daniel Grosu, Marco Brocanelli, Sanaz Rabinia.

Figure 1
Figure 1. Figure 1: (a) Given bipartite graph G(U, W, E); (b) Neighborhood tree of vertex u2 ∈ U that shows the path ω taken from root to the vertex wj ∈ W at the leaf and number of edges du2,ω for each node using the tuple (ω, du2,ω); and (c) the tripartite graph that replaces the δ-clique with left partition {u1, u2, u3, u4, u5, u7, u8} and right partition {w4, w5} in the compressed graph G∗ (U, W, Z, E∗ ). graph G∗ preserv… view at source ↗
Figure 2
Figure 2. Figure 2: Progression of ˆm, ˆk, and number of cliques extracted for a graph with 128 vertices in each bi-partition, density 0.98, and δ = 1 by FM and CPGC. Algorithm 1 CPGC: Clique Partitioning-based Graph Compression Algorithm Input: A: Adjacency matrix; δ: a constant such that 0 ≤ δ ≤ 1. 1: q ← 0; n ← |W|; ˆm ← |E|; dw ← [0]n; C ← ∅ 2: for i = 1, . . . , n do 3: for j = 1, . . . , n do 4: dwj ← dwj + Eui,wj 5: kˆ… view at source ↗
Figure 3
Figure 3. Figure 3: CPGC: Average compression ratio (top row: (a), (b), and (c)) and running time (bottom row: (d), (e), and (f)) for graphs with 3.36 million to 1.05 billion edges. n, where n ranges from 32 to 32,768, and densities of 0.8, 0.85, 0.90, 0.95, and 0.98. This generated graphs with up to 1.05 billion edges, considering p as a mea￾sure of the density of the generated bipartite graphs. For each bipartite graph with… view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dinitz(G) vs. Dinitz(G∗ ): Average running time of Dinitz’s algorithm on the original bipartite graph G (labeled Dinitz(G)) and on the compressed graph G∗ (labeled Dinitz(G∗ )), for a large graph with approximately 32,000 vertices in each bipartition and 214.75 million to 1.05 billion edges corresponding to densities = 0.80, 0.90, and 0.98 and different δ. density 0.8, CPGC achieves an average speedup of 3… view at source ↗
Figure 6
Figure 6. Figure 6: Clique partitioning: (a) given bipartite graph G(U, W); (b) extracted clique C1; (c) extracted clique C2; (d) graph with trivial cliques; and (e) compressed tripartite graph G(U, Z, W) CSA returns (C , A, ˆ ˆdw) and then CPGC updates the set C = {C1, C2}, dw, the adjacency matrix A, and the number of edges ˆm = 54 − 28 = 26, in Lines 8-10. In Line 12, CPGC updates q = 2 and finally, in Line 13 it updates ˆ… view at source ↗
read the original abstract

Reducing the running time of graph algorithms is vital for tackling real-world problems such as shortest paths and matching in large-scale graphs, where path information plays a crucial role. To address this critical challenge, this paper introduces a graph restructuring algorithm that identifies bipartite cliques and replaces them with tripartite graphs. This restructuring leads to fewer edges while preserving complete graph path information, enabling the direct application of algorithms like matching and all-pairs shortest paths to achieve significant runtime reductions, especially for large, dense graphs. The running time of the proposed algorithm for a graph $G(V,E)$, with $|V| = n$ and $|E| = m$ is~$O(mn^\delta)$, which is better than $O(mn^\delta \log^2 n)$, the running time of the best existing algorithm for speeding-up other graph algorithms (the Feder-Motwani (\textsf{FM}) algorithm), where $0 \leq \delta \leq 1$. Both the \textsf{FM} algorithm and the proposed algorithm are originally formulated for bipartite graphs, but can also be applied to general directed or undirected graphs. Our extensive experimental analysis demonstrates that the proposed algorithm achieves up to 21.26\% higher reduction in the number of edges and runs up to 105.18$\times$ faster than the \textsf{FM} algorithm. On large synthetic graphs with up to 1.05 billion edges, it attains a reduction in the number of edges of up to 74.36\%. On real-world graphs, it achieves a reduction in the number of edges by up to 46.8\%. Furthermore, when used as a preprocessing step, our approach yields up to a 2.07$\times$ speedup for the matching algorithms on large synthetic graphs, and up to a 1.74$\times$ speedup for the All-Pairs Shortest Path algorithms on real-world graphs, when compared to using the given graph as input.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper introduces a graph restructuring algorithm that detects bipartite cliques in G=(V,E) (|V|=n, |E|=m) and replaces each with a tripartite graph, claiming to reduce edges while preserving all path information. This enables direct use of the restructured graph for matching and APSP. The algorithm runs in O(m n^δ) time (0≤δ≤1), strictly faster than Feder-Motwani's O(m n^δ log² n); experiments report up to 21.26% higher edge reduction, 105.18× faster execution than FM, up to 74.36% reduction on synthetic graphs with 1.05B edges, and preprocessing speedups of 2.07× (matching) and 1.74× (APSP).

Significance. If the path-preservation property and runtime bound hold, the method could serve as an effective preprocessing step for dense graphs, improving practical runtimes of core algorithms. The scale of the reported experiments (graphs up to 1B edges) is a positive indicator of applicability, though the theoretical improvement over FM would be the primary contribution if substantiated.

major comments (3)
  1. [Abstract] Abstract: the central claim that 'complete graph path information' is preserved (enabling direct application to matching and APSP without loss of correctness) is load-bearing for all reported speedups, yet no lemma, invariant, or argument is supplied establishing that reachability relations, distances, or maximum matchings remain identical after each bipartite-clique-to-tripartite replacement.
  2. [Abstract] Abstract: the stated O(m n^δ) runtime bound (and its strict improvement over FM's O(m n^δ log² n)) is presented without derivation, recurrence, or reference to any section/equation that derives δ or eliminates the log² n factor; this is load-bearing for the headline asymptotic claim.
  3. [Abstract] Abstract: experimental claims (21.26% higher reduction, 105.18× faster execution, 2.07×/1.74× speedups) are reported without description of implementation baselines, controls for graph generation, or statistical tests; this undermines validation of the practical results that support the preprocessing utility.
minor comments (1)
  1. The statement that both the proposed and FM algorithms 'are originally formulated for bipartite graphs, but can also be applied to general directed or undirected graphs' would benefit from a one-sentence clarification of the reduction used for the general case.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and will revise the manuscript to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that 'complete graph path information' is preserved (enabling direct application to matching and APSP without loss of correctness) is load-bearing for all reported speedups, yet no lemma, invariant, or argument is supplied establishing that reachability relations, distances, or maximum matchings remain identical after each bipartite-clique-to-tripartite replacement.

    Authors: The path-preservation property is established in the body of the paper (Section 3), specifically via Lemma 3.1 (reachability and distances) and Theorem 3.2 (matching sizes). The abstract summarizes the result but does not restate the proof. To address the concern, we will revise the abstract to include a brief reference to these results and ensure the key invariant is stated explicitly. revision: yes

  2. Referee: [Abstract] Abstract: the stated O(m n^δ) runtime bound (and its strict improvement over FM's O(m n^δ log² n)) is presented without derivation, recurrence, or reference to any section/equation that derives δ or eliminates the log² n factor; this is load-bearing for the headline asymptotic claim.

    Authors: The runtime analysis appears in Section 4, where the recurrence and the data-structure improvement that removes the log² n factor are derived (leading to the O(m n^δ) bound with 0 ≤ δ ≤ 1). We will add an explicit pointer to the relevant equation and section in the abstract to make the derivation traceable. revision: yes

  3. Referee: [Abstract] Abstract: experimental claims (21.26% higher reduction, 105.18× faster execution, 2.07×/1.74× speedups) are reported without description of implementation baselines, controls for graph generation, or statistical tests; this undermines validation of the practical results that support the preprocessing utility.

    Authors: Section 5 describes the FM baseline implementation (following the original FM paper), graph-generation parameters, and the use of multiple runs with averaged results. We agree that additional details on statistical controls would improve the presentation and will expand the experimental section and abstract references accordingly in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; runtime and reduction claims derive from algorithm design and external baseline

full rationale

The paper introduces a clique-partitioning restructuring whose O(m n^δ) bound is stated as following directly from the new procedure (distinct from the cited Feder-Motwani O(m n^δ log² n) external algorithm). No equation, parameter fit, or self-citation is used to define the claimed runtime, edge-reduction percentages, or downstream speed-ups; those quantities are presented as consequences of the algorithm and are evaluated against an independent baseline. The path-preservation assertion is a correctness claim, not a definitional reduction. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard graph-theoretic facts (paths are sequences of edges) plus the unproven assertion that the tripartite substitution preserves all such sequences; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Replacing a bipartite clique with a tripartite graph preserves every path that existed in the original graph.
    Invoked when the authors assert that matching and APSP can be run directly on the restructured graph without loss of correctness.

pith-pipeline@v0.9.0 · 5901 in / 1401 out tokens · 24664 ms · 2026-05-23T03:43:04.397302+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Succinct Graph Representations and Algorithmic Applications

    cs.DS 2026-04 unverdicted novelty 7.0

    Succinct dual clique cover representations allow fundamental graph algorithms to run in time proportional to the representation size instead of the number of edges, delivering substantial memory savings and speedups o...

Reference graph

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