Title: Parallel and Distributed Expander Decomposition: Simple, Fast, and | Near-Optimal
Domain: Computer Science

Question: How to efficiently compute near-optimal φ-expander decompositions of undirected graphs in near-linear work and near-constant span, while implementing this process in the distributed Congest model and ensuring that only an $\tilde{O}(\phi)$ fraction of edges cross between partition sets, thereby surpassing the theoretical guarantees of existing parallel algorithms?

Background: - Expander decompositions have become a central framework in graph algorithms, offering a way to partition graphs into well-connected subgraphs (expanders) with few crossing edges. | | - Formally, for an undirected graph \( G=(V,E) \), a near-optimal \( \phi \)-expander decomposition partitions the vertex set \( V \) into \( V_1, V_2, \ldots, V_k \), such that each subgraph \( G[V_i] \) is a \( \phi \)-expander. A \( \phi \)-expander ensures that for any subset \( S \subseteq V \), the number of edges leaving \( S \) is large relative to the volume of the smaller side of the cut, i.e., \( |E_G(S, V \setminus S)| \geq \phi \cdot \min\{\vol_G(S), \vol_G(V \setminus S)\} \). When \( \phi \approx 1 \), the graph is highly connected with low diameter and spectral properties similar to a complete graph, while every simple connected graph is at least a \( \phi = 1/|V|^2 \) expander. | | - Recent practical implementations have demonstrated that expander decompositions can be computed for medium-sized graphs. However, to handle very large graphs with hundreds of millions or billions of edges, there is a need for either significantly improved algorithms or the utilization of parallelization techniques.

Definitions: - **Nearly Expander**: Given a graph \( G = (V, E) \) and a subset \( A \subseteq V \), the subgraph \( G[A] \) is considered a φ-nearly expander in \( G \) if for every subset \( S \subseteq A \) where \( \text{deg}_G(S) \leq \text{deg}_G(A \setminus S) \), the number of edges leaving \( S \) is at least \( \phi \cdot \text{deg}_G(S) \).

Methodology: - **Flow-based Techniques**: Employe a new parallel algorithm , leveraging flow-based techniques to enhance both the quality and efficiency of expander decomposition. The sequential algorithm framework is extended, utilizing a flow-based approach for trimming subgraphs to ensure each subgraph is a \(\phi\)-expander. | - **Parallelization Design**:developed a Parallel Unit Flow algorithm is , accelerating the solution of flow problems through multiple rounds of push and relabel operations. | - **Trimming Algorithm Optimization**: Introduce an optimized trimming algorithm, capable of converting a nearly expander into an expander within logarithmic iterations. By dynamically adjusting sink capacities, the number of flow problems to be solved is reduced, leading to a significant improvement in the algorithm's convergence speed.

Answer: Goal | | Input: undirected graph G=(V,E), |E|=m, parameter \varphi\in(0,1). | | Output: a partition V=V_1\cup\cdots\cup V_k such that each induced subgraph G[V_i] is a \Theta(\varphi)-expander, and the number of edges crossing between parts satisfies | \sum_{i<j} |E(V_i,V_j)| \;=\; \tilde{O}(\varphi\, m). | | Parallel guarantees: work \tilde{O}(m/\varphi^2), span \tilde{O}(1/\varphi^4). | Distributed Congest guarantees: rounds \tilde{O}(1/\varphi^4). | (Polylogarithmic factors in n are hidden by \tilde{O}(\cdot).) | | ⸻ | | Core ingredients (what to build and why it works) | | 1) A unit-flow primitive with parallel bounds | | We use a parallel push–relabel–style ParallelUnitFlow on the residual graph with: | • edge capacities bounded by \eta, | • per-vertex sink capacity \nabla(v)\ge \gamma\cdot \deg(v), | • bounded source load \Delta(v)\le \eta\cdot\deg(v) and \|\Delta\|_1\le 2m, | • height h. | | It returns a flow f and labels l:V\to\{0,\dots,h\} such that (i) edges with label gap >1 are saturated in the downhill direction, (ii) nodes with l(u)\ge1 have their sinks almost saturated, (iii) nodes with l(u)<h have no remaining excess. | | Parallel cost. Work \tilde{O}(m h \eta/\gamma); span \tilde{O}(h^2\eta/\gamma). | | Intuition: this is a degree-aware unit-flow. Because every node has sink capacity proportional to its degree, we can relate total progress to total excess and charge work to capacity used. This will certify expansion or expose a cut to trim. | | 2) A “trimming” routine that turns nearly-expanders into expanders | | Call a set A\subseteq V a \varphi-nearly expander in G if every S\subseteq A with \deg_G(S)\le \deg_G(A\!\setminus\!S) has | |E_G(S, V\!\setminus\!S)| \ge \varphi \cdot \deg_G(S). | (An induced expander implies nearly-expander, but not vice versa.) | | We run Trimming(G, A, \varphi): | • Give each edge in G[A] capacity \Theta(\varphi^2). | • Iteratively invoke ParallelUnitFlow on G[A]: the source encodes how much A “relies” on the outside, sinks are \propto \deg, and we use height h=\Theta(\frac{1}{\varphi}\log^2 n \log m). | • If all excess can be absorbed (feasible flow), then by a flow-certification lemma G[A] is a \Omega(\varphi)-expander. | • If not, the labels expose a small set S at top levels with too little internal capacity (witnessing where expansion fails). Trim S off: set A\gets A\setminus S and repeat. | | What this guarantees. If you begin with a \varphi-nearly expander A and the current boundary |E(A,V\setminus A)| is not too large, then one trimming pass produces A’\subseteq A with: | 1. G[A’] a \Theta(\varphi)-expander, | 2. \deg_G(A’) \ge \deg_G(A) - \tilde{O}\!\big(\tfrac{n}{|E(A,V\setminus A)|}\big) (so A’ is large), | 3. |E(A’,V\setminus A’)| \le 2\,|E(A,V\setminus A)| (boundary doesn’t blow up). | | Parallel cost per trimming: work \tilde{O}(m/\varphi^2), span \tilde{O}(1/\varphi^3). | | Why it works: If G[A] fails to be an expander, the failed unit-flow induces a short sequence of “ball-growing” label sets that identify a sparse internal cut. Removing only the ill-expanded fringe reduces excess substantially; repeating a bounded number of times certifies expansion. | | ⸻ | | The full decomposition algorithm | 1. Seed nearly-expanders. Start from V, repeatedly peel off sets A that are \varphi-nearly expanding (standard near-linear routines find such sets; or you can start with A=V if the boundary condition holds and shrink it). | 2. Trim to true expanders. For each nearly-expanding A, run Trimming(G, A, \varphi) to obtain A’ with G[A’] a \Theta(\varphi)-expander (by the unit-flow certification) and with boundary and size guarantees. | 3. Recurse on the remainder. Remove A’ from the graph and continue until all vertices are clustered. | 4. Stop condition and accounting. Each edge is charged only \tilde{O}(1/\varphi) times across the process (via capacities and absorbed mass). Summing the charges over all steps yields: | • total work \tilde{O}(m/\varphi^2), | • total span \tilde{O}(1/\varphi^4), | • total crossing edges \sum_{i<j}|E(V_i,V_j)|=\tilde{O}(\varphi m). | | Main theorem (parallel). With high probability, the above returns a \varphi-expander decomposition with error \tilde{O}(\varphi m), total work \tilde{O}(m/\varphi^2), and span \tilde{O}(1/\varphi^4). | | ⸻ | | Congest implementation (distributed) | | Every step above can be realized in Congest (synchronous rounds, O(\log n)-bit messages): | • ParallelUnitFlow becomes a distributed push–relabel with heights and local saturation checks; pushes stay on edges, relabels use only local degree and saturation; height h=\tilde{O}(1/\varphi) yields | \text{rounds} \;=\; \tilde{O}(h^2) \;=\; \tilde{O}(1/\varphi^2) | per unit-flow stage, and only a \tilde{O}(1) number of stages are needed, giving \tilde{O}(1/\varphi^4) rounds overall. | • Ball-growing / trimming uses local label thresholds (“keep nodes with l\ge h-j”), plus local checks of residual edges; each check is constant-hop and fits Congest messages. | | Main theorem (Congest). With high probability, we obtain the same \varphi-expander decomposition with error \tilde{O}(\varphi m) in \tilde{O}(1/\varphi^4) rounds. | | ⸻ | | Why this surpasses prior parallel guarantees | • Edge budget: We achieve a \tilde{O}(\varphi) fraction of cut edges, i.e., \tilde{O}(\varphi m) total. Many prior parallel routines either (a) certify only nearly-expanders (harder to use downstream), or (b) incur larger error factors (e.g., polylog blowups in \varphi) or higher span. | • Quality of parts: Every part is a true expander (not just nearly-expanding), simplifying subsequent applications (spectral routines, flow, sparsification). | • Parallel cost profile: The algorithm is near-linear work and near-constant span (polylog hidden), which improves on earlier constructions with larger depth/round complexity for comparable error. | | ⸻ | | Minimal, actionable recipe (what to implement) | 1. ParallelUnitFlow(G, c, Δ, ∇, h): | • Capacities: c(e)\le \eta, sinks \nabla(v)\ge \gamma\deg(v), sources \|\Delta\|_1\le 2m, \Delta(v)\le \eta\deg(v). | • Push from level j to j-1 until edges saturate or no excess; relabel nodes whose outgoing to j-1 fully saturated; repeat top-down; stage-doubling trick on sinks to halve unsettled excess. | • Return f,l. | Costs: work \tilde{O}(m h \eta/\gamma), span \tilde{O}(h^2\eta/\gamma). | 2. Trimming(G, A, φ): | • Set c\equiv \Theta(\varphi^2), h=\Theta((1/\varphi)\log^2 n\log m). | • Iterate: run ParallelUnitFlow on G[A] with sources from external-dependency gap and sinks \propto \deg. | • If all excess absorbed → certify G[A] as \Theta(\varphi)-expander; return A. | • Else use high-label set sequence to pick S with small internal cut; set A\leftarrow A\setminus S and continue. | • Guarantees: expander, size-loss bound, boundary growth bound. | 3. Decompose all of V: | • Find a large nearly-expanding region A (or start with V), call Trimming, remove its A’, recurse. | • Bookkeeping ensures at most \tilde{O}(1/\varphi) charging per edge → total inter-part edges \tilde{O}(\varphi m). | 4. Congest port: | • Implement pushes, relabels, residual checks with O(\log n)-bit messages; pipeline across levels. | • Achieve \tilde{O}(1/\varphi^4) rounds end-to-end. | | ⸻ | | Final guarantees (precise statements) | • Parallel expander decomposition. With high probability, | \text{work}=\tilde{O}(m/\varphi^2),\quad \text{span}=\tilde{O}(1/\varphi^4),\quad | \sum_{i<j}|E(V_i,V_j)|=\tilde{O}(\varphi m), | and every G[V_i] is a \Theta(\varphi)-expander. | • Distributed Congest. With high probability, the same decomposition is computed in \tilde{O}(1/\varphi^4) rounds. | | That’s the full method: a degree-aware unit-flow to certify expansion, a trim-if-needed loop to convert nearly-expanders into true expanders with controlled boundary growth, and a recursive cover to decompose the whole graph—all with the desired parallel and distributed complexities and the \tilde{O}(\varphi) cut-edge budget.

Checklist: 1. Problem setup
   - Input: undirected graph G=(V,E), |E|=m, parameter φ∈(0,1).
   - Output: partition V=⋃V_i, each G[V_i] is Θ(φ)-expander.
   - Crossing edges: ∑_{i<j}|E(V_i,V_j)| = Õ(φ m).

2. Parallel guarantees
   - Work = Õ(m / φ²).
   - Span = Õ(1 / φ⁴).
   - Polylog(n) factors hidden in Õ(·).

3. Distributed guarantees
   - Congest model implementation.
   - Rounds = Õ(1 / φ⁴).

4. Unit-flow primitive
   - Parallel push–relabel style with bounded capacities.
   - Guarantees saturation and excess elimination by labels.
   - Cost: work Õ(m h η / γ), span Õ(h² η / γ).

5. Trimming routine
   - Defines φ-nearly expander.
   - Iteratively runs unit-flow with height h=Θ((1/φ) log²n log m).
   - Guarantees: Θ(φ)-expander, large size retained, boundary growth ≤2×.
   - Cost: work Õ(m / φ²), span Õ(1 / φ³).

6. Decomposition algorithm
   - Seed nearly-expanders, then trim to true expanders.
   - Recurse on remainder until all vertices clustered.
   - Accounting: each edge charged Õ(1/φ) times.

7. Parallel theorem
   - With high probability: 
     • Work Õ(m / φ²).
     • Span Õ(1 / φ⁴).
     • Crossing edges Õ(φ m).
     • Each part a Θ(φ)-expander.

8. Congest theorem
   - With high probability: Õ(1 / φ⁴) rounds.
   - Same guarantees on cut size and expansion.

9. Improvements over prior work
   - Cut edges reduced to Õ(φ m).
   - True expanders obtained (not just nearly-expanders).
   - Parallel cost near-linear work and small span.

10. Minimal recipe
   - Step 1: ParallelUnitFlow with degree-aware sinks/sources.
   - Step 2: Trimming to certify or cut sparse fringe.
   - Step 3: Recursive decomposition with charging.
   - Step 4: Congest implementation via local pushes/relabels.