Approximation and Hardness Results for the Parallel-Block Construction Problem

Abstract

Several high-throughput blockchains, including Solana, Sui, and Aptos, execute transactions in parallel across multiple cores to improve throughput and reduce latency. However, transaction conflicts induced by shared state access fundamentally couple block construction with parallel scheduling. We formalize and study the Parallel-Block Construction problem: given transactions with execution times, rewards, and conflict relations, select and schedule a subset of transactions on p parallel cores within a runtime (gas) budget to maximize total reward.

We provide a comprehensive analysis of the complexity and approximation landscape for this problem. When the number of cores p is part of the input, we prove strong inapproximability via a reduction from Maximum Clique: unless NP = ZPP, no polynomial-time algorithm achieves an n1−ϵ -approximation for any ϵ > 0. For constant p and uniform processing times, we give a greedy algorithm achieving a tight (1 − 1/e)-approximation ratio. We further establish NP-completeness even for fixed p = 4 and unit-length transactions, and show that for p = 2 the problem admits an exact polynomial-time algorithm via a reduction to maximum-weight matching (with a cardinality constraint). In contrast, allowing heterogeneous processing times restores hardness: the problem becomes NP-complete for p = 2 with processing times in {1, 3} (even with unit rewards). For p = 2 and processing times in {1, 2}, we design a polynomial-time (2/3 − δ)-approximation algorithm (for any δ > 0) via a structural decomposition and a reduction to a budgeted matching problem.

We validate our theoretical results with experiments on Ethereum mainnet execution traces. In the homogeneous setting, our (1 − 1/e)-approximation algorithm almost always achieves rewards above 99% of the MILP-based optimal baseline, far exceeding the guaranteed factor of 1 − 1/e = 0.63. In the heterogeneous setting (p = 2, processing times in {1, 2}), the non-overlap variant achieves above 99% of the optimal across all tested configurations, confirming that the theoretical 2/3 bound is a pessimistic worst-case guarantee that does not reflect typical performance on real workloads.