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Lower Bounds for Non-adaptive Local Computation Algorithms

Amir Azarmehr, Soheil Behnezhad, Alma Ghafari, Madhu Sudan

TL;DR

The paper addresses whether non-adaptive LCAs can approach the efficiency of adaptive LCAs for constant-approximation problems like maximum matching, vertex cover, and MIS. It introduces a rigorous, formal notion of non-adaptive LCAs and builds a novel lower-bound framework that blends sublinear-time coupling techniques with a modified KMW construction to prove indistinguishability between key edge types. The main result shows a tight separation: any non-adaptive LCA achieving a constant-approximation requires $Δ^{Ω(\log Δ/ \log \log Δ)}$ queries, matching the prior black-box bound up to constants in the exponent and implying that modest gains would yield significant MPC improvements. This establishes a fundamental limit on non-adaptive LCAs and significantly informs the design of MPC and distributed approaches to these classic combinatorial problems.

Abstract

We study *non-adaptive* Local Computation Algorithms (LCA). A reduction of Parnas and Ron (TCS'07) turns any distributed algorithm into a non-adaptive LCA. Plugging known distributed algorithms, this leads to non-adaptive LCAs for constant approximations of maximum matching (MM) and minimum vertex cover (MVC) with complexity $Δ^{O(\log Δ/ \log \log Δ)}$, where $Δ$ is the maximum degree of the graph. Allowing adaptivity, this bound can be significantly improved to $\text{poly}(Δ)$, but is such a gap necessary or are there better non-adaptive LCAs? Adaptivity as a resource has been studied extensively across various areas. Beyond this, we further motivate the study of non-adaptive LCAs by showing that even a modest improvement over the Parnas-Ron bound for the MVC problem would have major implications in the Massively Parallel Computation (MPC) setting; It would lead to faster truly sublinear space MPC algorithms for approximate MM, a major open problem of the area. Our main result is a lower bound that rules out this avenue for progress. We prove that $Δ^{Ω(\log Δ/ \log \log Δ)}$ queries are needed for any non-adaptive LCA computing a constant approximation of MM or MVC. This is the first separation between non-adaptive and adaptive LCAs, and already matches (up to constants in the exponent) the algorithm obtained by the black-box reduction of Parnas and Ron. Our proof blends techniques from two separate lines of work: sublinear time lower bounds and distributed lower bounds. Particularly, we adopt techniques such as couplings over acyclic subgraphs from the recent sublinear time lower bounds of Behnezhad, Roghani, and Rubinstein (STOC'23, FOCS'23, STOC'24). We apply these techniques to a very different instance, (a modified version of) the construction of Kuhn, Moscibroda and Wattenhoffer (JACM'16) from distributed computing.

Lower Bounds for Non-adaptive Local Computation Algorithms

TL;DR

The paper addresses whether non-adaptive LCAs can approach the efficiency of adaptive LCAs for constant-approximation problems like maximum matching, vertex cover, and MIS. It introduces a rigorous, formal notion of non-adaptive LCAs and builds a novel lower-bound framework that blends sublinear-time coupling techniques with a modified KMW construction to prove indistinguishability between key edge types. The main result shows a tight separation: any non-adaptive LCA achieving a constant-approximation requires queries, matching the prior black-box bound up to constants in the exponent and implying that modest gains would yield significant MPC improvements. This establishes a fundamental limit on non-adaptive LCAs and significantly informs the design of MPC and distributed approaches to these classic combinatorial problems.

Abstract

We study *non-adaptive* Local Computation Algorithms (LCA). A reduction of Parnas and Ron (TCS'07) turns any distributed algorithm into a non-adaptive LCA. Plugging known distributed algorithms, this leads to non-adaptive LCAs for constant approximations of maximum matching (MM) and minimum vertex cover (MVC) with complexity , where is the maximum degree of the graph. Allowing adaptivity, this bound can be significantly improved to , but is such a gap necessary or are there better non-adaptive LCAs? Adaptivity as a resource has been studied extensively across various areas. Beyond this, we further motivate the study of non-adaptive LCAs by showing that even a modest improvement over the Parnas-Ron bound for the MVC problem would have major implications in the Massively Parallel Computation (MPC) setting; It would lead to faster truly sublinear space MPC algorithms for approximate MM, a major open problem of the area. Our main result is a lower bound that rules out this avenue for progress. We prove that queries are needed for any non-adaptive LCA computing a constant approximation of MM or MVC. This is the first separation between non-adaptive and adaptive LCAs, and already matches (up to constants in the exponent) the algorithm obtained by the black-box reduction of Parnas and Ron. Our proof blends techniques from two separate lines of work: sublinear time lower bounds and distributed lower bounds. Particularly, we adopt techniques such as couplings over acyclic subgraphs from the recent sublinear time lower bounds of Behnezhad, Roghani, and Rubinstein (STOC'23, FOCS'23, STOC'24). We apply these techniques to a very different instance, (a modified version of) the construction of Kuhn, Moscibroda and Wattenhoffer (JACM'16) from distributed computing.
Paper Structure (24 sections, 7 theorems, 71 equations, 4 figures, 1 algorithm)

This paper contains 24 sections, 7 theorems, 71 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Non-adaptive LCAs:
  3. Why Non-adaptive LCAs Matter
  4. Implications in the MPC model:
  5. Our Main Contribution
  6. Perspective: The Need for New Lower Bound Techniques
  7. Our new lower bound technique:
  8. Technical Overview
  9. Step 1: Protecting from extremely long walks by modifying KMW.
  10. Step 2 (main challenge): Protecting from moderately long walks.
  11. Additional modifications (cycles).
  12. Putting things together: indistinguishability through coupling.
  13. Paper Organization.
  14. Preliminaries
  15. The Construction
  16. ...and 9 more sections

Key Result

Theorem 1.1

Any (possibly randomized) non-adaptive LCA that given a graph $G$, returns an $O(1)$ approximation of maximum matching, an $O(1)$ approximation of minimum vertex cover, or a maximal independent set with constant probability, requires $\Delta^{\Omega(\log \Delta / \log \log \Delta)}$ queries.

Figures (4)

  • Figure 4.1: An illustration of the cluster tree (\ref{['def:cluster-tree']}) for $r = 3$. The shades represent the cluster colors. $C_0$ is the root. The edge labels are displayed only in the direction away from the root.
  • Figure 4.2: An example of distinguishing sequences, following label sequence $(\delta_0, \delta_0, \delta_0)$. The paths start at $C_0$ and $C_1$. One reaches a non-leaf cluster that has degree $\delta_0 + \delta_1 + \delta_2 + \delta_3$, whereas the other one reaches a leaf of degree $\delta_1$.
  • Figure 4.3: Another example of distinguishing sequences, following label sequence $(\delta_1, \delta_2, \delta_2)$. The paths start at $C_0$ and $C_1$. One reaches a non-leaf cluster that has degree $\delta_0 + \delta_1 + \delta_2 + \delta_3$, whereas the other one reaches a leaf of degree $\delta_3$.
  • Figure 4.4: An illustration of the full blueprint (\ref{['def:blueprint']}) for $r = 2$.

Theorems & Definitions (79)

  • Theorem 1.1
  • Definition 3.1: Non-adaptive LCA
  • Remark 3.2
  • Definition 3.3: Non-adaptive LCAs for maximum matching
  • Definition 4.1: The cluster tree KuhnMW16
  • Definition 4.2: Cluster degrees
  • Definition 4.3: Color
  • Claim 4.4
  • Claim 4.5
  • Claim 4.6
  • ...and 69 more