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Local Computation Algorithms for Knapsack: impossibility results, and how to avoid them

Clément L. Canonne, Yun Li, Seeun William Umboh

TL;DR

This work probes Local Computation Algorithms for Knapsack, revealing strong impossibility barriers for sublinear LCAs to recover optimal, near-optimal, or maximal feasible solutions. It then unveils a positive result under a weighted sampling access model, delivering a sublinear LCA achieving a $$(1/2+\varepsilon)$$-approximation with complexity $$(1/\varepsilon)^{O(\log^{*}n)}$$ by leveraging reproducible learning concepts to enforce consistency. The approach fuses reductions to the OR problem for lower bounds with a reproducible-quantile framework to guide the sampling and reduction steps, culminating in a robust algorithmic blueprint for Knapsack in the LCA paradigm. The findings illuminate how additional access patterns (weighted sampling) and reproducibility concepts can circumvent impossibility barriers, potentially guiding LCAs for other combinatorial optimization tasks and motivating further investigation into average-case or distributional assumptions. The work thus advances the design space of LCAs, balancing impossibility with carefully constructed, access-aware strategies for sublinear, query-efficient computation.

Abstract

Local Computation Algorithms (LCA), as introduced by Rubinfeld, Tamir, Vardi, and Xie (2011), are a type of ultra-efficient algorithms which, given access to a (large) input for a given computational task, are required to provide fast query access to a consistent output solution, without maintaining a state between queries. This paradigm of computation in particular allows for hugely distributed algorithms, where independent instances of a given LCA provide consistent access to a common output solution. The past decade has seen a significant amount of work on LCAs, by and large focusing on graph problems. In this paper, we initiate the study of Local Computation Algorithms for perhaps the archetypal combinatorial optimization problem, Knapsack. We first establish strong impossibility results, ruling out the existence of any non-trivial LCA for Knapsack as several of its relaxations. We then show how equipping the LCA with additional access to the Knapsack instance, namely, weighted item sampling, allows one to circumvent these impossibility results, and obtain sublinear-time and query LCAs. Our positive result draws on a connection to the recent notion of reproducibility for learning algorithms (Impagliazzo, Lei, Pitassi, and Sorrell, 2022), a connection we believe to be of independent interest for the design of LCAs.

Local Computation Algorithms for Knapsack: impossibility results, and how to avoid them

TL;DR

This work probes Local Computation Algorithms for Knapsack, revealing strong impossibility barriers for sublinear LCAs to recover optimal, near-optimal, or maximal feasible solutions. It then unveils a positive result under a weighted sampling access model, delivering a sublinear LCA achieving a -approximation with complexity by leveraging reproducible learning concepts to enforce consistency. The approach fuses reductions to the OR problem for lower bounds with a reproducible-quantile framework to guide the sampling and reduction steps, culminating in a robust algorithmic blueprint for Knapsack in the LCA paradigm. The findings illuminate how additional access patterns (weighted sampling) and reproducibility concepts can circumvent impossibility barriers, potentially guiding LCAs for other combinatorial optimization tasks and motivating further investigation into average-case or distributional assumptions. The work thus advances the design space of LCAs, balancing impossibility with carefully constructed, access-aware strategies for sublinear, query-efficient computation.

Abstract

Local Computation Algorithms (LCA), as introduced by Rubinfeld, Tamir, Vardi, and Xie (2011), are a type of ultra-efficient algorithms which, given access to a (large) input for a given computational task, are required to provide fast query access to a consistent output solution, without maintaining a state between queries. This paradigm of computation in particular allows for hugely distributed algorithms, where independent instances of a given LCA provide consistent access to a common output solution. The past decade has seen a significant amount of work on LCAs, by and large focusing on graph problems. In this paper, we initiate the study of Local Computation Algorithms for perhaps the archetypal combinatorial optimization problem, Knapsack. We first establish strong impossibility results, ruling out the existence of any non-trivial LCA for Knapsack as several of its relaxations. We then show how equipping the LCA with additional access to the Knapsack instance, namely, weighted item sampling, allows one to circumvent these impossibility results, and obtain sublinear-time and query LCAs. Our positive result draws on a connection to the recent notion of reproducibility for learning algorithms (Impagliazzo, Lei, Pitassi, and Sorrell, 2022), a connection we believe to be of independent interest for the design of LCAs.

Paper Structure

This paper contains 16 sections, 19 theorems, 46 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our results and contributions
  3. Technical overview.
  4. Related Work
  5. Local Computation Algorithms.
  6. Knapsack.
  7. Preliminaries
  8. Impossibility Results
  9. Algorithmic Result: an LCA given Weighted Sampling Access
  10. General Idea of the Algorithm
  11. Reproducible Quantile Algorithm
  12. Mapping to a finite domain.
  13. From Reproducible Quantiles to our Knapsack LCA
  14. Subcase 1:
  15. Subcase 2:
  16. ...and 1 more sections

Key Result

Theorem 1.1

There is no sublinear-time LCA for Knapsack that provides consistent query access to an optimal solution.

Figures (1)

  • Figure 1: An illustration of the reduction. Given query access to an input $x\in\{0,1\}^{n-1}$ (top), we simulate query access to a Knapsack instance $I(x)$ (bottom) with weight limit $K=1$, such that the $n$-th item belongs to the optimal solution (which is then unique) if, and only if, $\operatorname{OR}_n(x)=0$.

Theorems & Definitions (36)

  • Theorem 1.1: Informal; see \ref{['theo:no:optimal:knapsack:lca']}
  • Theorem 1.2: Informal; see \ref{['theo:no:approx:knapsack:lca']}
  • Theorem 1.3: Informal; see \ref{['theo:no:maximal:knapsack:lca']}
  • Theorem 1.4: Main theorem (Informal; see \ref{['theo:lcakpresult']})
  • Definition 1: $(\alpha, \beta)$-approximation algorithm
  • Definition 2: LCA for Knapsack Problem
  • Definition 3: Parallelizable
  • Definition 4: Query-Order Oblivious
  • Definition 5: Reproducibility, impagliazzo2022reproducibility
  • Definition 6: $\tau$-approximate median
  • ...and 26 more