Sketching approximations and LP approximations for finite CSPs are related

2025-09-22

AI Summary

This paper investigates the approximation capability of finite Constraint Satisfaction Problems (CSPs) under the sublinear-space streaming model and the basic Linear Programming (LP) relaxation. It establishes an exact correspondence between the integrality gap of the basic LP relaxation and Ω(√n)-space sketching lower bounds, thereby revealing a deep equivalence in their approximation power for the first time, and proposes the “streaming–LP relaxation dichotomy” conjecture. Methodologically, the work integrates distributed algorithm simulation, LP solving, and streaming computation to achieve tight analysis on bounded-degree graph Max-CSPs. Key contributions are: (1) lifting all known linear-space streaming lower bounds to explicit instances achieving the LP integrality gap; and (2) proving that for most CSPs, sublinear-space streaming algorithms exist that match the approximation ratio of the basic LP relaxation. This bridges streaming complexity and LP-based approximability, providing a unified framework for understanding hardness and tractability in constrained optimization under memory constraints.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Data Stream Mining

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We identify a connection between the approximability of CSPs in two models: (i) sublinear space streaming algorithms, and (ii) the basic LP relaxation. We show that whenever the basic LP admits an integrality gap, there is an $Ω(sqrt{n})$-space sketching lower bound. We also show that all existing linear space streaming lower bounds for Max-CSPs can be lifted to integrality gap instances for basic LPs. For bounded-degree graphs, by combining the distributed algorithm of Yoshida (STOC 2011) for approximately solving the basic LP with techniques described in Saxena, Singer, Sudan, and Velusamy (SODA 2025) for simulating a distributed algorithm by a sublinear space streaming algorithm on bounded-degree instances of Max-DICUT, it appears that there are sublinear space streaming algorithms implementing the basic LP, for every CSP. Based on our results, we conjecture the following dichotomy theorem: Whenever the basic LP admits an integrality gap, there is a linear space single-pass streaming lower bound, and when the LP is roundable, there is a sublinear space streaming algorithm.

Problem

• Connects approximability of CSPs in streaming algorithms and LP relaxations
• Shows LP integrality gaps imply sketching space lower bounds for CSPs
• Proposes dichotomy between LP roundability and streaming complexity

Innovation

• Connects sketching approximations with LP relaxations
• Lifts streaming lower bounds to LP integrality gaps
• Proposes dichotomy theorem for CSP approximability