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k-SUM Hardness Implies Treewidth-SETH

Michael Lampis

TL;DR

This work establishes a fundamental link between Sum-based hardness (k-SUM/k-XOR) and SAT hardness parameterized by treewidth (tw-SETH). It develops randomized reductions from k-SUM and k-XOR to SAT on formulas with treewidth near (k/2)·log n by combining interactive hashing with pathwidth reductions and then extends these ideas to derive tight lower bounds for classic DP problems parameterized by treewidth. The approach strengthens confidence in treewidth-based lower bounds by deriving them from Sum-based hypotheses and provides a new bridge between the Sum and SAT branches of fine-grained complexity. The results have practical implications for the optimality of DP algorithms for Independent Set, Max Cut, and k-Coloring under these alternative hypotheses, broadening the landscape of assumptions that underpin lower bounds in parameterized algorithms.

Abstract

We show that if k-SUM is hard, in the sense that the standard algorithm is essentially optimal, then a variant of the SETH called the Primal Treewidth SETH is true. Formally: if there is an $\varepsilon>0$ and an algorithm which solves SAT in time $(2-\varepsilon)^{tw}|φ|^{O(1)}$, where $tw$ is the width of a given tree decomposition of the primal graph of the input, then there exists a randomized algorithm which solves k-SUM in time $n^{(1-δ)\frac{k}{2}}$ for some $δ>0$ and all sufficiently large $k$. We also establish an analogous result for the k-XOR problem, where integer addition is replaced by component-wise addition modulo $2$. As an application of our reduction we are able to revisit tight lower bounds on the complexity of several fundamental problems parameterized by treewidth (Independent Set, Max Cut, $k$-Coloring). Our results imply that these bounds, which were initially shown under the SETH, also hold if one assumes the k-SUM or k-XOR Hypotheses, arguably increasing our confidence in their validity.

k-SUM Hardness Implies Treewidth-SETH

TL;DR

This work establishes a fundamental link between Sum-based hardness (k-SUM/k-XOR) and SAT hardness parameterized by treewidth (tw-SETH). It develops randomized reductions from k-SUM and k-XOR to SAT on formulas with treewidth near (k/2)·log n by combining interactive hashing with pathwidth reductions and then extends these ideas to derive tight lower bounds for classic DP problems parameterized by treewidth. The approach strengthens confidence in treewidth-based lower bounds by deriving them from Sum-based hypotheses and provides a new bridge between the Sum and SAT branches of fine-grained complexity. The results have practical implications for the optimality of DP algorithms for Independent Set, Max Cut, and k-Coloring under these alternative hypotheses, broadening the landscape of assumptions that underpin lower bounds in parameterized algorithms.

Abstract

We show that if k-SUM is hard, in the sense that the standard algorithm is essentially optimal, then a variant of the SETH called the Primal Treewidth SETH is true. Formally: if there is an and an algorithm which solves SAT in time , where is the width of a given tree decomposition of the primal graph of the input, then there exists a randomized algorithm which solves k-SUM in time for some and all sufficiently large . We also establish an analogous result for the k-XOR problem, where integer addition is replaced by component-wise addition modulo . As an application of our reduction we are able to revisit tight lower bounds on the complexity of several fundamental problems parameterized by treewidth (Independent Set, Max Cut, -Coloring). Our results imply that these bounds, which were initially shown under the SETH, also hold if one assumes the k-SUM or k-XOR Hypotheses, arguably increasing our confidence in their validity.

Paper Structure

This paper contains 22 sections, 21 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Executive Summary
  3. Background
  4. Our results
  5. Comparison with previous results
  6. Techniques and Proof Overview
  7. Open problems
  8. Other related work
  9. Preliminaries
  10. $k$-XOR implies $\textrm{tw}$-SETH
  11. Tools
  12. Subset (XOR)-Sum to Pathwidth SAT
  13. Linear Hash Functions
  14. The Reduction
  15. Proof of Correctness
  16. ...and 7 more sections

Key Result

Theorem 1

The following statements hold: More strongly, we have the following: if the $\textrm{tw}$-SETH is false, then there exists an $\varepsilon>0$ such that for sufficiently large $k$, both $k$-SUM and $k$-XOR admit randomized algorithms running in time $n^{(1-\varepsilon)\frac{k}{2}}$.

Figures (1)

  • Figure 1: Summary of the relations between some notable fine-grained hypotheses with the new implications of this paper marked in red. In particular, the $k$-SUM and $k$-XOR Hypotheses now imply tight lower bounds for several standard problems parameterized by treewidth. We give definitions of all hypotheses and references to all other relations in Section \ref{['sec:prelim']}, but mention briefly that SCC refers to the Set Cover Conjecture and LD-C-SETH refers to the SETH for $n$-input boolean circuits of size $s$ and depth $\varepsilon n$.

Theorems & Definitions (21)

  • Theorem 1: Informal
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • ...and 11 more