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Wiring Switches to Light Bulbs

Stephen M. Buckley, Anthony G. O'Farrell

TL;DR

The paper studies the extremal behavior of a wiring model where pressing button i toggles bulb i and at most m others, formalized via M(W,c) and minima mu(n,m), mu^*(n,m), nu(n,m), nu^*(n,m) over admissible binary matrices W. It derives exact formulas for m=2 and a precise, partitioned description for m=3, employing graph-theoretic and linear-algebraic tools, including pivoting and decomposition into augmented complete subgraphs, to prove tight lower bounds on the number of bulbs that can be lit and to characterize extremal wirings. Connections to MAX-XOR-SAT and Hadamard-code ideas are highlighted, and asymptotic behavior for larger m is discussed, with notable sublinear growth phenomena and structural characterizations. The results provide explicit extremal values and constructive examples, advancing understanding of XOR-based switch networks and their combinatorial limits with potential implications for related SAT and coding problems.

Abstract

Given n buttons and n bulbs so that the ith button toggles the ith bulb and at most two other bulbs, we compute the sharp lower bound on the number of bulbs that can be lit regardless of the action of the buttons.

Wiring Switches to Light Bulbs

TL;DR

The paper studies the extremal behavior of a wiring model where pressing button i toggles bulb i and at most m others, formalized via M(W,c) and minima mu(n,m), mu^*(n,m), nu(n,m), nu^*(n,m) over admissible binary matrices W. It derives exact formulas for m=2 and a precise, partitioned description for m=3, employing graph-theoretic and linear-algebraic tools, including pivoting and decomposition into augmented complete subgraphs, to prove tight lower bounds on the number of bulbs that can be lit and to characterize extremal wirings. Connections to MAX-XOR-SAT and Hadamard-code ideas are highlighted, and asymptotic behavior for larger m is discussed, with notable sublinear growth phenomena and structural characterizations. The results provide explicit extremal values and constructive examples, advancing understanding of XOR-based switch networks and their combinatorial limits with potential implications for related SAT and coding problems.

Abstract

Given n buttons and n bulbs so that the ith button toggles the ith bulb and at most two other bulbs, we compute the sharp lower bound on the number of bulbs that can be lit regardless of the action of the buttons.

Paper Structure

This paper contains 22 sections, 7 theorems, 19 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Origins
  3. Notation
  4. Results
  5. Notation and terminology
  6. Graphs
  7. Edge function
  8. Matrix reformulation
  9. Degree
  10. Connection to coding
  11. Augmented complete graphs
  12. Relationship to Satisfiability
  13. Formulae for $\nu$ and $\nu^*$
  14. Trivial bounds
  15. A lower bound for $M(W,c)$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $n\in\mathbb{N}$.

Figures (14)

  • Figure 1: A Wiring
  • Figure 2: Graph for the wiring in Figure \ref{['Figure1']}
  • Figure 3: $n=9$
  • Figure 4: $n=9$, $m=3$
  • Figure 5: $n=11$, $m=4$
  • ...and 9 more figures

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 5.1
  • proof
  • ...and 4 more