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Pizza Sharing is PPA-hard

Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos

Abstract

We study the computational complexity of finding a solution for the straight-cut and square-cut pizza sharing problems. We show that computing an $\varepsilon$-approximate solution is PPA-complete for both problems, while finding an exact solution for the square-cut problem is FIXP-hard. Our PPA-hardness results apply for any $\varepsilon < 1/5$, even when all mass distributions consist of non-overlapping axis-aligned rectangles or when they are point sets, and our FIXP-hardness result applies even when all mass distributions are unions of squares and right-angled triangles. We also prove that the decision variants of both approximate problems are NP-complete, while the decision variant for the exact version of square-cut pizza sharing is $\exists\mathbb{R}$-complete.

Pizza Sharing is PPA-hard

Abstract

We study the computational complexity of finding a solution for the straight-cut and square-cut pizza sharing problems. We show that computing an -approximate solution is PPA-complete for both problems, while finding an exact solution for the square-cut problem is FIXP-hard. Our PPA-hardness results apply for any , even when all mass distributions consist of non-overlapping axis-aligned rectangles or when they are point sets, and our FIXP-hardness result applies even when all mass distributions are unions of squares and right-angled triangles. We also prove that the decision variants of both approximate problems are NP-complete, while the decision variant for the exact version of square-cut pizza sharing is -complete.

Paper Structure

This paper contains 19 sections, 27 theorems, 22 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hardness results
  4. Hardness of approximate Straight-Pizza-Sharing
  5. The reduction.
  6. Hardness of approximate Square-Pizza-Sharing
  7. The reduction.
  8. Hardness of discrete Straight-Pizza-Sharing and Square-Pizza-Sharing
  9. Hardness of exact Square-Pizza-Sharing
  10. Membership results
  11. Membership results for approximate Straight-Pizza-Sharing
  12. Membership results for approximate Square-Pizza-Sharing
  13. Membership results for discrete Square-Pizza-Sharing
  14. $\exists\mathbb{R}$ membership for the decision variant of exact Square-Pizza-Sharing
  15. Conclusions
  16. ...and 4 more sections

Key Result

Theorem 1

Let $f : S^d \to \mathbb{R}\xspace^d$ be a continuous function, where $S^d$ is a $d$-dimensional sphere. Then, there exists an $x \in S^d$ such that $f(x)=f(-x)$.

Figures (9)

  • Figure 1: An example with 4 masses and various partitions of the plane into two regions, namely the shaded and non-shaded one. In a solution, each region contains half the area of each mass.
  • Figure 2: Some examples of valuation functions.
  • Figure 3: Placing the big-tiles on the $y=x^2$ curve. The $j$-th, $(j+1)$-st and $(j+2)$-nd big-tiles are centered on the curve. Their size is small enough to prevent any straight line (red/dashed) from intersecting more than two big-tiles.
  • Figure 4: The construction for a part of an instance with four mass distributions.
  • Figure 5: An example of the reduction from Consensus-Halving to Square-Pizza-Sharing.
  • ...and 4 more figures

Theorems & Definitions (52)

  • Theorem 1: Borsuk-Ulam
  • Definition 2
  • Definition 3
  • Claim 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • ...and 42 more