Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents

Michaela Borzechowski, John Fearnley, Spencer Gordon, Rahul Savani, Patrick Schnider, Simon Weber

TL;DR

We address the equivalence of three promise search problems—$P$-LCP, USOs, and a restricted $\alpha$-Ham Sandwich variant—by showing that two binary choices suffice via reductions that pass through a new intermediate problem, $P$-Lin-Bellman. The key methodological advance is the Lin-Bellman framework, a Bellman-like system that captures max/min over two affine expressions, with a promise guaranteeing a unique solution; this connects to discounted stochastic games and enables polynomial-time reductions between $P$-LCP and $P$-Lin-Bellman, and between $P$-GLCP and $P$-Lin-Bellman. Consequently, $P$-GLCP is equivalent to $P$-LCP and grid USOs are equivalent to cube USOs; the Colorful Tangent formulation of the $α$-Ham Sandwich problem is also reducible to and from $P$-LCP through the intermediate problem. The work places these problems in the promise UEOPL setting and highlights a unifying toolkit for translating non-binary instances into binary ones across algebraic, combinatorial, and geometric domains. Overall, the paper offers a novel intermediate formulation that clarifies the relationships among these core problems and paves the way for identifying natural UEOPL-complete problems.

Abstract

We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the $α$-Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our $α$-Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs.

Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents

TL;DR

We address the equivalence of three promise search problems—-LCP, USOs, and a restricted -Ham Sandwich variant—by showing that two binary choices suffice via reductions that pass through a new intermediate problem, -Lin-Bellman. The key methodological advance is the Lin-Bellman framework, a Bellman-like system that captures max/min over two affine expressions, with a promise guaranteeing a unique solution; this connects to discounted stochastic games and enables polynomial-time reductions between -LCP and -Lin-Bellman, and between -GLCP and -Lin-Bellman. Consequently, -GLCP is equivalent to -LCP and grid USOs are equivalent to cube USOs; the Colorful Tangent formulation of the -Ham Sandwich problem is also reducible to and from -LCP through the intermediate problem. The work places these problems in the promise UEOPL setting and highlights a unifying toolkit for translating non-binary instances into binary ones across algebraic, combinatorial, and geometric domains. Overall, the paper offers a novel intermediate formulation that clarifies the relationships among these core problems and paves the way for identifying natural UEOPL-complete problems.

Abstract

We provide polynomial-time reductions between three search problems from three distinct areas: the P-matrix linear complementarity problem (P-LCP), finding the sink of a unique sink orientation (USO), and a variant of the -Ham Sandwich problem. For all three settings, we show that "two choices are enough", meaning that the general non-binary version of the problem can be reduced in polynomial time to the binary version. This specifically means that generalized P-LCPs are equivalent to P-LCPs, and grid USOs are equivalent to cube USOs. These results are obtained by showing that both the P-LCP and our -Ham Sandwich variant are equivalent to a new problem we introduce, P-Lin-Bellman. This problem can be seen as a new tool for formulating problems as P-LCPs.
Paper Structure (2 sections, 2 equations, 2 figures)

This paper contains 2 sections, 2 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A New Intermediate Problem: P-Lin-Bellman

Figures (2)

  • Figure 1: Red: Reductions we show in this paper. Black: Existing reductions and trivial inclusions.
  • Figure 2: A 3-dimensional USO. The marked vertex denotes its unique global sink.

Theorems & Definitions (2)

  • Definition 1
  • Definition 2