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Computing the EHZ capacity is NP-hard

Karla Leipold, Frank Vallentin

TL;DR

The paper proves that computing the EHZ capacity $c_{\mathrm{EHZ}}$ is NP-hard, even for simplices, by reducing the NP-complete feedback-arc set problem in complete bipartite tournaments to a decision version of the capacity. The core reduction uses a block matrix construction to form a simplex $P(\tilde{B},\mathbf{e})$ and shows that $c_{\mathrm{EHZ}}(P(\tilde{B},\mathbf{e}))$ encodes a permutation-based objective tied to a maximum acyclic subgraph of an auxiliary digraph derived from the graph. A key step interprets the EHZ-capacity optimization as solving a maximum acyclic subgraph problem via a skew-symmetric weight matrix $W$, with a constant $\Delta$ accounting for fixed edges, and an extra vertex to ensure Eulerian structure. The authors further relate the augmented graph back to the original FAS instance, demonstrating that the hardness carries to the decision problem for $c_{\mathrm{EHZ}}$ of simplices. This establishes a rigorous complexity barrier for computing symplectic capacities and clarifies the intractability landscape in symplectic geometry.

Abstract

The Ekeland-Hofer-Zehnder capacity (EHZ capacity) is a fundamental symplectic invariant of convex bodies. We show that computing the EHZ capacity of polytopes is NP-hard. For this we reduce the feedback arc set problem in bipartite tournaments to computing the EHZ capacity of simplices.

Computing the EHZ capacity is NP-hard

TL;DR

The paper proves that computing the EHZ capacity is NP-hard, even for simplices, by reducing the NP-complete feedback-arc set problem in complete bipartite tournaments to a decision version of the capacity. The core reduction uses a block matrix construction to form a simplex and shows that encodes a permutation-based objective tied to a maximum acyclic subgraph of an auxiliary digraph derived from the graph. A key step interprets the EHZ-capacity optimization as solving a maximum acyclic subgraph problem via a skew-symmetric weight matrix , with a constant accounting for fixed edges, and an extra vertex to ensure Eulerian structure. The authors further relate the augmented graph back to the original FAS instance, demonstrating that the hardness carries to the decision problem for of simplices. This establishes a rigorous complexity barrier for computing symplectic capacities and clarifies the intractability landscape in symplectic geometry.

Abstract

The Ekeland-Hofer-Zehnder capacity (EHZ capacity) is a fundamental symplectic invariant of convex bodies. We show that computing the EHZ capacity of polytopes is NP-hard. For this we reduce the feedback arc set problem in bipartite tournaments to computing the EHZ capacity of simplices.
Paper Structure (6 sections, 2 theorems, 29 equations)

This paper contains 6 sections, 2 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Notation from combinatorial optimization
  3. Outline of the polynomial time reduction; Proof of Theorem \ref{['thm:main']}
  4. From the feedback arc set problem to the EHZ capacity
  5. Interpretation of the EHZ capacity
  6. Removing the extra vertex

Key Result

Theorem 1.1

The following decision problem is $\mathop{\mathrm{NP}}\nolimits$-complete: Given the rational inputs $\gamma \in \mathbb{Q}$, a matrix $B \in \mathbb{Q}^{k \times (2n)}$, a vector $c \in \mathbb{Q}^k$. Suppose that $P(B,c)$ is a $2n$-dimensional convex polytope having $k$ facets. Is the EHZ capacit

Theorems & Definitions (7)

  • Theorem 1.1
  • Example 2.1
  • Example 3.1
  • Example 3.2
  • Lemma 4.1
  • proof
  • Example 4.2