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Partial Minimum Branching Program Size Problem is ETH-hard

Ludmila Glinskih, Artur Riazanov

TL;DR

The paper establishes ETH-hardness of the Partial Minimum Branching Program Size Problem (MBPSP*) by reducing from the (n × n)-BPIS problem via once-appearance branching programs and a carefully constructed partial function γ_G. The core technique shows that any mbpsp* solution would yield a polynomial-time method to solve BPIS, contradicting ETH, hence MBPSP* requires $N^{\Omega(\log\log N)}$ time. The authors extend the framework to partial minimization across restricted BP classes (e.g., read-k BPs, OBDDs) and provide unconditional 1-NBP lower bounds for total minimization problems, while also proving NP-hardness of compressing BPs. Additionally, they derive corollaries for weaker models and outline future directions for strengthening lower bounds and extending to nondeterministic settings. Overall, the work advances understanding of ETH-based hardness in BP minimization and its implications for broader circuit/minimization problems such as MCSP.

Abstract

We show that assuming the Exponential Time Hypothesis, the Partial Minimum Branching Program Size Problem (MBPSP*) requires superpolynomial time. This result also applies to the partial minimization problems for many interesting subclasses of branching programs, such as read-k branching programs and OBDDs. Combining these results with the recent unconditional lower bounds for MCSP [Glinskih, Riazanov'22], we obtain an unconditional superpolynomial lower bound on the size of Read-Once Nondeterministic Branching Programs (1-NBP) computing the total versions of the minimum BP, read-k-BP, and OBDD size problems. Additionally we show that it is NP-hard to check whether a given BP computing a partial Boolean function can be compressed to a BP of a given size.

Partial Minimum Branching Program Size Problem is ETH-hard

TL;DR

The paper establishes ETH-hardness of the Partial Minimum Branching Program Size Problem (MBPSP*) by reducing from the (n × n)-BPIS problem via once-appearance branching programs and a carefully constructed partial function γ_G. The core technique shows that any mbpsp* solution would yield a polynomial-time method to solve BPIS, contradicting ETH, hence MBPSP* requires time. The authors extend the framework to partial minimization across restricted BP classes (e.g., read-k BPs, OBDDs) and provide unconditional 1-NBP lower bounds for total minimization problems, while also proving NP-hardness of compressing BPs. Additionally, they derive corollaries for weaker models and outline future directions for strengthening lower bounds and extending to nondeterministic settings. Overall, the work advances understanding of ETH-based hardness in BP minimization and its implications for broader circuit/minimization problems such as MCSP.

Abstract

We show that assuming the Exponential Time Hypothesis, the Partial Minimum Branching Program Size Problem (MBPSP*) requires superpolynomial time. This result also applies to the partial minimization problems for many interesting subclasses of branching programs, such as read-k branching programs and OBDDs. Combining these results with the recent unconditional lower bounds for MCSP [Glinskih, Riazanov'22], we obtain an unconditional superpolynomial lower bound on the size of Read-Once Nondeterministic Branching Programs (1-NBP) computing the total versions of the minimum BP, read-k-BP, and OBDD size problems. Additionally we show that it is NP-hard to check whether a given BP computing a partial Boolean function can be compressed to a BP of a given size.
Paper Structure (29 sections, 24 theorems, 2 equations, 1 figure)

This paper contains 29 sections, 24 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Techniques
  3. Corollaries
  4. Weaker Classes
  5. Unconditional Results for a Weaker Model
  6. NP-hardness of Compressing BPs
  7. Organization
  8. Preliminaries
  9. Once-Appearance Branching Programs
  10. Bipartite Permutation Independent Set Problem
  11. Proof of the Main Theorem
  12. Structure of oaBP Computing gamma prime
  13. Structure of gamma prime Conditioned on conditions
  14. Proof of Lemma order
  15. Proof of adjacent mapping
  16. ...and 14 more sections

Key Result

Theorem 1

Every deterministic algorithm computing $\hbox{\textmd{MBPSP}}^{*}$ on inputs of length $N$ requires time $N^{\Omega(\log\log(N))}$ assuming the Exponential Time Hypothesis.

Figures (1)

  • Figure 1: Fixed edges between $\{z_a, y_a, x_b, y_c, z_c\}$.

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Theorem 8: LMS18
  • ...and 38 more