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Does Subset Sum Admit Short Proofs?

Michał Włodarczyk

TL;DR

It is shown that Subset Sum in permutation groups is at least as hard for nondeterministic computation as 3Coloring in bounded-pathwidth graphs.

Abstract

We investigate the question whether Subset Sum can be solved by a polynomial-time algorithm with access to a certificate of length poly(k) where k is the maximal number of bits in an input number. In other words, can it be solved using only few nondeterministic bits? This question has motivated us to initiate a systematic study of certification complexity of parameterized problems. Apart from Subset Sum, we examine problems related to integer linear programming, scheduling, and group theory. We reveal an equivalence class of problems sharing the same hardness with respect to having a polynomial certificate. These include Subset Sum and Boolean Linear Programming parameterized by the number of constraints. Secondly, we present new techniques for establishing lower bounds in this regime. In particular, we show that Subset Sum in permutation groups is at least as hard for nondeterministic computation as 3Coloring in bounded-pathwidth graphs.

Does Subset Sum Admit Short Proofs?

TL;DR

It is shown that Subset Sum in permutation groups is at least as hard for nondeterministic computation as 3Coloring in bounded-pathwidth graphs.

Abstract

We investigate the question whether Subset Sum can be solved by a polynomial-time algorithm with access to a certificate of length poly(k) where k is the maximal number of bits in an input number. In other words, can it be solved using only few nondeterministic bits? This question has motivated us to initiate a systematic study of certification complexity of parameterized problems. Apart from Subset Sum, we examine problems related to integer linear programming, scheduling, and group theory. We reveal an equivalence class of problems sharing the same hardness with respect to having a polynomial certificate. These include Subset Sum and Boolean Linear Programming parameterized by the number of constraints. Secondly, we present new techniques for establishing lower bounds in this regime. In particular, we show that Subset Sum in permutation groups is at least as hard for nondeterministic computation as 3Coloring in bounded-pathwidth graphs.
Paper Structure (18 sections, 31 theorems, 1 equation, 2 figures)

This paper contains 18 sections, 31 theorems, 1 equation, 2 figures.

Table of Contents

  1. Introduction
  2. When is certification easy or hard?
  3. Constructive vs. non-constructive proofs.
  4. The problems under consideration
  5. Our focus: Subset Sum.
  6. Integer Linear Programming.
  7. Other problems parameterized by the number of relevant bits.
  8. Our contribution
  9. Organization of the paper.
  10. Preliminaries
  11. Pathwidth.
  12. Group theory.
  13. Equivalences
  14. Permutation Subset Sum
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem 3

The following parameterized problems are equivalent with respect to NPPT:

Figures (2)

  • Figure 1: The matrix $A'$ in \ref{['lem:ss:zeroilp']}.
  • Figure 2: An illustration to \ref{['lem:perm:homo']}. The three permutations are $\pi_1, \pi_0, \pi_z \in S_{10}$. The first one acts as $g$ on the upper set and as identity on the lower set. In the second one these roles are swapped whereas $\pi_z$ acts as symmetry between the two sets. The permutations $\widehat{\Gamma}(1), \widehat{\Gamma}(-1)$ in \ref{['lem:perm:run-group-perm']} are obtained as $\pi_0 \circ \pi_z$ and $\pi_1 \circ \pi_z$. Multiplying a sequence of permutations from $\{\widehat{\Gamma}(1), \widehat{\Gamma}(-1)\}$ yields a permutation acting as identity on the upper set if and only if the arguments are alternating 1s and -1s.

Theorems & Definitions (34)

  • Conjecture 1
  • Theorem 3
  • Theorem 4: $\bigstar$
  • Theorem 5
  • Lemma 5: $\bigstar$
  • Definition 6
  • Lemma 7
  • Definition 8
  • Lemma 9
  • Lemma 10
  • ...and 24 more