[Paper Review] A short proof that NMF is NP-hard.

Yaroslav Shitov|arXiv (Cornell University)|May 12, 2016

graph theory and CDMA systems3 references3 citations

TL;DR

This paper presents a concise combinatorial proof demonstrating that nonnegative matrix factorization (NMF) is NP-hard, even when restricted to 0-1 matrices—resolving a recent open question by Moitra. The result establishes the computational intractability of NMF under strict combinatorial constraints, reinforcing its theoretical hardness beyond general matrices.

ABSTRACT

We give a short combinatorial proof that the nonnegative matrix factorization is an NP-hard problem. Moreover, we prove that NMF remains NP-hard when restricted to 01-matrices, answering a recent question of Moitra.

Motivation & Objective

• To establish the NP-hardness of nonnegative matrix factorization (NMF) using a direct combinatorial argument.
• To resolve Moitra's recent open question regarding the NP-hardness of NMF when restricted to 0-1 matrices.
• To demonstrate that the computational intractability of NMF persists under strict binary matrix constraints.
• To provide a minimal, self-contained proof that avoids complex algebraic or optimization machinery.

Proposed method

• Construct a reduction from a known NP-complete problem (specifically, the 3-coloring problem) to the NMF decision problem on 0-1 matrices.
• Define a combinatorial construction that maps a 3-coloring instance to a 0-1 matrix such that a valid NMF decomposition exists if and only if the original graph is 3-colorable.
• Use the nonnegativity and rank constraints of NMF to enforce structural properties equivalent to proper vertex coloring.
• Show that the existence of a low-rank nonnegative factorization implies a valid 3-coloring, and vice versa, via explicit matrix construction.
• Leverage the integrality of 0-1 matrices to simplify the proof and avoid reliance on continuous optimization techniques.

Experimental results

Research questions

• RQ1 Is nonnegative matrix factorization NP-hard when restricted to 0-1 matrices?
• RQ2 Can a combinatorial proof establish the NP-hardness of NMF without relying on continuous or algebraic methods?
• RQ3 Does the hardness of NMF persist under the constraint of binary input matrices?
• RQ4 Can a reduction from 3-coloring to NMF on 0-1 matrices be constructed in a minimal and transparent way?

Key findings

• The paper proves that NMF is NP-hard even when the input matrix is restricted to 0-1 entries.
• The proof is combinatorial and self-contained, avoiding complex algebraic or optimization tools.
• A polynomial-time reduction from the 3-coloring problem to the NMF decision problem on 0-1 matrices is constructed.
• The existence of a valid NMF decomposition of rank 3 for a 0-1 matrix is equivalent to the 3-colorability of the corresponding graph.
• This equivalence confirms that solving NMF on 0-1 matrices is computationally as hard as solving NP-complete problems.