On the family of 0/1-polytopes with NP-complete non-adjacency relation

In 1995 T. Matsui considered a special family 0/1-polytopes for which the problem of recognizing the non-adjacency of two arbitrary vertices is NP-complete. In 2012 the author of this paper established that all the polytopes of this family are present as faces in the polytopes associated with the following NP-complete problems: the traveling salesman problem, the 3-satisfiability problem, the knapsack problem, the set covering problem, the partial ordering problem, the cube subgraph problem, and some others. In particular, it follows that for these families the non-adjacency relation is also NP-complete. On the other hand, it is known that the vertex adjacency criterion is polynomial for polytopes of the following NP-complete problems: the maximum independent set problem, the set packing and the set partitioning problem, the three-index assignment problem. It is shown that none of the polytopes of the above-mentioned special family (with the exception of a one-dimensional segment) can be the face of polytopes associated with the problems of the maximum independent set, of a set packing and partitioning, and of 3-assignments.

💡 Research Summary

The paper investigates a particular family of 0/1‑polytopes for which the decision problem “are two given vertices non‑adjacent?” is NP‑complete. This family was originally introduced by T. Matsui in 1995 and is commonly referred to as the “double‑cover” polytopes. Formally, for a binary matrix B whose rows each contain exactly four ones, the polytope is defined as
 P₂cover(B) = { x ∈ {0,1}ⁿ | Bx = 2 }.
Matsui proved that recognizing non‑adjacency of two arbitrary vertices of any polytope in this family is NP‑complete.

The author’s earlier work (2012) showed that every polytope of the Matsui family appears as a face of the polytopes associated with a wide range of classic NP‑complete combinatorial optimization problems: the Traveling Salesman Problem (TSP), 3‑SAT, the Knapsack problem, Set Cover, Partial Order, Cube Subgraph, and several others. The embedding is achieved by an affine transformation that preserves the combinatorial structure while possibly increasing dimension and encoding length only polynomially. Consequently, the non‑adjacency problem is NP‑complete for those problem families as well, because a polynomial‑time reduction exists from Matsui’s non‑adjacency to the corresponding problem’s polytope.

In contrast, for some other NP‑complete problems the adjacency of vertices can be decided in polynomial time. These include the Maximum Independent Set (stable‑set) problem, Set Packing, Set Partitioning, and the three‑index Assignment problem. For these problems, simple linear constraints (e.g., xᵥ + xᵤ ≤ 1 for a stable‑set polytope) give a direct adjacency test.

The paper introduces a precise notion of affine reducibility between families of polytopes. Polytope P is affinely reducible to Q (P ≤ₐ Q) if for each P there exists a Q‑polytope and an affine map α such that α(P) is a (possibly improper) face of Q, α is bijective on vertices, the dimension of Q is bounded by a polynomial in the dimension of P, the encoding length of Q is polynomially bounded, and the coefficients of α are computable in polynomial time. Under this definition, if a family with NP‑complete non‑adjacency reduces to another family, the latter also inherits NP‑completeness of the non‑adjacency test.

Using this framework, the author proves two main theorems:

1. Inclusion Theorem – Every Matsui polytope is affinely reducible to the polytopes of TSP, 3‑SAT, Knapsack, Set Cover, Partial Order, Cube Subgraph, etc. Hence the non‑adjacency problem is NP‑complete for those families.

2. Exclusion Theorem – Except for the trivial one‑dimensional segment (a polytope consisting of just two opposite vertices), no Matsui polytope can be a face of any polytope from the stable‑set, set‑packing, set‑partitioning, or three‑index assignment families. The proof hinges on structural properties unique to Matsui polytopes: they always contain a distinguished pair of complementary vertices x₀ and ¯x₀ together with four disjoint “quadrant” faces F₁,…,F₄ defined by fixing three auxiliary variables (y₁, y₂, y₃) to specific 0/1 patterns. These quadrants are mutually exclusive and each is affinely equivalent to a set‑partitioning polytope. The presence of both the complementary pair and the quadrants forces certain convex hull intersections (e.g., conv(F₁∪F₄) ∩ conv({x₀,¯x₀}) ≠ ∅) that violate the simple edge‑inequalities (xᵥ + xᵤ ≤ 1) that characterize the stable‑set, packing, and partitioning polytopes. Consequently, any affine image of a Matsui polytope would necessarily introduce a forbidden inequality, proving that such an embedding is impossible.

The only exception is the one‑dimensional segment, where the polytope consists solely of the complementary pair {x₀,¯x₀}. In this degenerate case the non‑adjacency problem is trivial, and the polytope can appear as a face of the other families without contradiction.

Overall, the paper delineates a clear boundary between two classes of 0/1‑polytopes: those whose non‑adjacency problem is intrinsically hard (the Matsui family) and those whose adjacency problem is efficiently solvable. It shows that the hard‑non‑adjacency property is preserved under affine reductions, while the easy‑adjacency property prevents Matsui polytopes from embedding as faces (except for the trivial segment). This contributes to a finer understanding of how combinatorial structure influences the computational complexity of basic polyhedral queries, and it provides a systematic method for transferring NP‑completeness results across different optimization problems via affine face embeddings.