A counterexample to the Foregger-Sinkhorn tie-point conjecture

Abstract: The Foregger-Sinkhorn tie-point conjecture asserts that if a nearly decomposable doubly stochastic matrix minimizes the permanent on a face and the permanental cofactor at a prescribed zero is larger than its permanent, then that zero is a tie point. We give a counterexample in dimension eight.

• The paper constructs an explicit order-eight counterexample in which a nearly decomposable support has a unique global permanent minimizer, yet the Foregger–Sinkhorn implication fails.
• The authors parametrize the complete face of the Birkhoff polytope and use a multiaffine reduction to prove global uniqueness, obtaining a minimizer with permanent approximately 0.01628248.
• For the zero position (1,5), the permanental cofactor exceeds the permanent by more than 0.024, but deleting an original edge leaves the augmented support fully indecomposable, proving that the position is not a tie point.

A counterexample to the Foregger–Sinkhorn tie-point conjecture

Problem setting

The paper “A counterexample to the Foregger–Sinkhorn tie-point conjecture” (2608.13025) refutes a longstanding implication concerning permanent minimization over faces of the Birkhoff polytope. Let $\Omega_n$ denote the polytope of $n \times n$ doubly stochastic matrices. For a prescribed set of zero positions $Z$, the face $\Omega_n(Z)$ consists of all doubly stochastic matrices vanishing on $Z$.

The conjecture concerns a nearly decomposable support. A support is fully indecomposable when its associated bipartite graph is connected, every support edge belongs to a perfect matching, and no permissible single-edge deletion produces a partly decomposable support. Given a zero position $g$ outside a nearly decomposable support $S$, the position is called a tie point if adding $g$ and then deleting any original support edge always yields a partly decomposable support.

The Foregger–Sinkhorn conjecture asserted that if a nearly decomposable matrix $A$ minimizes the permanent over the corresponding face and a prescribed zero $(i,j)$ satisfies

$n \times n$0

then $n \times n$1 must be a tie point. The paper constructs an explicit counterexample in dimension eight. The result is a direct separation between a strict permanental-cofactor inequality and the combinatorial condition imposed by tie-point structure.

The order-eight support

The counterexample is based on the zero-one support matrix

$n \times n$2

The support has 19 edges and exactly six perfect matchings, represented by the row-to-column strings

$n \times n$3

These matchings establish total support because their union contains every support edge. The paper also exhibits a spanning tree of the associated 16-vertex bipartite graph, proving connectedness. Consequently, $n \times n$4 is fully indecomposable.

To prove near decomposability, the authors verify that every support edge is essential. For each edge $n \times n$5, they identify another edge $n \times n$6 such that every perfect matching containing $n \times n$7 also contains $n \times n$8. Deleting $n \times n$9 therefore leaves $Z$0 outside every perfect matching, destroying total support. Hartfiel’s single-edge deletion criterion then implies that every such deletion produces a partly decomposable support. Thus $Z$1 is nearly decomposable.

The construction also lies outside the “complex” or “multiplex” family for which the tie-point implication had previously been verified. In the present support, four vertices have degree greater than two, rather than the two distinguished vertices characteristic of that special family. This distinction is important: the counterexample does not merely expose a gap in a proof for the known family, but demonstrates that the conjecture fails once its restrictive incidence pattern is removed.

Parametrization of the entire face

A central technical component is the explicit parametrization of every matrix in the closed face $Z$2. Each matrix has the form

$Z$3

where

$Z$4

The row and column constraints determine all remaining entries. The six free path parameters correspond to six length-three paths grouped into three bundles:

• $Z$5, joining $Z$6 to $Z$7;
• $Z$8, joining $Z$9 to $\Omega_n(Z)$0;
• $\Omega_n(Z)$1, joining $\Omega_n(Z)$2 to $\Omega_n(Z)$3.

Each path has edge weights $\Omega_n(Z)$4. The equality of the two endpoint weights is forced by the doubly stochastic constraints, and it permits the permanent to be expressed through bundle-level quantities.

For a bundle with path parameters $\Omega_n(Z)$5, define

$\Omega_n(Z)$6

The contribution when neither bundle hub is used is

$\Omega_n(Z)$7

whereas the contribution when both hubs are used is

$\Omega_n(Z)$8

The global graph contains one additional edge, $\Omega_n(Z)$9, and its cycle structure restricts perfect matchings to two global states. The permanent therefore becomes

$Z$0

where

$Z$1

This reduction is more than a convenient calculation. It makes the optimization over the complete face tractable, including all boundary strata. The authors explicitly avoid restricting attention to matrices with exactly the prescribed support.

Multiaffine reduction and global minimization

For fixed bundle sums, the products satisfy

$Z$2

After normalizing these intervals, the permanent is affine in each normalized product coordinate separately. Hence it is multiaffine in three variables and can be written as a convex combination of its eight corner values.

This observation is crucial. It does not assert that the permanent is globally convex; rather, it exploits coordinatewise affinity to reduce the optimization to eight explicitly analyzable configurations for every fixed $Z$3.

Seven corner values are shown to exceed $Z$4. The remaining corner, corresponding to equality of the two path parameters in every bundle, is

$Z$5

Its derivative factors as

$Z$6

On $Z$7, the first two factors are negative, so the sign of the derivative is determined by

$Z$8

The derivative of $Z$9 is strictly positive on the real line because its discriminant is negative and its leading coefficient is positive. Moreover,

$g$0

Therefore $g$1 has a unique root $g$2 in $g$3, and $g$4 decreases strictly up to $g$5 and increases strictly afterward. The unique global minimizer on the entire closed face is consequently obtained at

$g$6

where $g$7 is the unique root in the stated interval of

$g$8

The resulting matrix $g$9 has permanent

$S$0

The uniqueness claim is stronger than merely exhibiting a local stationary point or a minimizer in the relative interior. The multiaffine convex-combination argument proves global optimality over every boundary and interior stratum of $S$1, while the strict separation of the seven noncentral corners and the uniqueness of the root $S$2 establish uniqueness of the minimizer.

Failure of the tie-point implication

The prescribed zero is

$S$3

The paper computes the corresponding permanental cofactor and obtains

$S$4

The numerator is strictly decreasing on the isolating interval for $S$5, and its value at $S$6 is positive. Consequently,

$S$7

Thus the conjecture’s numerical hypothesis holds with a substantial strict margin.

The combinatorial conclusion nevertheless fails. After adjoining $S$8 and deleting the original support edge $S$9, the resulting support has five perfect matchings:

$g$0

Their union is the entire augmented-and-deleted support, and the paper gives a spanning tree of the associated bipartite graph. The support therefore remains connected and has total support, hence remains fully indecomposable. Since one original edge can be deleted without producing a partly decomposable support, $g$1 is not a tie point.

This establishes all components of the counterexample simultaneously: $g$2 is the unique global permanent minimizer on the relevant face, $g$3 is nearly decomposable, the cofactor exceeds the permanent, and the prescribed zero is not a tie point.

Theoretical implications

The result invalidates the Foregger–Sinkhorn conjecture in its general form and shows that the cofactor inequality does not encode sufficient information about the edge-critical structure of the support. The inequality compares two weighted matching aggregates: the permanent of the full matrix and the permanent of a deleted-row/deleted-column minor. Tie-point status, by contrast, is a universal structural condition over all original support edges. The counterexample demonstrates that these analytic and combinatorial properties can decouple even at a unique global minimizer.

The proof also illustrates the importance of analyzing the closed face rather than only the relative interior. Boundary points correspond to degenerations in which one path parameter vanishes or two parameters become unequal at fixed sum. Because the permanent is not generally convex on the Birkhoff polytope, a direct convexity argument would be unavailable. The multiaffine decomposition provides a more precise substitute for this particular support geometry.

The order eight should not be interpreted as a proven minimal dimension. The paper explicitly leaves open whether counterexamples exist in smaller dimensions. Determining the minimum order would require an exhaustive or structurally constrained classification of nearly decomposable supports, their face parametrizations, and the corresponding permanent minima.

Practical and future directions

For extremal permanent problems, the result cautions against inferring support-criticality from first-order or cofactor comparisons alone. Algorithms that search for permanent minimizers and then classify zeros through cofactor inequalities will require an independent combinatorial verification of tie-point properties. The support graph, its perfect-matching lattice, and the effect of single-edge deletions must be treated as separate data.

Several directions follow naturally. One is the systematic enumeration of nearly decomposable supports by order, with particular attention to graph families containing multiple high-degree hubs. Another is the development of symbolic or certified computational methods for minimizing permanents on low-dimensional faces. The present proof is especially amenable to such methods because it reduces the permanent to a multiaffine polynomial and isolates the decisive one-variable factor.

More broadly, the counterexample suggests that any valid replacement for the conjecture will need additional hypotheses. Possible restrictions could involve the structure of the support’s ear decomposition, the number and arrangement of high-degree vertices, uniqueness properties of perfect matchings, or stronger relations between the minimizing weights and the support’s edge-criticality. The verified complex/multiplex family indicates that graph-theoretic restrictions may recover a valid theorem even though the unrestricted implication is false.

Conclusion

The paper gives an explicit order-eight counterexample to the Foregger–Sinkhorn tie-point conjecture. It constructs a nearly decomposable support, proves that a specified matrix is the unique global permanent minimizer on the complete associated face, establishes a strict inequality between a permanental cofactor and the permanent, and independently proves that the corresponding zero is not a tie point. The result separates permanent-minimization data from tie-point structure and provides a concrete basis for revising the conjecture through additional combinatorial hypotheses.