The Yolk

An interactive demonstration of the 2-dimensional case of Craig Tovey's polynomial-time algorithm for computing the yolk — the smallest ball that intersects every median hyperplane of a point set.

Live demo: https://glesav.github.io/Yolk_Demo/

Contents

• What is a yolk?
• Definitions
• Constraints and KKT conditions
• Complexity in R^2
• The demo
• State of the art
• References

What is a yolk?

Given a single category of qualitative information, finding the middle value is straightforward. If there are an odd number of items in the category, you can sort them and take the middle value (the median). If there are an even number, you can find the middle two values and pick some compromise between them (usually their average).

When we extend to multiple categories, finding the middle value within a (policy) space becomes difficult. When each category is represented in its own dimension, there does not always exist a single point that satisfies a "middleness" property across all categories simultaneously. The compromise is the yolk.

Given m categories of interrelated yet distinct values, a set of points can be plotted in the space R^m. The yolk identifies the middle-value region between the categories.

"The yolk [...] is a key solution concept in the Euclidean spatial model as the region of policies where a dynamic voting game will tend to reside [...] it is the smallest ball intersecting all median hyperplanes [of a set]." [1]

A hyperplane is a subspace one dimension lower than the original ("ambient") space; in R^3 it is a plane, and in R^2 it is a line. A hyperplane partitions the space into two halves.

A median hyperplane is one that partitions the space so that each closed half-space contains at least half of the points.

There are infinitely many median hyperplanes for a given set of points, and finding the yolk is NP-hard in an arbitrary number of dimensions. In [1], Tovey gives — and proves — a polynomial-time algorithm for the yolk in any fixed dimension. This page explores the 2-dimensional case in particular.

Definitions

• Yolk: the smallest ball intersecting all median hyperplanes.
• Hyperplane: a subspace one dimension lower than its ambient space, dividing that space into two half-spaces.
• Median split: any pair of subsets whose union is the complete set and each of whose cardinality is at least half of the complete set.
• Median hyperplane: a hyperplane that divides the points so that the two resulting subsets form a median split.
• Extremal median hyperplane: a median hyperplane that contains m points of the set in R^m.
• Binding points: points of the set that lie on a hyperplane.
• Affine combination [2]: a linear combination of distinct points whose coefficients sum to one.
• Affine set: a set containing every affine combination of the points within it.
• Affine hull: the smallest affine set containing all of a set of points.
• Determining hyperplane: a hyperplane containing the points of the set (at most m of them) that span Aff(B).

Constraints and Karush-Kuhn-Tucker conditions

A great deal of Tovey's paper concerns the constraints of the yolk problem. In the proof leading to his theorem he makes several key observations (and, in his own words, gets "lucky" along the way):

• Finding the yolk can be framed as a nonlinear optimization problem: minimize, over candidate centers, the distance to the furthest median hyperplane.
• This optimization can be reformulated to search through the exponentially large set of all median hyperplanes relative to the origin.
• Despite there being infinitely many median hyperplanes, they share identical sets of solutions "which [are] only polynomially large".
• This reduces the constraints to a single quadratic one, to which the KKT conditions can be applied.
• The Karush-Kuhn-Tucker conditions let inequality constraints be handled within an optimization problem.
• They reduce the search from an infinite set to a finite (though exponentially large) collection of polynomially sized sets of hyperplanes.
• They let the problem focus on the determining hyperplanes: the hyperplanes bounded by binding points and normal to a given vector from the origin. There are a finite (and polynomial) number of these.
• Together, these constraints yield Theorem 1 and Corollary 1.

Complexity in R^2

Theorem 1. The determining median hyperplanes suffice to determine the radius of the 0-centered yolk.

Corollary 1. The radius r(x) of the x-centered yolk can be determined in polynomial time O(n^(m+1)) for any fixed dimension m.

In the 2-dimensional case, finding the determining median hyperplanes can be done in O(n^3). There are C(n,1) + C(n,2) ways to choose a hyperplane bounded by either 1 or 2 points of the set — which is O(n^2) candidates — and checking whether a given hyperplane is median takes linear time, for a total of O(n^3).

Theorem 2. For any fixed m, the yolk of n nondegenerate points can be computed in polynomial time O(n^((m+1)^2)).

For R^2

1. Find the determining hyperplanes. Count the bounded median-hyperplane combinations; this is O(n^2).

2. Build the candidate set E. For each 3-tuple of determining hyperplanes, compute the points equidistant from the affine hulls of the three hyperplanes. The dimension is fixed, so each is found in constant time. Tovey calls this set E and bounds it by |E| = O(n^(m(m+1))), which is O(n^6) for m = 2.

3. Locate the yolk. The yolk center lies in E. For each point e in E, find the distance to the furthest median hyperplane; the e minimizing that distance is the yolk center, and that minimized maximum is the yolk radius — the smallest radius touching every median hyperplane. By Corollary 1 this costs O(|E| · n^(m+1)) = O(n^(m(m+1)) · n^(m+1)) = O(n^((m+1)^2)), i.e. O(n^9) for m = 2.

The demo

The demo walks through the construction one stage at a time and lets you build your own point sets.

Controls

• Click in the canvas to place a point; click a point again to remove it.
• Next / Back (or the ← / → arrow keys) step through the construction.
• Random drops a sample set; Reset (or R) clears everything; Clear removes the points but keeps you on the input step.
• The sidebar shows the live count of median hyperplanes and, on the final step, the yolk radius.

Running it locally

Just double-click index.html — it opens straight in your browser. No server, no build step, no install. (Or skip that entirely and use the live demo.)

Notes and simplifications

• The demo assumes an odd number of points in general position. The adjustments for even counts and degenerate configurations are given in [1]; the sidebar warns when an even count is placed.
• The yolk is computed from the B = 2 extremal median hyperplanes (those pinned to two points). For each 3-tuple of them, the triangle's incenter and three excenters give the candidate set E — the points equidistant from three median lines. The yolk center is the candidate that minimizes the distance to the furthest median line: a min-max, not a plain maximum. A convex sub-gradient refinement then snaps the center to the exact optimum, which also keeps the answer correct when the candidate enumeration is capped on large inputs.
• The B = 1 hyperplanes (pinned to a single point) are drawn as an illustrative fan of median lines but, as in the original write-up, are not folded into the final yolk computation.
• Performance. The construction is built from O(n^2) median-line work plus the capped candidate pass, and is recomputed only on demand rather than every frame. It stays smooth well past the handful of points that exhausted memory in the first version, handling dozens of points comfortably.

State of the art

Significant progress has been made on the yolk problem over the past 30 years. [4] shows that the 2-dimensional yolk can be solved in O(n log n) by exploiting duality and the zones induced by the set of hyperplanes.

References

[1] Craig Tovey, "A polynomial-time algorithm for computing the yolk in fixed dimension," Mathematical Programming 57, March 1992, pp. 259–277.

[2] Stephen Boyd, Lieven Vandenberghe, Convex Optimization, Cambridge University Press, 2009, pp. 21–55.

[3] Geoff Gordon, Ryan Tibshirani, "Karush-Kuhn-Tucker conditions," Optimization 10-725. Available: https://www.cs.cmu.edu/~ggordon/10725-F12/slides/16-kkt.pdf

[4] Timothy M. Chan, Sariel Har-Peled, Mitchell Jones, "Optimal Algorithms for Geometric Centers and Depth," May 2021. Available: https://arxiv.org/pdf/1912.01639.pdf