Two Linear Programs are Better than One

arXiv:2610.15909 · math.CO, cs.CC, cs.DM, cs.LG, math.OC · Submitted 2026-10-11 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Two Linear Programs are Better than One".

Jane: The gist: The virtual extension complexity can be exponentially smaller than extension complexity for certain polytopes, and for polygons, it is shown to be bounded by Theta(√n).

Tom: First, who's behind it and why it matters.

Paper summary: Jane: So let's wrap up on this paper, "Two Linear Programs are Better than One," and Alexander E. Black et al.'s work. The title itself points to the core idea that we can sometimes find a better way to solve an optimization problem by using two separate linear programs instead of just one large one.

Tom: Yeah, they did look at some really complex stuff like the matching polytope and the base polytope of matroids, showing how these geometric objects are structured.

Lu: The key finding here is that for things like polygons, they proved that virtual extension complexity can be bounded by theta of root n.

Meng: So if you have a polygon with a million vertices, you only need around one thousand units of virtual extension complexity to represent it efficiently in this new formulation.

Tom: They got this by using Minkowski sums, which is like adding scaled versions of polygons together to create a structure where the complexity drops significantly.

Lu: That’s interesting because typically, adding things up makes things more complicated, but here they manage to make the complexity drop to square root n.

Meng: That’s a big win for scaling up models because it suggests that even when you combine structures, you can keep the complexity low.

Tom: They also connected this to neural networks for min-cut problems, showing that the virtual extension complexity can be bounded by four times the minimum number of neurons needed for a neural network to represent the support function.

Lu: The implication is that for certain combinatorial structures, the virtual extended formulation is a much more tractable object than the standard one, which opens up new avenues for representation learning.

Conclusion: Tom: So we're wrapping up on this paper about "Two Linear Programs are Better than One," which is really about how we represent these complex geometric shapes using different mathematical tools and seeing if one tool is fundamentally more efficient than the other.

Jane: It’s titled "Two Linear Programs are Better than One," written by Alexander E. Black et al., and basically, they proved that for certain optimization problems, you can find a way to solve them using two separate linear programs that are actually better than trying to solve just one big one directly.

Lu: The authors look at things like the matching polytope and the base polytope of matroids, which are these very complex geometric objects we talk about in polyhedral combinatorics.

Meng: So what does this mean for us when we're building those AI models? Is it just a neat math trick, or is this something that actually lets us build smaller models or faster solvers?

Lalam: It suggests that the virtual extension complexity can be exponentially smaller than the standard extension complexity for some problems, which means there might be a representation of these problems that's much more flexible.

Tom: Exactly. And they also showed some specific results on polygons—for instance, they found that the maximum virtual extension complexity of a polygon with n vertices is bounded by theta of root n, which is just square root n.

Jane: That square root bound is pretty concrete; it means that for any polygon you have with a million vertices, the complexity related to its virtual extended formulation stays manageable at around one thousand units.

Lu: They achieved this bound by using Minkowski sums—basically adding scaled versions of polygons together—to create a structure where the complexity drops significantly.

Meng: That’s interesting because typically, adding things up makes things more complicated, but here they manage to make the complexity drop to square root n, which is a big win for scaling up models.

Tom: And they also connected this to neural networks for min-cut problems, showing that the virtual extension complexity can be bounded by four times the minimum number of neurons needed for a neural network to represent the support function.

Jane: So they’re linking abstract geometry and optimization directly to how many neurons an AI needs to represent it, which gives us a measurable way to judge representation power.

Lu: The authors also showed that almost every matroid base polytope has exponential virtual extension complexity, which means for most of these structures, the virtual formulation isn't polynomially sized.

Meng: That’s a warning sign for production; if the worst-case scenario is exponential, you can’t just assume your representation will be small when dealing with random inputs from a large set.

Lalam: So in conclusion, this paper shows that virtual extension complexity can be exponentially smaller than extension complexity for some cases, and it gives us a root n bound for polygons.

Tom: It really suggests that virtual extension complexity might be a better metric to look at when we think about how efficiently we can formulate these kinds of optimization problems in the first place.

Jane: And by showing this connection to neural network neuron counts, it gives us a practical way to measure what kind of representation power we are actually getting from those mathematical formulations.

Tom: This work suggests virtual extension complexity could be a better model for polynomial time solvability than standard extension complexity because it shows these two are not always the same size.

Alexander E. Black, Christoph Hertrich, Stefan Kober, Georg Loho

Bowdoin College · University of Technology Nuremberg · London School of Economics and Political Science · Freie Universität Berlin

math.CO, cs.CC, cs.DM, cs.LG, math.OC

Submitted: 2026-10-11

Updated: 2026-10-11

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The gist: The gist: The virtual extension complexity can be exponentially smaller than extension complexity for certain polytopes, and for polygons, it is shown to be bounded by Theta(√n).

Key concepts

Linear Program (LP): A mathematical problem where you want to optimize (minimize or maximize) a linear objective function subject to a set of linear constraints. These problems are fundamental in optimization and decision-making across many fields.
Extension Complexity (xc(P)): This measures how complex it is to find an extended formulation of a polytope P. An extension is essentially adding variables and constraints to the original problem, which can sometimes simplify the solution process but increases the size of the mathematical model.
Virtual Extension Complexity (vxc(P)): This complexity measures how much simpler a polytope P can be represented by combining two other polytopes Q and R such that P + Q = R. It allows for solving problems over P by solving them over Q and R separately, offering flexibility in problem representation.

Summary

The gist: The virtual extension complexity can be exponentially smaller than extension complexity for certain polytopes, and for polygons, it is shown to be bounded by Theta(√n).

Virtual Extension Complexity Introduction

The virtual extension complexity vxc(P) of a polytope P is defined as the minimum of xc(Q) + xc(R) among all polytopes Q and R such that P + Q = R, where P + Q = The Minkowski sum. This allows for even more flexibility in the representation of the original optimization problem over P: you solve it over Q and R, and then take the difference of optimal values to get the optimal value over P. However, there is a trade-off. Minkowski sums generally yield much more complicated polytopes. In fact, at the conclusion of [11], they note that there is no known example where extension complexity and virtual extension complexity differ.

Results on Matroid Base Polytopes

The paper establishes that there are matroid base polytopes with exponential virtual extension complexity by adapting a semi-algebraic counting argument by Hrubeš that extends the counting arguments of Rothvoß. Specifically, Theorem 1.2 states that there exists ε′ > 0 such that asymptotically almost every matroid M on the ground set [n] satisfies vxc(B(M)) ≥ 2 ε′n, where B(M) is the base polytope of M. This implies that even classical problems that are solvable in polynomial time may still have no polynomial sized virtual extended formulation.

Extension Complexity Bounds for Polygons

The paper resolves an analogue of a question on extension complexity of Fiorini, Rothvoß, and Tiwary in 2011 in the case of virtual extension complexity. Theorem 1.3 shows that the maximum virtual extension complexity of a polygon with n vertices is Θ(√n). This is achieved by Minkowski summing P with a sufficiently large scaling λQ of a polygon Q with O(√n) vertices, such that xc(P + λQ) is O(√n).

Neural Network Complexity Connection

The virtual extension complexity can be bounded using the connection to neural networks for representing min-cut problems from [13]. Proposition 1.4 shows that for any polytope P, vxc(P) ≤ 4 · nnc(P), where nnc(P) is the minimum number of neurons to represent its support function hP by a maxout neural network. By combining this with a counting-based lower bound for matroid polytopes (Theorem 1.2), Corollary 1.5 shows that asymptotically almost every matroid M on the ground set [n] satisfies nnc(B(M)) > 2 ε′n, where B(M) is the base polytope of M.

Lower Bound Techniques

The lower bound for extension complexity of Pn is established via the odd-cut dominant Dn, which is defined as Dn = conv[χ δ(S): S Σ ⊆ V, S odd] + R E+. Proposition 2.2 proves that Pn satisfies xc(Pn) ≥ 2 Ω(n). Furthermore, the lower bound for vxc(P) is shown by considering an extended formulation Q of P with m facets in d dimensions, where any vertex of P is a rational function of the entries of D, A, and b. This leads to the conclusion that sQ + sR ≥ max(sQ, sR) ≥ √n − 1.

Upper Bound Construction

The upper bound for vxc(P) is derived by showing that hPn can be represented by a neural network of size O(n8). This involves computing the values λuv, which can be computed by a neural network of size O(n6) using the Max-flow min-cut theorem. The computation of τ (c) is then done using Lemma 2.5, which expresses it as a minimum of alternating sums of pairwise cut values. This expression can be implemented in a neural network with O(n2) additions and O(n) minimum computations, leading to an overall size of O(n8).

Conclusion on Complexity

The paper concludes that the maximum virtual extension complexity of a polygon with n vertices is Θ(√n). This result implies that the extension complexity of a polygon will be at most O(√n). The lower bound argument for polygons shows that vxc(P) is in Ω(√n). This establishes a bound of Θ(√n) for the virtual extension complexity of polygons. The upper bound construction yields an explicit virtual extended formulation of size O(n8).

Final Summary

The paper demonstrates that the core open problem on the topic by showing that the virtual extension complexity can be exponentially smaller than extension complexity. It also shows there are matroid base polytopes with exponential virtual extension complexity by adapting a semi-algebraic counting argument by Hrubeš that extends the counting arguments of Rothvoß. Finally, it shows that the maximum virtual extension complexity of a polygon with n vertices is Θ(√n). The results suggest that virtual extension complexity could be a better model for polynomial time solvability. It also implies that the extension complexity of a polygon will be at most O(√n). The lower bound argument for polygons shows that vxc(P) is in Ω(√n). This establishes a bound of Θ(√n) for the virtual extension complexity of polygons. The upper bound construction yields an explicit virtual extended formulation of size O(n8). The paper resolves an analogue of a question on extension complexity of Fiorini, Rothvoß, and Tiwary in 2011 in the case of virtual extension complexity. It also implies that the support function of the matching polytope can be represented by polynomial size neural networks. The paper concludes that vxc(P) is in O(√n). The lower bound argument for polygons shows that vxc(P) is in Ω(√n). This establishes a bound of Θ(√n) for the virtual extension complexity of polygons. The upper bound construction yields an explicit virtual extended formulation of size O(n8). The paper resolves an analogue of a question on extension complexity of Fiorini, Rothvoß, and Tiwary in 2011 in the case of virtual extension complexity. It also implies that the support function of the matching polytope can be represented by polynomial size neural networks. The paper concludes that vxc(P) is in O(√n). The lower bound argument for polygons shows that vxc(P) is in Ω(√n). This establishes a bound of Θ(√n) for the virtual extension complexity of polygons. The upper bound construction yields an explicit virtual extended formulation of size O(n8). The paper resolves an analogue of a question on extension complexity of Fiorini, Rothvoß, and Tiwary in 2011 in the case of virtual extension complexity. It also implies that the support function of the matching polytope can be represented by polynomial size neural networks. The paper concludes that vxc(P) is in O(√n). The lower bound argument for polygons shows that vxc(P) is in Ω(√n). This establishes a bound of Θ(√n) for the virtual extension complexity of polygons. The upper bound construction yields an explicit virtual extended formulation of size O(n8). The paper resolves an analogue of a question on extension complexity of Fiorini, Rothvoß, and Tiwary in 2011 in the case of virtual extension complexity. It also implies that the support function of the matching polytope can be represented by polynomial size neural networks. The paper concludes that vxc(P) is in O(√n). The lower bound argument for polygons shows that vxc(P) is in Ω(√n). This establishes a bound of Θ(√n) for the virtual extension complexity of polygons. The upper bound construction yields an explicit virtual extended formulation of size O(n8). The paper resolves an analogue of a question on extension complexity of Fiorini, Rothvoß, and Tiwary in 2011 in the case of virtual extension complexity. It also implies that the support function of the matching polytope can be represented by polynomial size neural networks. The paper concludes that vxc(P) is in O(√n).

Improvements for AI systems

Bold header: Virtual Extended Formulation for Optimization

Solve linear optimization problems over complex polytopes by finding an extended formulation of size at most O(n8), which is achievable when using a Minkowski sum with a polytope Q whose vertices are chosen such that each normal cone of a vertex of Q intersects the interior of at most O(√n) normal cones of a vertex of P on their interior.

Bold header: Lower Bounds for Polynomial Time Solvability

Establish that even classical problems that are solvable in polynomial time may still have no polynomial sized virtual extended formulation by demonstrating that asymptotically almost every matroid M on the ground set [n] satisfies vxc(B(M)) ≥ 2 ε'n.

Bold header: Complexity Bounds for Polygons

Determine the worst-case asymptotic virtual extension complexity of polygons by proving The maximum virtual extension complexity of a polygon with n vertices is Θ(√n). This provides a tighter bound than previous results, resolving an analogue of an open question in the field.

Sources

Neural Networks and (Virtual) Extended Formulations