Title: The Parameterized Complexity of Computing the VC-Dimension
Venue: NeurIPS 2025
Accepted: yes
Keywords: VC-dimension

parameterized complexity

hypergraphs

graphs

treewidth
Abstract: The VC-dimension is a well-studied and fundamental complexity measure of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In particular, given a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$, we prove that the naive $2^{\mathcal{O}(|\mathcal{V}|)}$-time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a $1$-additive fixed-parameter approximation algorithm when parameterized by the maximum degree of $\mathcal{H}$ and a fixed-parameter algorithm when parameterized by its dimension, and that these are essentially the only such exploitable structural parameters. Lastly, we consider a generalization of the problem, formulated using graphs, which captures the VC-dimension of both set systems and graphs. We design a $2^{\mathcal{O}(\texttt{tw}\cdot \log \texttt{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\texttt{tw}$ (which, for a set system, applies to the treewidth of its incidence graph). This is in contrast with closely related problems that require a double-exponential dependency on the treewidth (assuming the ETH).


Peer reviews:
comments: SUMMARY:
This paper presents a systematic parameterized-complexity study of computing the VC-dimension of set systems (hypergraphs). Building on earlier results that established W[1]-hardness when parameterized by the solution size or the hypergraph degeneracy, the authors obtain several new findings:

- **Maximum degree.** It is 1-additive approximation FPT algorithm when parameterized by the maximum degree of the hypergraph.
- **Maximum hyperedge size.** It is FPT when parameterized by the largest hyperedge size.
- **Treewidth.** It is FPT when parameterized by the treewidth of the corresponding bipartite incidence graph.
- **Hypertree width / transversal number.** It is *not* FPT when parameterized by hypertree width or transversal number, unless LogNP=P

The paper also explores several variants of the VC-dimension problems.

STRENGTHS AND WEAKNESSES:
**Strengths:** By combining new algorithms with matching hardness for a wide range of structural parameters, this paper provides a comprehensive picture on the parameterized complexity of VC-dimension. The results are clearly presented and well organized.

**Weaknesses & Suggestions:** Several questions remain to be explored. For examples,  do the lower bounds still hold when one aims to approximate, rather than exact compute, the VC-dimension?  Clarifying this approximability landscape would further strengthen the contribution.

QUESTIONS:
See "weaknesses"

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper is a theoretical investigation of the complexity of computing the VC dimension. VC dimension is a key quantity with far-reaching implications so understanding the complexity is a fundamental task and with a long history.\n\nThe authors establish several results:\n\n-Under ETH, the simple algorithm that runs in 2^n time is optimal. This algorithm, given a hypergraph on V vertices, will simply check all subsets of V to see if it's shattered. The main technique here is a clever reduction from the 3-coloring problem and relying on known hardness for the latter.\n\n-Given this, the authors turn to fixed parameter approaches, and show that VC computation depends on the types of parameters used. For some natural choices of them, like maximum degree ∆ of hyperedges, they almost solve the problem by providing a additive-1 approximation. They also analyze the dimension D of the hypergraph, showing FPT algorithms, and also on the negative, other core structural hypergraph parameters (hypertree, width and transversal number) do not yield FPT algorithms.\n\n-Finally, the authors focus on treewidth parameterization and show positive results. The define a slight generalization called GEN-VC-DIMENSION and show it is FPT parameterized by the treewidth tw of G (Theorem 17), in particular they design a 2^O(tw·log tw) · |V |^O(1)-time algorithm. They also show a lower bound based on vertex cover number vc which is always at least as large as tw.\n\nSTRENGTHS AND WEAKNESSES:\nSTRENGTHS:\n-they authors solve fundamental problems in the area that should be of broad interest\n-deep set of results combining FPT knowledge and connecting it to VC dimension\n-clever reduction from 3-Coloring to VC dimension computation, this leads to Theorem 9. This is satisfying result given that the naive algorithm appears to be best under ETH in some sense\n-clever usage of treewidth in an algorithmic way to determine the VC dimension\n\nWEAKNESSES:\n-literature comparison at a technical level would be appreciated, see brief question below.\n\nQUESTIONS:\nminor question that would improve the presentation and highlight novelties in your reductions: compared to the previous works doing FPT algos for VC dimension, it would be nice to have a paragraph or two where you compare not just the results from previous works to yours, but also the techniques as well. For example, the 3-coloring reduction, were there known connections between the two problems?\n\nLIMITATIONS:\n-it is a theoretical work so yes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce the VC dimension for graphs and hypergraphs and show a lower bound on the complexity of computing the VC dimension for graphs conditioned on the exponential time hypothesis.\nThey show this via an exponential time reduction from the 3-Coloring problem.\nThey further show results in parameterized complexity, for the parameters maximum degree and treewidth.\nFor the treewidth result, they provide an explicit dynamic programming approach.\n\nSTRENGTHS AND WEAKNESSES:\n+: \nThe paper provides a very nice introduction to the subject, such that even those outside the field can follow. \nThe proofs are written clearly, in a step-by-step manner, that is reasonably easy to follow, even considering their at times quite technical nature.\n\n-:\nWhile much of the paper is very clearly communicated, the reasoning and exact nature for the kind of exponential time reduction they use is not explicitly provided.\nI believe that it would be very helpful to the reader to devote a short paragraph to explaining, why exponential time is needed, what kind of exponential time you are considering and why such a reduction gives you the result in the end.\nFurther, a longer conclusion, considering possible impact and tying your work to other results in the area could benefit the paper, although I do realize that space is limited.\n\nminor comments:\nl. 126: Maybe don't use T for the tree, since you used T as a symbol for a subset of V in the sentence before. Using another symbol for either of the Ts could help avoid confusion.\nl. 199: it should be \"i \\in [k]\" instead of \"i \\in [j]\", right?\nl. 294: \"thesesubsets\" -> \"these subsets\"\nl. 300f: I don't understand the sentence inside the parentheses.\nl. 369: \"when it exists\" -> \"if it exists\"\n\nQUESTIONS:\nCould you please elaborate further on the exponential time reduction? Why can't we just solve 3Col in the reduction? Please elaborate why the use of an exponential time reduction is not problematic.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of computing the VC dimension of a set system. Let us recall the set up. A set system can be represented as a hypergraph or the corresponding bipartite incidence graph. A subset S of the universe is called \"shattered\" if, for every subset T of S, there is a set in the system whose intersection with S is exactly T. So, in some sense, every subset of S \"appears\" if we restrict the universe (of the set system) to be S. \n\nThe VC dimension is the size of largest shattered set. The computational problem is to compute the VC dimension given an explicit representation of the set system. Note that the largest possible shattered set has size log H, where H is the number of sets in the system. So that automatically gives H^(log H) time algorithms. This is essentially equivalent to also trying out every subset of the system and checking it is shattered.\n\nThere are a series of results:\n \nResult 1. Assuming the Exponential Time Hypothesis, all exact algorithms require 2^{(\\Omega(|V|)} time, where V is the universe.\nResult 2. There is a Delta^D poly(|H|) time algorithm that gives an additive 1-approximation to the VC dimension. Delta is the max degree of the incidence graph.\nResult 3. There is a (tw)^(tw) poly(H) algorithm, where tw is the treewidth of the incidence graph.\n\nResult 1 comes from a reduction from 3-coloring. It goes via a graph VC dimension problem, where the set system is given by the neighborhoods.\nResult 2 comes from ordering ideas (reminiscent of graph degeneracy).\nResult 3 is the most technical, and essentially constructs a dynamic programming solution using the tree decomposition (the typical paradigm).\n\nSTRENGTHS AND WEAKNESSES:\nThe VC dimension is a fundamental quantity of interest, and it's nice to have a suite of results. The first result on the exponential hardness sets the stage for parameterized algorithms. And both the parameters (degree and treewidth) are natural enough in this setting.\n\nMy primary complaints about the paper are as follows:\n - (Major) Is this the right problem/setting to study? In most VC dimension settings, the number of sets is extremely large and implicitly defined (think halfspaces). So there is often a circuit that describes the set systems. The algorithms described do not (or do they?) port to that setting. It is very rare that one has an explicit set system and wishes to compute the VC dimension.\n - (Minor) It's incorrect to say that the 2^O(|V|) algorithm has been shown to be optimal, assuming ETH. The hardness does not rule out exponential time algorithms with a smaller exponent, which would certainly be useful. The hardness shows that it is unlikely, we can get subexponential dependence.\n\nQUESTIONS:\nMajor:\n\n1. Are there any use cases/applications where one wants to compute the VC dimension of an explicit set system? I think the paper would be stronger with a more compelling case for this problem. I consider NeurIPS a non (purely) theoretical venue. So a purely theoretical justification (as a TCS paper) doesn't seem enough.\n2. Can one improve the Delta^D poly(|H|) to get a dependence on the degeneracy? Certainly seems possible.\n\nMinor:\n\n1. I contest the point that this paper shows that the trivial 2^{(O(|V|)} algorithm is optimal. I suggest that be reworded.\n2. Prop 1 \"Unless the ETH fails, there exists an..\" has double negatives and is hard to parse. I suggest: \"Assume the ETH. Then there exists a constant \\eps such that all algorithms have running time \\Omega(...).\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}