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Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension

Among classes of tournaments for which neither hardness nor polynomial-time solvability of isomorphism was known, bounded VC dimension stood out as an open problem of Neuen and Grohe. Resolved: isomorphism of tournaments of VC dimension $d$ is decidable in time $n^{O(d \log d)}$, so automorphism groups of bounded-VC tournaments are computable in polynomial time; isomorphism of tournaments of bounded chromatic number is also polynomial-time decidable.

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theoretical-computer-scienceAug 14, 2026Significance 12/100Registry: unreviewed

Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension

Prior state unknown→proved

Among classes of tournaments for which neither hardness nor polynomial-time solvability of isomorphism was known, bounded VC dimension stood out as an open problem of Neuen and Grohe. Resolved: isomorphism of tournaments of VC dimension $d$ is decidable in time $n^{O(d \log d)}$, so automorphism groups of bounded-VC tournaments are computable in polynomial time; isomorphism of tournaments of bounded chromatic numb…

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Among classes of tournaments for which neither hardness nor polynomial-time solvability of isomorphism was known, bounded VC dimension stood out as an open problem of Neuen and Grohe. Resolved: isomorphism of tournaments of VC dimension $d$ is decidable in time $n^{O(d \log d)}$, so automorphism groups of bounded-VC tournaments are computable in polynomial time; isomorphism of tournaments of bounded chromatic number is also polynomial-time decidable.

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