4-to-1 Games Conjecture Gets a Claimed Perfect-Completeness Proof
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A newly posted manuscript claims to settle the longstanding 4-to-1 Games Conjecture at perfect completeness, where earlier work cited in the manuscript covered a related 2-to-1 problem with almost-perfect completeness.
A claimed hardness result with perfect completeness
The manuscript states that, for every ε greater than zero, there is a bounded alphabet size for which it is NP-hard to distinguish a 4-to-1 game that can be satisfied perfectly from one whose value is at most δ. It says this confirms the 4-to-1 Games Conjecture.
The authors also list implications for the hardness of coloring 3-colorable graphs and finding large independent sets in 2-colorable 3-uniform hypergraphs. NP-hardness is a complexity-theory classification for problems believed not to have efficient general solutions.
According to the manuscript, the strongest earlier result addressed an almost-perfect-completeness version for the stricter 2-to-1 Games problem.
A manuscript now claims a proof of the 4-to-1 Games Conjecture with perfect completeness, extending beyond the earlier result described by its authors.
The manuscript’s claimed perfect completeness means that intended positive instances can be satisfied with certainty. If the proof holds, it affects the longstanding conjecture and hardness consequences for graph coloring and hypergraph independent sets.
Sources
- Hacker NewsThe Hardness of 4-to-1 Games with Perfect CompletenessRead source →