المفاهيم الأساسية
Sequential local algorithms with certain local rules fail to solve the random k-XORSAT problem when the clause density exceeds the clustering threshold, even though solutions exist with high probability.
الملخص
The content discusses the limitations of sequential local algorithms in solving the random k-XORSAT problem, a random constraint satisfaction problem where each clause is a Boolean linear equation of k variables.
Key highlights:
- There exist two distinct thresholds rcore(k) < rsat(k) for the random k-XORSAT problem:
- For r < rsat(k), the random instance has solutions with high probability.
- For rcore(k) < r < rsat(k), the solution space shatters into an exponential number of clusters.
- The authors prove that for any r > rcore(k), sequential local algorithms with certain local rules fail to solve the random k-XORSAT problem with high probability. This includes:
- The algorithm using Unit Clause Propagation as the local rule for k ≥ 9.
- Algorithms using any local rule that can calculate the exact marginal probabilities of variables in instances with tree-like factor graphs, for k ≥ 13.
- The best known linear-time algorithm succeeds with high probability for r < rcore(k), suggesting that rcore(k) is the sharp threshold for the existence of a linear-time algorithm.
- The authors introduce a notion of "freeness" for sequential local algorithms and show that if an algorithm is strictly 2μ(k,r)-free, it fails to find a solution for the random k-XORSAT instance when the clause density is beyond the clustering threshold.
- The authors prove that the Unit Clause Propagation algorithm and algorithms using Belief Propagation or Survey Propagation as local rules satisfy the strict 2μ(k,r)-freeness condition, leading to their failure for certain values of k.
الإحصائيات
There are no key metrics or important figures used to support the author's key logics.
اقتباسات
There are no striking quotes supporting the author's key logics.