Reset sequences for monotonic automata
SIAM Journal on Computing
On the Approximation of Shortest Common Supersequencesand Longest Common Subsequences
SIAM Journal on Computing
Synchronizing finite automata on Eulerian digraphs
Theoretical Computer Science - Mathematical foundations of computer science
Synchronizing generalized monotonic automata
Theoretical Computer Science - Insightful theory
Algorithmic construction of sets for k-restrictions
ACM Transactions on Algorithms (TALG)
An algorithmic approach to the automated design of parts orienters
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Synchronizing Automata and the Černý Conjecture
Language and Automata Theory and Applications
Approximating the Minimum Length of Synchronizing Words Is Hard
Theory of Computing Systems
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We consider the problem of finding minimum reset sequences in synchronizing automata. The well-known Černý conjecture states that every n-state synchronizing automaton has a reset sequence with length at most (n - 1)2. While this conjecture gives an upper bound on the length of every reset sequence, it does not directly address the problem of finding the shortest reset sequence. We call this the MINIMUM RESET SEQUENCE (MRS) problem. We give an O(kmnk + n4/k)-time ⌈n-1/k-1⌉-approximation for the MRS problem for any k ≥ 2. We also show that our analysis is tight. When k = 2 our algorithm reduces to Eppstein's algorithm and yields an (n-1)-approximation. When k = n our algorithm is the familiar exponential-time, exact algorithm. We define a nontrivial class of MRS which we call STACK COVER. We show that STACK COVER naturally generalizes two classic optimization problems: MIN SET COVER and SHORTEST COMMON SUPERSEQUENCE. Both these problems are known to be hard to approximate, although at present, SET COVER has a slightly stronger lower bound. In particular, it is NP-hard to approximate SET COVER to within a factor of c ċ log n for some c 0. Thus, the MINIMUM RESET SEQUENCE problem is as least as hard to approximate as SET COVER. This improves the previous best lower bound which showed that it was NP-hard to approximate the MRS on binary alphabets to within any constant factor. Our result requires an alphabet of arbitrary size.