Approximating the Minimum Length of Synchronizing Words Is Hard

Berlinkov, Mikhail V.

doi:10.1007/s00224-013-9511-y

Cited by 25 publications

(32 citation statements)

References 15 publications

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“…We mention also the observation which, as far as we know, was first made in [5]: if there exists an upper bound of the form O(n 2 ) for the reset threshold of synchronizing n-automata with two input letters, then a bound of the same magnitude (but probably with a worse constant) exists also for the reset threshold of synchronizing n-automata with any fixed size of the input alphabet.…”

Section: The Role Of the Alphabet Sizementioning

confidence: 85%

Primitive digraphs with large exponents and slowly synchronizing automata

2013

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Section: The Role Of the Alphabet Sizementioning

confidence: 85%

Primitive digraphs with large exponents and slowly synchronizing automata

2013

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“…Recently, it has been shown that the problem of finding the length of the shortest reset word (the reset length, in short) is FP NP[log] -complete, and the related decision problem is both NP-and coNP-hard (Olschewski and Ummels 2010) [cf. also (Berlinkov 2010) and (Martyugin 2009(Martyugin , 2011]. On the other hand, there are several theoretical and experimental results showing that most automata are synchronizing (Berlinkov 2013) and most of them have relatively short reset words (Ananichev et al 2010;Skvortsov and Tipikin 2011).…”

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confidence: 99%

Computing the shortest reset words of synchronizing automata

2013

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In this paper we give the details of our new algorithm for finding minimal reset words of finite synchronizing automata. The problem is known to be computationally hard, so our algorithm is exponential in the worst case, but it is faster than the algorithms used so far and it performs well on average. The main idea is to use a bidirectional breadth-first-search and radix (Patricia) tries to store and compare subsets. A good performance is due to a number of heuristics we apply and describe here in a suitable detail. We give both theoretical and practical arguments showing that the effective branching factor is considerably reduced. As a practical test we perform an experimental study of the length of the shortest reset word for random automata with up to n = 350 states and up to k = 10 input letters. In particular, we obtain a new estimation of the expected length of the shortest reset word ≈ 2.5 √ n − 5 for binary automata and show that the error of this estimate is sufficiently small. Experiments for automata with more than two input letters show certain trends with the same general pattern.

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“…Berlinkov showed that finding an O(1)-approximation is NP-hard by giving a combinatorial reduction from SAT [6]. Later, Gerbush and Heeringa [11] used the log n−approximation hardness of SetCover [9] to prove that O(log n)-approximation of the shortest reset word is NP-hard.…”

Section: Previous Work and Our Resultsmentioning

confidence: 99%

“…The construction is not the simplest possible, nor the most efficient in the number of states of the resulting automaton, but it provides good intuitions for the further proofs. For a simpler construction in this spirit see [6].…”

Section: Simple Hardness Resultsmentioning

confidence: 99%

Strong Inapproximability of the Shortest Reset Word

Gawrychowski

Straszak

2015

Mathematical Foundations of Computer Science 2015

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TheČerný conjecture states that every n-state synchronizing automaton has a reset word of length at most (n−1) 2 . We study the hardness of finding short reset words. It is known that the exact version of the problem, i.e., finding the shortest reset word, is NP-hard and coNP-hard, and complete for the DP class, and that approximating the length of the shortest reset word within a factor of O(log n) is NP-hard [Gerbush and Heeringa, CIAA'10], even for the binary alphabet [Berlinkov, DLT'13]. We significantly improve on these results by showing that, for every ε > 0, it is NP-hard to approximate the length of the shortest reset word within a factor of n 1−ε . This is essentially tight since a simple O(n)-approximation algorithm exists.

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Approximating the Minimum Length of Synchronizing Words Is Hard

Cited by 25 publications

References 15 publications

Primitive digraphs with large exponents and slowly synchronizing automata

Primitive digraphs with large exponents and slowly synchronizing automata

Computing the shortest reset words of synchronizing automata

Strong Inapproximability of the Shortest Reset Word

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