Maximum-Likelihood Decoding of Reed–Solomon Codes is NP-Hard

V. Guruswami, A. Vardy
2005 IEEE Transactions on Information Theory  
Maximum-likelihood decoding is one of the central algorithmic problems in coding theory. It has been known for over 25 years that maximum-likelihood decoding of general linear codes is NP-hard. Nevertheless, it was so far unknown whether maximum-likelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure. In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes. We moreover show that maximum-likelihood
more » ... ding of Reed-Solomon codes remains hard even with unlimited preprocessing, thereby strengthening a result of Bruck and Naor. Instance: An m ¢ n matrix H over £ q , a target vector s ¤ ¥ £ m q , and an integer w ¦ 0. Question: Is there a vector v ¤ § £ n q of weight¨w, such that Hv t © s t ? Berlekamp, McEliece, and van Tilborg [4] proved in 1978 that this problem is NP-complete using a reduction from THREE-DIMENSIONAL MATCHING, a well-known NP-complete problem [9, p. 50]. Since 1978, the complexity of maximum-likelihood decoding of general linear codes has been extensively studied. Bruck and Naor [5] and Lobstein [16] showed in 1990 that the problem remains hard even if the code is known in advance, and can be preprocessed for as long as desired in order to devise a decoding algorithm. Arora, Babai, Stern, and Sweedyk [1] proved that MLD-Linear is NP-hard to approximate within any constant factor. Downey, Fellows, Vardy, and Whittle [7] proved that MLD-Linear remains hard even if the parameter w is a constant -it is not fixed-parameter tractable unless FPT © W 1 . Recently, the complexity of approximating MLD-Linear with unlimited preprocessing was studied by Feige and Micciancio [8] and by Regev [19] -this work strengthens the results of both [5, 16] and [1] by showing that MLD-Linear is NP-hard to approximate within a factor of 3 ε for any ¦ 0, even if unlimited preprocessing is allowed. While the papers surveyed in the foregoing paragraph constitute a significant body of work, all these papers deal with the general class of linear codes. This leads to a somewhat incongruous situation. On one hand, there is no nontrivial useful family of codes for which a polynomial-time maximum-likelihood decoding algorithm is known (such a result would, in fact, be regarded a breakthrough). On the other hand, the specific codes used in the reductions of [1, 4, 5, 7, 8, 16, 19] are unnatural, and the problem of showing NP-hardness of maximum-likelihood decoding for any useful class of codes with nontrivial algebraic structure remains open, despite repeated calls for its resolution. For example, the survey of algorithmic complexity in coding theory [22] says: Although we have, by now, accumulated a considerable amount of results on the hardness of MAXIMUM-LIKELIHOOD DECODING, the broad worst-case nature of these results is still somewhat unsatisfactory. [...] Thus it would be worthwhile to establish the hardness of MAX-IMUM-LIKELIHOOD DECODING in the average sense, or for more narrow classes of codes. The first step along these lines was taken by Alexander Barg [2, Theorem 4], who showed that maximum-likelihood decoding is NP-hard for the class of product (or concatenated) Note that MAXIMUM-LIKELIHOOD DECODING OF LINEAR CODES is NP-complete over all finite fields q . Berlekamp, McEliece, and van Tilborg [4] only proved this result for the special case q 2. The easy extension to arbitrary prime powers can be found, for instance, in [2, Proposition 2]. 1 codes, namely codes of type © ¢ ¡ ¤ £ ¦ ¥ , where ¡ and ¥ are linear codes over £ q . Barg writes in [2] that this result is ... the first statement about the decoding complexity of a somewhat more restricted class of codes than just the "general linear codes." Observe, however, that the code © § ¡ £ ¦ ¥ does not have any algebraic structure unless ¡ and ¥ are further restricted in some manner. Furthermore, it is intuitively clear that the decoding problem for this code cannot be much simpler than the decoding problem for its factors ¡ and ¥ , which are, again, general linear codes. In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes. Let q © 2 m and let £ q X denote the ring of univariate polynomials over £ q . Reed-Solomon codes are obtained by evaluating certain subspaces of £ q
doi:10.1109/tit.2005.850102 fatcat:huoqs7n4gvebxkbgvmgai7xate