On the Efficiency of the Hamming C-Centerstring Problems

Amir, Amihood; Ficler, Jessica; Roditty, Liam; Shalom, Oren Sar

doi:10.1007/978-3-319-07566-2_1

Cited by 2 publications

(12 citation statements)

References 17 publications

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“…We remark that this algorithm cannot be significantly improved assuming the ETH. 1 It is known that there is no o( ) • n O (1) -time algorithm for the special case Closest String unless the ETH fails [14]. The running time of our algorithm matches this lower bound (up to a constant in the exponent) and therefore there is presumably no substantially faster algorithm with respect to .…”

Section: Parameter Number Of Columnsmentioning

confidence: 52%

“…They also proved that dropping any of c, d, or r results in parameterized intractability even for binary alphabet. Amir et al [1] showed that the clustering version of MinLRMC on complete matrices with unbounded alphabet size is NP-hard when restricted to only two columns. Note that fixed-parameter tractability for the clustering variant implies fixed-parameter tractability for MinRMC and MinLRMC with respect to d + r. Indeed, to reach for (more) practical algorithms, we consider an alternative parameterization by the maximum number k of missing entries in any row vector (which can be much smaller than r).…”

Section: Parameter Minrmcmentioning

confidence: 99%

“…We sometimes use the O * -notation which suppresses polynomial factors in the input size. A (1) time for an arbitrary computable function f . An algorithm with such a running time is called a fixed-parameter algorithm.…”

Section: Parameterized Complexitymentioning

confidence: 99%

“…We use Algorithm 1 to overcome this issue (Line 3). Algorithm 3 also considers the columns where the first row vector has missing entries (recall that P * (S [1]) denotes column indices j with S[1, j] = * ) in the return the output of Algorithm 1 on (S , d 2 , . .…”

Section: Is Given In Algorithm 2 First It Determines Whether the Firs...mentioning

confidence: 99%

“…We claim that there exists a j ∈ R with v[j] = S[i, j] for every choice of R. Otherwise, v[j] = S[i, j] and S[i, j] = * holds for all j ∈ R and we have δ(v, S[i]) > d i (a contradiction). Note that S[:, [ ] \ {j}] has exactly one less missing entry if j ∈ P * (S [1]) and that d…”

Section: Is Given In Algorithm 2 First It Determines Whether the Firs...mentioning

confidence: 99%

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Parameterized Algorithms for Matrix Completion With Radius Constraints

Koana,

Froese,

Niedermeier

2020

Preprint

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Considering matrices with missing entries, we study NP-hard matrix completion problems where the resulting completed matrix shall have limited (local) radius. In the pure radius version, this means that the goal is to fill in the entries such that there exists a 'center string' which has Hamming distance to all matrix rows as small as possible. In stringology, this problem is also known as Closest String with Wildcards. In the local radius version, the requested center string must be one of the rows of the completed matrix.Hermelin and Rozenberg [CPM 2014, TCS 2016 performed parameterized complexity studies for Closest String with Wildcards. We answer one of their open questions, fix a bug concerning a fixed-parameter tractability result in their work, and improve some upper running time bounds. For the local radius case, we reveal a computational complexity dichotomy. In general, our results indicate that, although being NP-hard as well, this variant often allows for faster (fixed-parameter) algorithms.

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Section: Parameter Number Of Columnsmentioning

confidence: 52%

Section: Parameter Minrmcmentioning

confidence: 99%

Section: Parameterized Complexitymentioning

confidence: 99%

Section: Is Given In Algorithm 2 First It Determines Whether the Firs...mentioning

confidence: 99%

Section: Is Given In Algorithm 2 First It Determines Whether the Firs...mentioning

confidence: 99%

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Parameterized Algorithms for Matrix Completion With Radius Constraints

Koana,

Froese,

Niedermeier

2020

Preprint

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Combinatorial n-fold integer programming and applications

2019

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Many fundamental NP-hard problems can be formulated as integer linear programs (ILPs). A famous algorithm by Lenstra solves ILPs in time that is exponential only in the dimension of the program, and polynomial in the size of the ILP. That algorithm became a ubiquitous tool in the design of fixedparameter algorithms for NP-hard problems, where one wishes to isolate the hardness of a problem by some parameter. However, in many cases using Lenstra's algorithm has two drawbacks: First, the run time of the resulting algorithms is often doubly-exponential in the parameter, and second, an ILP formulation in small dimension cannot easily express problems involving many different costs.Inspired by the work of Hemmecke, Onn and Romanchuk [Math. Prog. 2013], we develop a singleexponential algorithm for so-called combinatorial n-fold integer programs, which are remarkably similar to prior ILP formulations for various problems, but unlike them, also allow variable dimension. We then apply our algorithm to a few representative problems like Closest String, Swap Bribery, Weighted Set Multicover, and obtain exponential speedups in the dependence on the respective parameters, the input size, or both.Unlike Lenstra's algorithm, which is essentially a bounded search tree algorithm, our result uses the technique of augmenting steps. At its heart is a deep result stating that in combinatorial n-fold IPs, existence of an augmenting step implies existence of a "local" augmenting step, which can be found using dynamic programming. Our results provide an important insight into many problems by showing that they exhibit this phenomenon, and highlights the importance of augmentation techniques.1 Kannan's algorithm in parameterized complexity: they modeled Closest String with k input strings as an ILP of dimension k O(k) , and thereby concluded with the first fixed-parameter algorithm for Closest String. This success led Niedermeier [Nie04] to propose in his book:[...] It remains to investigate further examples besides Closest String where the described ILP approach turns out to be applicable. More generally, it would be interesting to discover more connections between fixed-parameter algorithms and (integer) linear programming.Since then, many more applications of Lenstra's and Kannan's algorithm for parameterized problems have been proposed. However, essentially all of them [BFN + 15, DS12, FLM + 08, HR15, Lam12, MW15] share a common trait with the algorithm for Closest String: they have a doubly-exponential run time dependence on the parameter. Moreover, it is difficult to find ILP formulations with small dimension for problems whose input contains many objects with varying cost functions, such as in Swap Bribery [BCF + 14, Challenge #2].

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On the Efficiency of the Hamming C-Centerstring Problems

Cited by 2 publications

References 17 publications

Parameterized Algorithms for Matrix Completion With Radius Constraints

Parameterized Algorithms for Matrix Completion With Radius Constraints

Combinatorial n-fold integer programming and applications

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