Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

Hajiaghayi, MohammadTaghi; Jain, Kamal; Lau, Lap Chi; Măndoiu, Ion; Russell, Alexander; Vazirani, Vijay V.

doi:10.1007/11758525_102

Cited by 20 publications

(21 citation statements)

References 20 publications

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“…To handle precedence constraints we need to adapt the knapsack-related subroutine of our algorithm to the problem with a given partial order of jobs. This subproblem coincides with the partially ordered knapsack problem (POK) which is strongly NP-hard [19] and also hard to approximate [13]. On the positive side, FPTASes exist for several POK problems with special partial orders, including directed out-trees, two-dimensional orders, and the complement of chordal bipartite orders [19,25].…”

Section: Universal Scheduling With Precedence Constraintsmentioning

confidence: 94%

Universal Sequencing on an Unreliable Machine

et al. 2012

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Abstract. We consider scheduling on an unreliable machine that may experience unexpected changes in processing speed or even full breakdowns. Our objective is to minimize w j f (C j ) for any nondecreasing, nonnegative, differentiable cost function f (C j ). We aim for a universal solution that performs well without adaptation for all cost functions for any possible machine behavior. We design a deterministic algorithm that finds a universal scheduling sequence with a solution value within 4 times the value of an optimal clairvoyant algorithm that knows the machine behavior in advance. A randomized version of this algorithm attains in expectation a ratio of e. We also show that both performance guarantees are best possible for any unbounded cost function. Our algorithms can be adapted to run in polynomial time with slightly increased cost. When jobs have individual release dates, the situation changes drastically. Even if all weights are equal, there are instances for which any universal solution is a factor of Ω(log n/ log log n) worse than an optimal sequence for any unbounded cost function. Motivated by this hardness, we study the special case when the processing time of each job is proportional to its weight. We present a nontrivial algorithm with a small constant performance guarantee.

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Section: Universal Scheduling With Precedence Constraintsmentioning

confidence: 94%

Universal Sequencing on an Unreliable Machine

et al. 2012

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show abstract

“…To handle precedence constraints we need to adapt the knapsack related subroutine of our algorithm to the problem with a given partial order of jobs. This subproblem coincides with the partially ordered knapsack problem (POK) which is strongly NP-hard [18] and also hard to approximate [13]. On the positive side, FPTASes exist for several POK problems with special partial orders, including directed out-trees, two dimensional orders, and the complement of chordal bipartite orders [18,24].…”

Section: Universal Scheduling With Precedence Constraintsmentioning

confidence: 95%

Universal Sequencing on an Unreliable Machine

Mestre

Megow

2015

Encyclopedia of Algorithms

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We consider scheduling on an unreliable machine that may experience unexpected changes in processing speed or even full breakdowns. We aim for a universal solution that performs well without adaptation for any possible machine behavior. Our objective is to minimize w j f (C j ) for any non-decreasing, nonnegative, differentiable cost function f (C j ). We design a deterministic algorithm that finds a universal scheduling sequence with a solution value within 4 times the value of an optimal clairvoyant algorithm that knows the machine behavior in advance. A randomized version of this algorithm attains in expectation a ratio of e. We also show that both results are best possible among all universal solutions. Our algorithms can be adapted to run in polynomial time with slightly increased cost.When jobs have individual release dates, the situation changes drastically. Even if all weights are equal, there are instances for which any universal solution is a factor of Ω(log n/ log log n) worse than an optimal sequence for any unbounded cost function. Motivated by this hardness, we study the special case when the processing time of each job is proportional to its weight. We present a non-trivial algorithm with a small constant performance guarantee.

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“…Notice, however, that, as long as every integer in S is not bounded by a polynomial in the length of the input, none of the approximation results of [12] and [13] applies.…”

Section: Hardnessmentioning

confidence: 99%

“…Covering a set of strings S with a set X of substrings in S is indeed the Minimum Generating Set problem for unary alphabet under the unary encoding scheme. To narrow the context, notice that, given a set of binary strings S, finding a minimum cardinality set X of substrings in S such that every string in S can be written as a concatenation of at most two substrings in X is NP-complete (the proof being an easy binary alphabet encoding of the result of Néraud [15]) Finally, Hajiaghayi et al [12] considered the Minimum Multicolored Subgraph problem, which can be seen as a generalization of Minimum 2-Generating Set when every integer in the input set is bounded by a polynomial in the length of the input. This paper is organized as follows: we first recall basic definitions in Section 2, and we then formally introduce the considered problem.…”

Section: Introductionmentioning

confidence: 99%

On Finding Small 2-Generating Sets

Fagnot

Fertin

2009

Lecture Notes in Computer Science

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Abstract. Given a set of positive integers S, we consider the problem of finding a minimum cardinality set of positive integers X (called a minimum 2-generating set of S) such that every element of S is an element of X or is the the sum of two (non-necessarily distinct) elements of X. We give some elementary properties of 2-generating sets and prove that finding a minimum cardinality 2-generating set is hard to approximate within ratio 1 + ε for any ε > 0. We then prove our main result, which consists in a standard representation lemma for minimum cardinality 2-generating sets.

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Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping

Cited by 20 publications

References 20 publications

Universal Sequencing on an Unreliable Machine

Universal Sequencing on an Unreliable Machine

Universal Sequencing on an Unreliable Machine

On Finding Small 2-Generating Sets

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