### Table 2. NP-hard problems

1998

"... In PAGE 3: ... Hence, only these two problems are proved to be ordinarily NP-hard. The other NP-hard problems in Table2 as well as the NP-hard J2jno wait; rj; pij = 1jCmax, which is equivalent to J2jno wait; pij = 1jLmax by symmetry, and J2jno wait; pij = 1j Tj are open for the ordinary or strong NP-hardness. The letter C in the machine environment eld denotes a cycle shop, a special case of a job shop, where all the jobs have the same route passing through the machines like in a ow shop but repetitions of machines in the route are allowed.... ..."

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### Table 4: NP-hard subclasses of V.

"... In PAGE 9: ... The relations included in each of these algebras can be found in Table 3. Further, let VNP denote the set of subalgebras listed in Table4 . We have... In PAGE 18: ... Proof: V17 s = DV(V17 f ). 2 2 NP-Hardness Results This section provides NP-hardness proofs for the subclasses of V presented in Table4 . The reductions are mostly made from di erent subalgebras of Allen apos;s interval algebra.... ..."

### Table 4. Reductions proving the NP-hardness

1998

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### Table 1: Complexity status of some unrelated machine problems

2005

Cited by 3

### Table 2.1: Complexity status of some unrelated machine problems.

### Table 2. Problem dimensions. Recall that the dimension of each BMI is equal to m, hence depends on the order of the plant. One can see that the number of BMIs quickly becomes very large as the order of the controller increases. Considering controllers of higher orders, or with more design parameters than those proposed in Table 1, would have led to systems of 256 BMIs or more that must be discarded for obvious practical reasons. This is a main hindrance to the design of high-order controllers, and helps to pinpoint the origin of the di culty of robust stabilization of interval plants. In [17], the authors showed that many of the problems considered in the robust control literature can be formulated as BMIs. In view of Theorem 3, this is also the case for the robust stabilization of interval plants. Unfortunately, BMIs are highly non-convex optimization problems and solving a general BMI was shown to be NP-hard [29].

2001

"... In PAGE 7: ... Theorem 3 Any vector z solution to the BMIs n X j=0 n X k=0 zjzkHijk gt; 0; i = 1; : : : ; N (3) parametrizes a pair of polynomials x(s) and y(s) giving rise to a robustly stabilizing con- troller for interval plant (1). In Table2 , we report the number of BMIs that occur for di erent controller architectures, together with the number of decision variables, or design parameters.... ..."

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### Table 1 Formula depth Approximation algorithm NP-hardness factor

2003

### Table 1: Actions in the proof of NP-hardness of S-concurrent executability

"... In PAGE 61: ... Moreover, we add an action which completely clears the database, such that every feasible permutation leads to the empty database. The actions and their descriptions are given in the Table1 below. There Ati;j = VAL(xk; 1) if the j-th literal of clause Ci is xk, and Ati;j = VAL(xk; 0) if it is :xk.... ..."

### Table 1. Performance of bidding rules against optimal in known environments.

2005

"... In PAGE 8: ... The NP-hardness of optimizing the performance did not allow us to solve larger multi-robot exploration tasks. Table1... ..."

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### Table 1: Energy potentials for alphabets HP, HP apos; and HPNX. Lattice Protein Folding is NP-hard 25. A large variety of approximation al- gorithms was therefore developed 15;26;27. Most of these are not fast enough to investigate large ensembles of structures and stochastic optimization techniques (see e.g. 28) are not useful either to study ensemble properties of speci cally folded single chainsj. Hart and Istrail29 recently presented an algorithm for the HP model that guarantee folding within at least 3=8 of the optimum energy. It iThe frequency of Hs is the same as in the HP model, such that a random distribution of the HX subset corresponds exactly to the HP model. janother reasons is given in the next section 2.2

1997

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