Risk averse single machine scheduling - complexity and approximation

In this paper a class of single machine scheduling problems is considered. It is assumed that job processing times and due dates can be uncertain and they are specified in the form of discrete scenario set. A probability distribution in the scenario set is known. In order to choose a schedule some risk criteria such as the value at risk (VaR) an conditional value at risk (CVaR) are used. Various positive and negative complexity results are provided for basic single machine scheduling problems. In this paper new complexity results are shown and some known complexity results are strengthen.


Introduction
Scheduling under risk and uncertainty has attracted considerable attention in recent literature. In practical applications of scheduling models the exact values of input parameters, such as job processing times or due dates, are often unknown in advance. Hence, a solution must be computed, before the true realization of the input data reveals. Typically, a scenario set U is a part of the input, which contains all possible realizations of the problem parameters, called scenarios. If the probability distribution in U is unknown, then robust optimization framework can be applied and solution performance in a worst case is optimized. First robust scheduling problems have been discussed in [11,22,41]. Two uncertainty representations, namely a discrete and interval ones were considered. In the former, scenario set U contains a finite number of distinct scenarios. In the latter, for each uncertain parameter an interval of its possible values is specified and U is the Cartesian product of these intervals. In order to compute a solution the minmax and minmax regret criteria can be applied. Minmax (regret) scheduling problems have various complexity properties, depending on the cost function and the uncertainty representation (see, e.g., [5,26,17,1,12]). For a survey of minmax (regret) scheduling problems we refer the reader to [19].
The robust scheduling models have well known drawbacks. Minimizing the maximum cost can lead to very conservative solutions. The reason is that the probability of occurrence of the worst scenario may be very small and the information connected with the remaining scenarios is ignored while computing a solution. One method of overcoming this drawback was given in [20], where the OWA criterion, proposed in [40], was applied to compute an optimal schedule. In this approach, a set of weights is specified by the decision maker, which reflect his attitude towards a risk. The OWA operator contains the maximum, average and Hurwicz criteria as special cases. However, it does not take into account a probabilistic information, which may be available for scenario set U .
In the case, when a probability distribution in U is known, the stochastic scheduling models are considered. The parameters of scheduling problem are then random variables with known probability distributions. Under this assumption, the expected solution performance is typically optimized (see, e.g., [28,33,39,38]). However, this criterion assumes that the decision maker is risk neutral and leads to solutions that guarantee an optimal long run performance. Such a solution may be questionable, for example, if it is implemented only once (see, e.g., [22]). In this case, the decision maker attitude towards a risk should be taken into account.
In [23] a criterion called conditional value at risk (CVaR) was applied to a stochastic portfolio selection problem. Using this criterion, the decision maker provides a parameter α ∈ [0, 1), which reflects his attitude towards a risk. When α = 0, then CVaR becomes the expectation. However, for greater value of α, more attention is paid to the worst outcomes, which fits into the robust optimization framework. The conditional value at risk is closely connected with the value at risk (VaR) criterion (see, e.g., [32]), which is just the α-quantile of a random outcome. Both risk criteria have attracted considerable attention in stochastic optimization (see, e.g., [29,9,30,31]). This paper is motivated by the recent papers [36] and [4], in which the following stochastic scheduling models were discussed. We are given a scheduling problem with discrete scenario set U . Each scenario ξ i ∈ U is a realization of the problem parameters (for example, processing times and due dates), which can occur with a known positive probability Pr[ξ i ]. The cost of a given schedule is a discrete random variable with the probability distribution induced by the probability distribution in U . The VaR and CVaR criteria, with a fixed level α, are used to compute a best solution.
In [36] and [4] solution methods based on mixed integer programming models were proposed to minimize VaR and CVaR in scheduling problems with the total weighted tardiness criterion. The aim of this paper is to analyze the models discussed in [36] and [4] from the complexity point of view. We will consider the class of single machine scheduling problems with basic cost functions, such as the maximum tardiness, the total flow time, the total tardiness and the number of late jobs. We will discuss also the weighted versions of these cost functions. We provide a picture of computational complexity for all these problems by proving some positive and negative complexity results. Since VaR and CVaR generalize the maximum criterion, we can use some results known from robust minmax scheduling. The complexity results for minmax versions of single machine scheduling problems under discrete scenario set were obtained in [1,3,11,27]. In this paper we will show that some of these results can be strengthen. This paper is organized as follows. In Section 2 we recall the definitions of the VaR and CVaR criteria and show their properties, which will be used later on. In Section 3 the problems discussed in this paper are defined. In Section 4 some general relationships between the problems with various risk criteria are shown. Finally, Sections 5 and 6 contain some new negative and positive complexity results for the the considered problems. These results are summarized in the tables presented in Section 3.

The risk criteria
Let Y be a random variable. We will consider the following risk criteria [32,35]: • Conditional value at risk : and CVaR α [Y] can be computed by using the following programs, respectively (see, e.g., [4,31,35]): where M ≥ max{b 1 , . . . , b K } and [K] = {1, . . . , K}. Notice that (1)b is a linear programming problem. In the following, we will use the following dual to (1)b: The equality constraint in (2) follows from the fact that all b k , k ∈ [K], are nonnegative. Substituting r k = q k /(1 − α) into (2), we get the following equivalent formulation for CVaR α [Y]: Program (3) can be solved by using a greedy method, which is illustrated in Figure 1. Namely, we fix the optimal values of q k by greedily distributing the amount 1 − α among the largest values of b i . It is easy to see that CVaR for sufficiently small ǫ > 0 and any probability distribution.
We now show several properties of the risk measures which will be used later on in this paper. Lemma 1. Let Y be a discrete random variable which can take K nonnegative values b 1 , . . . , b K . The following inequalities hold for each α ∈ [0, 1): where follows directly from the definition of the expected value and the conditional value at risk. We now prove the second inequality. Let r * 1 , . . . r * k be the optimal values in (2). Then the inequality Pr[Y=b k ] , and the lemma follows. Lemma 2. Let X and Y be two discrete random variables taking nonnegative values a 1 , . . . , a K , and b 1 , . . . , b K , respectively, with Proof. Let us compute CVaR α [X] by using (2) and denote by r * k , k ∈ [K], the optimal values in (2).

Problem formulations
We are given a set J of n jobs, which can be partially ordered by some precedence constraints. Namely, i → j means that job j cannot start before job i is completed. For each job j ∈ J a nonnegative processing time p j , a nonnegative due date d j and a nonnegative weight w j can be specified. A schedule π is a feasible (i.e. preserving the precedence constraints) permutation of the jobs and Π is the set of all feasible schedules. We will use C j (π) to denote the completion time of job j in schedule π. Obeying the standard notation, we will use T j (π) = [C j (π) − d j ] + to define the tardiness of j in π, and U j (π) = 1 if C j (π) > d j (job j is late in π) and U j (π) = 0 (job j is on-time in π), otherwise. In the deterministic case we seek a schedule π ∈ Π that minimizes a given cost function f (π). The basic cost functions are the total flow time j∈J C j (π), the total tardiness j∈J T j (π), the maximum tardiness max j∈J T j (π) and the total number of late jobs j∈J U j (π). We can also consider the weighted versions of these functions. Scheduling problems P will be denoted by means of the standard Graham's notation (see, e.g., [8]).
In this paper we assume that job processing times and due dates can be uncertain. The uncertainty is modeled by a discrete scenario set U = {ξ 1 , ξ 2 , . . . , ξ K }. Each realization of the parameters ξ ∈ U is called a scenario. For each scenario ξ ∈ U a probability Pr[ξ] of its occurrence is known (without loss of generality we can assume Pr[ξ] > 0). We will use p j (ξ) and d j (ξ) to denote the processing time and due date of job j under scenario ξ ∈ U , respectively. We will denote by C j (π, ξ), T j (π, ξ) and U j (π, ξ) the completion time, tardiness and unit penalty of job π, respectively, under scenario ξ ∈ U . Also, f (π, ξ) stands for the cost of schedule π under scenario ξ ∈ U . Given a feasible schedule π ∈ Π, we denote by F(π) a random cost of π. Notice that F(π) is a discrete random variable with the probability distribution induced by the probability distribution in U .
In this paper we will study the problems Min-VaR α P, Min-CVaR α P, Min-Exp P, and Min-Max P, in which we minimize the corresponding performance measure for a fixed α and a specific single machine scheduling problem P, under a given scenario set U . Notice that the robust Min-Max P problem is a special case of both Min-VaR α P and Min-CVaR α P. Also, Min-Exp P is a special case of Min-CVaR α P.
In the next sections we provide a number of new positive and negative complexity and approximation results for basic single machine scheduling problems P. Tables 1-3 summarize the known and new results. In Table 1, the negative results for uncertain due dates and deterministic processing times are shown. In Table 2, the negative results for uncertain processing times and deterministic due dates are presented. Finally, in Table 3, some positive results are shown. f (π; ξ 1 ) = 13 str. NP-hard str. NP-hard str. NP-hard (assignment) not at all appr. for any α ∈ (0, 1) not appr. for any for any α ∈ (0, 1) constant γ > 1 1|| Uj NP-hard as above as above as above 1|pj = 1| Tj poly sol. str. NP-hard str. NP-hard str. NP-hard (assignment) not at all appr. for any α ∈ (0, 1) not appr. within for any α ∈ (0, 1) str. NP-hard as above as above as above

Some general properties
In this section we will show some general relationships between the problems with various performance criteria. These properties will be used later to establish some positive and negative complexity results for particular problems.
Theorem 1. The following statements are true:

If
Min-Exp P with K-scenarios is NP-hard and hard to approximate within ρ > 1, then Min-CVaR α P with K + 1 scenarios is also NP-hard and hard to approximate within ρ for each constant α ∈ [0, 1).
. Denote by F ′ (π) the random cost of π under the new scenario set U ′ . For each schedule π we get (see Figure 3): Hence there is a cost preserving reduction from Min-Exp P with K scenarios to Min-CVaR α P with K + 1 scenarios and the theorem follows.
Theorem 2. Assume that w j = 1 for each job j ∈ J in problem P. If Min-Max P with K ≥ 2 scenarios is NP-hard and hard to approximate within ρ > 1, then 1. Min-VaR α P with K + 1 scenarios is NP-hard and hard to approximate within ρ > 1 for each constant α ∈ (0, 1).
Proof. Choose an instance of the Min-Max P problem with U = {ξ 1 , . . . , ξ K }, K ≥ 2. Fix α ∈ (0, 1) and create U ′ by adding to U a dummy scenario ξ ′ such that the cost of each schedule under ξ ′ equals M and M ≥ f (π, ξ i ) for each i ∈ [K] and each π ∈ Π. It is enough to fix p j (ξ ′ ) = p max and d j (ξ ′ ) = d min for each job j ∈ J, where p max = max j∈J,i∈[K] p j (ξ i ) is the maximum job processing time and d min = min j∈J,i∈[K] d j (ξ i ) is the minimum due date over all scenarios. For each of the two assertions, we define an appropriate probability distribution in U ′ . We will use F ′ (π) to denote the random cost of π under U ′ .
In order to prove the statement 1, we fix Pr[ξ ′ ] = 1−α and Pr[ξ i ] = α K for each i ∈ [K] (see Figure 4a). The equality VaR α [F ′ (π)] = Max[F(π)] holds. Hence, there is a cost preserving reduction from Min-Max P with K scenarios to Min-VaR α P with K + 1 scenarios and the statement follows. To prove the statement 2, we fix Pr[ξ ′ ] = γ and Pr[ξ i ] = β for each i ∈ [K], where γ and β satisfy the following system of equations (see Figure 4b): β β : : : Figure 4: Illustration of the proof of Theorem 2.
schedule π we get where β, γ and M are numbers depending on K and α. Hence Min-Max P and the corresponding instance of Min-CVaR α P have the same optimal solutions and the theorem follows.

Negative complexity results
In this section we will prove some negative complexity results for basic single machine scheduling problems. These results are summarized in Tables 1 and 2.

Uncertain due dates
We first address the problem of minimizing the value at risk criterion. The following theorem characterizes the complexity of some basic problems: Theorem 3. For each α ∈ (0, 1), Min-Var α P is strongly NP-hard and not at all approximable, when Proof. Consider an instance of the following strongly NP-hard Min 3-Sat problem [21,6]. We are given boolean variables x 1 , . . . , x n , a collection of clauses C 1 , . . . C m , where each clause is a disjunction of at most 3 literals (variables or their negations) and we ask if there is an assignment to the variables which satisfies at most L < m clauses. We can ask equivalently, if there is an assignment to the variables for which at least l = m − L clauses are not satisfied.
Given an instance of Min 3-Sat, we create two jobs J x i and J x i for each variable x i , i ∈ [n]. A due date scenario ξ i corresponds to clause C i = (l 1 ∨ l 2 ∨ l 3 ) and is formed as follows. For each q = 1, 2, 3, if l q = x j , then the due date of J x j is 2j − 1 and the due date of J x j is 2j; if l q = x j , then the due date of J x j is 2j and the due date of J x j is 2j − 1; if neither x j nor x j appears in C i , then the due dates of J x j and J x j are set to 2j. An example is shown in Table 4. Table 4: The set of jobs and the due date scenarios for the formula ( Let us define a subset of the schedules Π ′ ⊆ Π such that each schedule π ∈ Π ′ is of the . Observe that Π ′ contains exactly 2 n schedules and each such a schedule corresponds to the assignment to the variables such that x j = 0 if J x j is processed before J x j and x j = 1 otherwise. Note that this correspondence is one-to-one. In the following we assume that f (π, ξ i ) is the maximum tardiness, or the total tardiness, or the sum of unit penalties in π under ξ i . The reasoning will be the same for each of these cost functions. If π / ∈ Π ′ , then f (π, ξ i ) > 0 for each scenario ξ i . Indeed, suppose that π / ∈ Π ′ and let J j (J ′ j ) be the last job in π which is not placed properly, i.e. J j , is late under all scenarios. On the other hand, if π ∈ Π ′ , then the number of scenarios under which no job is late is equal to the number of unsatisfiable clauses for the assignment corresponding to π. Fix α ∈ (0, 1). We will add to U one additional scenario ξ ′ and define a probability distribution in U , depending on the fixed α, so that the answer to Min 3-Sat is yes if and only if there is schedule π for which VaR α [F(π)] ≤ 0. This will prove the stated result. We consider two cases: 1. l/m ≥ α. We create dummy scenario ξ ′ under which the due date of all jobs is equal to 0. The probability of this scenario is equal to l−αm l . The probability of each of the remaining scenarios is equal to 1 Assume that the answer to Min 3-Sat is yes. So, there is an assignment to the variables which satisfies at most m − l clauses. By the above construction, there is a schedule π ∈ Π ′ whose cost is positive under at most m − l scenarios plus the dummy one. It holds Assume that the answer to Min 3-Sat is no. Then, for every schedule π there are more than m − l scenarios under which the cost of π is positive plus the dummy one. Hence Pr[F(π) > 0] > (1 − α) and Pr[F(π) ≤ 0] < α. In consequence, VaR α [F(π)] > 0.
2. l/m < α. We create dummy scenario ξ ′ under which the due date of each job equals 2n. The probability of the dummy scenario is mα−l m−l . The probability of each of the remaining scenarios is equal to 1 Assume that the answer to Min 3-Sat is yes. So, there is an assignment to the variables which satisfies at most m − l clauses. By the construction, there is a schedule π whose cost is positive under at most m − l scenarios. Hence Assume that the answer to Min 3-Sat is no. Then for each assignment more than m−l clauses are satisfied. By the construction, for every schedule π there are more than m − l scenarios under which the cost π is positive. Therefore It follows from Theorem 3 that the problem discussed in [4] is strongly NP-hard and not at all approximable even in the very restrictive case in which all job processing times are equal to 1. It was shown in [20] that Min-Exp 1|p j = 1|T max is strongly NP-hard and hard to approximate within 7/6 − ǫ for any ǫ > 0. Hence, we immediately get from Theorem 1 that for each constant α ∈ [0, 1), Min-CVaR α 1|p j = 1|T max is strongly NP-hard and hard to approximate within 7/6 − ǫ for any ǫ > 0.
We consider now the problem with the total tardiness criterion. The deterministic 1|| T j problem is known to be NP-hard [25]. However, 1|p j = 1| T j is polynomially solvable(see, e.g., [8]). The following result characterizes the complexity of the minmax version of this problem: Theorem 4. Min-Max 1|p j = 1| T j is strongly NP-hard and not approximable within 5 4 −ǫ for any ǫ > 0.
Proof. We will show a reduction from the strongly NP-complete 3-Sat problem, in which we are given boolean variables x 1 , . . . , x n , a collection of clauses C 1 , . . . C m , where each clause is a disjunction of at most 3 literals (variables or their negations) and we ask if there is an assignment to the variables which satisfies all clauses (see, e.g., [14]). Given an instance of 3-Sat, we create two jobs J x j and J x j for each variable x j , j ∈ [n], |J| = 2n. A due date scenario ξ i corresponding to clause C i = (l 1 ∨ l 2 ∨ l 3 ) is created in the same way as in the proof of Theorem 3. Additionally, for each variable x j we create scenario ξ ′ j under which the due dates of J x j and J x j are 2(j − 1) + 1 2 and the due dates of the remaining jobs are set to 2n (see Table 5). We first show that the answer to 3-Sat is yes if and only if there is a schedule π such that max ξ∈U j∈J T j (π, ξ) ≤ 2.
corresponds to a truth assignment. Assume that the answer to 3-Sat is yes. Consider schedule π = (J 1 , J ′ 1 , J 2 , J ′ 2 , . . . , J n , J ′ n ), where J j , J ′ j ∈ {J x j , J x j }. Furthermore J x j is processed before J x j if and only if x j = 1. Since in every clause at least one literal is true, at most two jobs in π are late under each scenario ξ i ∈ U . The tardiness of each job in π under any ξ i ∈ U is at most 1. Furthermore, the total tardiness in π under any ξ ′ j is exactly 2. In consequence, max ξ∈U j∈J T j (π, ξ) ≤ 2. Assume that there is a schedule π, such that max ξ∈U j∈J T j (π, ξ) ≤ 2. We claim that Suppose that this is not the case, and let J k (J ′ k ) be the last job in π which is not placed properly. The completion time of J k (J ′ k ) is at least 2k + 1. So, its tardiness under ξ ′ k is at least 2k + 1 − (2k − 2 + 1 2 ) = 2.5. Let x j = 1 if and only if J x j is processed before J x j in π. Since only two jobs can be late under any ξ i , this assignment satisfies all clauses and the answer to 3-Sat is yes.
In order to prove the lower approximation bound, it is enough to observe that if the answer to 3-Sat is no, then each schedule has the total tardiness 3 under some scenario ξ i or 2.5 under some scenario ξ ′ j , which gives a gap at least 5 4 . From the fact that 1|| T j is weakly NP-hard (see [13]), we get immediately that more general Min-Exp 1|| T j problem is weakly NP-hard as well. The next theorem strengthens this result.
Proof. We will show a polynomial time reduction from the deterministic 1|| w j T j problem, which is known to be strongly NP-hard [25]. Consider an instance of 1|| w j T j . Let W = j∈J w j > 0 and P = j∈J p j . We build an instance of Min-Exp 1|| T j with the same set of jobs J and job processing times p j , j ∈ J. We create K = |J| = n due date scenarios as follows. Under scenario ξ j , j ∈ [n], job j has due date equal to d j and all the remaining jobs have due dates equal to P . We also fix Pr[ξ i ] = w i /W , i ∈ [n]. For any schedule π, we get In consequence 1|| w j T j and Min-Exp 1|| T j have the same optimal solutions and the theorem follows.
It was shown in [1] that Min-Max 1|p j = 1| U j with uncertain due dates is strongly NP-hard. The following theorem strengthens this result: Theorem 6. Min-Max 1|p j = 1| U j is not approximable within any constant factor unless P=NP.
Proof. Consider the following Min-Max 0-1 Selection problem. We are given a set of items E = {e 1 , e 2 , . . . , e n } and an integer q ∈ [n]. For each item e j , j ∈ [n], there is a cost c j (ξ i ) ∈ {0, 1} under scenario ξ i , i ∈ [K]. We seek a selection X ⊆ E of exactly q items, |X| = q, which minimizes the maximum cost over all scenarios, i.e. the value of max i∈[K] e i ∈X c j (ξ i ). This problem was discussed in [18], where it was shown that it is not approximable within any constant factor γ ≥ 1. We will show that there is a cost preserving reduction from Min-Max 0-1 Selection to the considered scheduling problem, which will imply the stated result.
Given an instance of Min-Max 0-1 Selection, we build the corresponding instance of Min-Max 1|p j = 1| U j as follows. We create a set of jobs J = E, |J| = n, with deterministic unit processing times. For each i ∈ [K], if c j (ξ i ) = 1 then d j (ξ ′ i ) = n − q, and if c j (ξ i ) = 0, then d j (ξ ′ i ) = n. So, we create K due date scenarios that correspond to the cost scenarios of Min-Max 0-1 Selection.
Suppose that there is a solution X to Min-Max 0-1 Selection such that e i ∈X c j (ξ i ) ≤ C for each i ∈ [K]. Hence X contains at most C items, C ≤ q, with the cost equal to 1 under each scenario. In the corresponding schedule π, we first process n − q jobs from J \ X and then the jobs in X in any order. It is easily seen that there are at most C late jobs in π under each scenario ξ ′ i , hence the maximum cost of schedule π is at most C. Conversely, let π be a schedule in which there are at most C late jobs under each scenario ξ ′ i . Clearly C ≤ q since the first n − q jobs in π must be on-time in all scenarios. Let us form solution X by choosing the items corresponding to the last q jobs in π. Among these jobs at most C are late under each scenario, hence the cost of X is at most C under each scenario ξ i . Thus, by Theorem 2, Min-CVaR α 1|p j = 1| U j is strongly NP-hard for any α ∈ (0, 1) (notice that p max = 1 in the proof of Theorem 2 and in the new scenario set U ′ still only due dates are uncertain).
Proof. Choose the deterministic 1|| w j U j problem, which is known to be NP-hard [16]. The reduction from this problem to Min-Exp 1|| U j is the same as the one in the proof of Theorem 5.
It is worth noting that in the proof of Theorem 7 we require an arbitrary probability distribution in the scenario set and we have shown that the problem is only weakly NP-hard. Its complexity for uniform probability distribution is open.

Uncertain processing times
In this section we characterize the complexity of the problems under consideration when only processing times are uncertain. It has been shown in [22] that Min-Max 1|| C j is strongly NP-hard. Furthermore, this problem is also hard to approximate within 6 5 − ǫ for any ǫ > 0 [27]. Using Theorem 2, we can immediately conclude that the same negative result holds for Min-VaR α 1|| C j for any α ∈ (0, 1]. Also, strong NP-hardness of the min-max problem implies that Min-CVaR α 1|| C j is strongly NP-hard for each fixed α ∈ (0, 1). Observe that the boundary case α = 0 (i.e. Min-Exp 1|| C j ) is polynomially solvable, as it easily reduces to the deterministic 1|| C j problem. Since 1|| T j is a special case of 1|| C j , with d j = 0 for each j ∈ J, the same negative results are true for the problem with the total tardiness criterion. Observe, however that Min-Exp 1|| T j is also NP-hard, since the deterministic 1|| T j problem is known to be weakly NP-hard [13].
It has been shown in [3] that Min-Max 1|| U j with uncertain processing times and deterministic due dates is NP-hard. The following theorem strengthens this result: Theorem 8. Min-Max 1|| U j is strongly NP-hard. This assertion remains true even when all the jobs have a common deterministic due date.
Proof. We show a polynomial time reduction from the 3-Sat problem (see the proof of Theorem 4). Given an instance of 3-Sat, we create an instance of Min-Max 1|| U j in the following way. For each variable x i we create two jobs J x i and J x i , so J contains 2n jobs. The due dates of all these jobs are the same under each scenario and equal 2. For each clause C j = (l 1 , l 2 , l 3 ) we construct processing time scenario ξ i , under which the jobs J l 1 , J l 2 , J l 3 have processing time equal to 1 and all the remaining jobs have processing times equal to 0. Then, for each pair of jobs J x i , J x i we construct scenario ξ ′ i under which the processing times of J x i , J x i are 2 and all the remaining jobs have processing times equal to 0. A sample reduction is shown in Table 6. We will show that the answer to 3-Sat is yes if and only if there is a schedule π such that max ξ∈U j∈J U (π, ξ) ≤ n.
corresponds to a satisfying truth assignment.
Assume that the answer to 3-Sat is yes. Then there exists a truth assignment to the variables which satisfies all the clauses. Let us form schedule π by processing first the jobs corresponding to true literals in any order and processing then the remaining jobs in any order. From the construction of the scenario set it follows that the completion time of the nth job in π under each scenario is not greater than 2. In consequence, at most n jobs in π are late under each scenario and max ξ∈U j∈J U (π, ξ) ≤ n.
Assume that there is a schedule π such that j∈J U (π, ξ) ≤ n for each ξ ∈ U , which means that at most n jobs in π are late under each scenario. Observe first that J x i and J x i cannot appear among the first n jobs in π for any i ∈ [n]; otherwise more than n jobs would be late in π under ξ ′ i . Hence the first n jobs in π correspond to a truth assignment to the variables x 1 , . . . , x n , i.e. when J l is among the first n jobs, then the literal l is true. Since f (π, ξ i ) ≤ n, the completion time of the n-th job in π under ξ i is not greater than 2. We conclude that at most two jobs among the first n job have processing time equal to 1 under ξ i , so there are at most two false literals for each clause and the answer to 3-Sat is yes.

Positive complexity results
In this section we establish some positive complexity results. Namely, we provide several polynomial and approximation algorithms for particular problems. A summary of the results can be found in Table 3.

Problems with uncertain due dates
Consider the Min-Exp 1|p j = 1| w j U j problem with uncertain due dates. We introduce variables x ij ∈ {0, 1}, i ∈ [n], j ∈ [n], where x ij = 1 if j ∈ [n] is the ith job in the schedule constructed. The variables satisfy the assignment constraints, i.e. i∈[n] x ij = 1 for each j ∈ [n] and j∈[n] x ij = 1 for each i ∈ [n]. If x ij = 1, then the completion time of job j equals i. Define c ijk = w j if i > d j (ξ k ) and c ijk = 0 otherwise, for each i, j ∈ [n] and k ∈ [K]. If the variables x ij describe π, then Hence the problem is equivalent to the Minimum Assignment with the cost matrix c * ij . The same result holds for Min-Exp 1|p j = 1| w j T j . It is enough . We thus get the following results: Theorem 9. Min-Exp P is polynomially solvable, when P ∈ {1|p j = 1| w j U j , 1|p j = 1| w j T j }.
From Theorems 9 and 1, we immediately get the following approximation result: Since Min-Max 1|p j = 1| w j U j and Min-Max 1|p j = 1| w j T j are special cases of the min-max version of Minimum Assignment, which is approximable within K (see, e.g., [2]), both problems are approximable within K as well.
We now study the Min-Exp 1|| w j T j problem with uncertain due dates and deterministic processing times. This problem is strongly NP-hard since 1|| w j T j is strongly NP-hard. The expected cost of π can be rewritten as We thus get a single machine scheduling problem 1|| f j with job-dependent cost functions of form f j (C j (π)) = i∈ Note also that these functions are nonnegative and nondecreasing with respect to C j (π). The same analysis can be done for the Min-Exp 1|| w j U j problem with uncertain due dates and deterministic processing times. Hence and from [10], where a (4 + ǫ)-approximation algorithm, for any ǫ > 0, for this class of problems was provided, we get the following result (see also Theorem 1): Theorem 11. If P ∈ {1|| w j U j , 1|| w j T j }, then Min-Exp P is approximable within 4 + ǫ and Min-CVaR α P is approximable within min{ 4+ǫ Pr min , 4+ǫ 1−α }, ǫ > 0, for any ǫ > 0 and each constant α ∈ [0, 1) When the probability distribution in U is uniform, then the approximation ratio in Theorem 1 can be improved to min{(4 + ǫ)K, 4+ǫ 1−α }. Since Min-Max P is a special case of Min-CVar α P with uniform probability distribution and α sufficiently large, we get that Min-Max P, P ∈ {1|| w j U j , 1|| w j T j }, is approximable within (4 + ǫ)K for any ǫ > 0.

The total weighted flow time criterion
In this section we focus on the problems with the total weighted flow time criterion. We start by recalling a well known property (see, e.g., [27]), which states that every such a problem with uncertain processing times and deterministic weights can be transformed into an equivalent problem with uncertain weights and deterministic processing times. This transformation goes as follows. For each processing time scenario ξ i , i ∈ [K], we invert the role of processing times and weights obtaining the weight scenario ξ ′ i . Formally, p j = w j and w j ( . We also invert the precedence constraints, i.e. if i → j in the original problem, then j → i in the new one. Given a feasible schedule π = (π(1), . . . , π(n)), let π ′ = (π(n), . . . , π(1)) be the corresponding inverted schedule. Of course, schedule π ′ is feasible for the inverted precedence constraints. It is easy to verify that f (π, is the random cost of π ′ for scenario set U ′ . Hence, the original problem with uncertain processing times and the new one with uncertain weights have the optimal solutions with the same performance measure. From now on we make the assumption that the jobs have deterministic processing times p j , j ∈ J and w j (ξ i ) is the weight of job j under scenario ξ i , i ∈ [K]. The value of CVaR α [F(π)], for a fixed schedule π, can be computed by solving the following optimization problem (see the formulation (1)b): Let δ ij ∈ {0, 1}, i, j ∈ [n], be binary variables such that δ ij = 1 if job i is processed before job j in a schedule constructed. The vectors of all feasible job completion times (C 1 , . . . , C n ) can be described by the following system of constraints [34]: Let us denote by V C ′ the relaxation of V C, in which the constraints δ ij ∈ {0, 1} are relaxed with 0 ≤ δ ij ≤ 1. It has been proved in [37,15] that each vector (C 1 , . . . , C n ) that satisfies V C ′ also satisfies the following inequalities: The formulations (7) and (6) lead to the following mixed integer programming model for Min-CVaR α 1|prec| w j C j with uncertain weights: We now solve the relaxation of (9), in which V C is replaced with V C ′ . Let (C * 1 , . . . , C * n ) be the relaxed optimal job completion times and z * be the optimal value of the relaxation. Consider discrete random variable Y, which takes the value j∈J w j (ξ i )C * j with probability Pr[ξ i ], i ∈ [K]. The equality z * = CVaR α [Y] holds. We relabel the jobs so that C * 1 ≤ C * 2 ≤ · · · ≤ C * n and form schedule π = (1, 2, . . . , n) in nondecreasing order of C * j . Since the vector (C * j ) satisfies V C ′ it must also satisfy (8). Hence, setting I = {1, . . . , j}, we get 2 and, finally C j = j i=1 p j ≤ 2C * j for each j ∈ J -this reasoning is the same as in [37]. For each scenario ξ i ∈ U , the inequality f (π, ξ i ) = j∈J w j (ξ i )C j ≤ 2 j∈J w j (ξ i )C * j holds. By Lemma 2, we have CVaR α [F(π)] ≤ 2 · CVaR α [Y ] = 2z * . Since z * is a lower bound on the value of an optimal solution, π is a 2-approximate schedule. Let us summarize the obtained result.
This result can be refined when the deterministic 1|prec| w j C j problem is polynomially solvable (for example, when the precedence constraints form an sp-graph, see, e.g., [8]). In this case Min-Exp 1|prec| w j C j is polynomially solvable, and we can also apply Theorem 1, which leads to the following result: Theorem 13. If 1|prec| w j C j is polynomially solvable, then Min-CVaR α 1|prec| w j C j is approximable within min{ 1 1−α , 2} for each α ∈ [0, 1).
Observe that 1 1−α < 2 for each α < 0.5. Let us consider Min-VaR α 1|prec| w j C j problem. The value of VaR α [F(π)], for a fixed schedule π, can be computed by solving the following MIP problem (see (1)a): where M k is an upper bound on the schedule cost under scenario ξ k , k ∈ [K]. Using the formulation (7) together with (1), we can get a mixed integer programming formulation for Min-VaR α 1|prec| w j C j . By replacing the constraints V C with relaxed V C ′ in the constructed model, we get a mixed integer problem with K binary variables. This problem can be solved in polynomial time when K is a constant. The same analysis as for Min-CVaR α 1|prec| w j C j (we also use Lemma 2) leads to the following result: Theorem 14. If the number of scenarios is constant, then Min-VaR α 1|prec| w j C j is approximable within 2 for each α ∈ (0, 1].

The bottleneck objective
In this section we address a class of single machine scheduling problems with a bottleneck objective, i.e. in which f (π) = max j∈J f j (C j (π)), where f j (t) is the cost of completing job j at time t. An important and well known example is 1|prec| max w j T j , in which the maximum weighted tardiness is minimized. This problem can be solved in O(n 2 ) time by Lawler's algorithm [24]. We will use the fact that the minmax versions of the bottleneck problems are polynomially solvable for a wide class of cost functions [7,20]. In particular, the minmax version of 1|prec| max w j T j with uncertain processing times and uncertain due dates can be solved in O(Kn 2 ) time by using the algorithm constructed in [20]. In the following, we will assume that f (π, ξ) = max j∈J w j T j (π, ξ) for a given scenario ξ ∈ U . We also assume that job processing times and due dates are nonnegative integers under all scenarios and job weights are positive integers. In consequence, the value of f (π, ξ) is a nonnegative integer for each ξ. Let f max be an upper bound on the schedule cost over all scenarios. Let h : Q K + → Q + be a nondecreasing function with respect to Q K + . Suppose that h can be evaluated in g(K) time for a given vector t t t = (t 1 , . . . , t K ) ∈ Z K + . Consider the corresponding scheduling problem PS, in which we seek a feasible schedule π ∈ Π minimizing H(π) = h(f (π, ξ 1 ), . . . , f (π, ξ K )). We can find such a schedule by solving a number of the following auxiliary problems: given a vector t t t ∈ Z K + , check if Π(t t t) = {π ∈ Π : f (π, ξ i ) ≤ t i , i ∈ [K]} is nonempty, and if so, return any schedule π t t t ∈ Π(t t t). From the monotonicity of the function h, it follows that for each π ∈ Π(t t t) the inequality h(f (π, ξ 1 ), . . . , f (π, ξ K )) ≤ h(t t t) is true. Thus, in order to solve the problem PS, it suffices to enumerate all possible vectors t t t = (t 1 , . . . , t K ), where t i ∈ {0, 1, . . . , f max }, i ∈ [K], and compute π t t t ∈ Π(t t t) if Π(t t t) is nonempty. A schedule π t t t with the minimum value of H(π t t t ) is returned.
The crucial step in this method is solving the auxiliary problem. We now show that this can be done in polynomial time for the bottleneck problem with the maximum weighted tardiness criterion. Given any t t t ∈ Z K + , we first form scenario set U ′ by specifying the following parameters for each ξ i ∈ U and j ∈ J: The scenario set U ′ can be built in O(Kn) time. We then solve the minmax problem with scenario set U ′ , which can be done in O(Kn 2 ) time by using the algorithm constructed in [20].
If the maximum cost of the schedule π returned is 0, then π t t t = π; otherwise Π(t t t) is empty. Since all the risk criteria considered in this paper are nondecreasing functions with respect to schedule costs over scenarios (see Lemma 2 for γ = 1) and g(K) is negligible in comparison with Kn 2 , we get the following result: Theorem 15. Min-Exp P, Min-VaR α P and Min-CVaR α P are solvable in O(f K max (Kn 2 )) time, when P is 1|prec| max w j T j .
The above running time is pseudopolynomial if K is constant. Notice that the special cases, when P is 1|prec, p j = 1|T max are solvable in O(Kn K+2 ) time, which is polynomial if K is constant (as we can fix f max = n).

Conclusions and open problems
In this paper we have discussed a wide class of single machine scheduling problems with uncertain job processing times and due dates. This uncertainty is modeled by a discrete scenario set with a known probability distribution. In order to compute a solution we have applied the risk criteria, namely, the value at risk and conditional value at risk. The expectation and the maximum criteria are special cases of the risk measures. We have provided a number of negative and positive complexity results for problems with basic cost functions. Moreover, we have sharpened some negative ones obtained in [1,3]. The picture of the complexity is presented in Tables 1-3. Obviously, the negative results obtained remain true for more general cases, for instance, for the problems with more than one machine.
There is still a number of interesting open problems on the models discussed. The negative results for uncertain due dates assume that the number of due dates scenarios is a part of input. The complexity status of the problems when the number of due date scenarios is fixed (in particular, equals 2) is open. For uncertain processing times, an interesting open problem is Min-Exp 1|| U j (see Table 2). There is still a gap between the positive and negative results, in particular, we conjecture that the negative results for Min-Var P for uncertain processing times (see Table 2) can be strengthen. Now they are just the same as for the Min-Max P.