Quantum Speed-ups for Single-machine Scheduling Problems

Grover search is currently one of the main approaches to obtain quantum speed-ups for combinatorial optimization problems. The combination of Quantum Minimum Finding (obtained from Grover search) with dynamic programming has proved particularly efficient to improve the worst-case complexity of several NP-hard optimization problems. Specifically, for these problems, the classical dynamic programming complexity (ignoring the polynomial factors) in O* (cn) can be reduced by a bounded-error hybrid quantum-classical algorithm to O* (cnquant) for cquant < c. In this paper, we extend the resulting hybrid dynamic programming algorithm to three examples of single-machine scheduling problems: minimizing the total weighted completion time with deadlines, minimizing the total weighted completion time with precedence constraints, and minimizing the total weighted tardiness. The extension relies on the inclusion of a pseudo-polynomial term in the state space of the dynamic programming as well as an additive term in the recurrence.


INTRODUCTION
The interest in quantum computing to solve combinatorial optimization problems has been growing for several years in the operational research community.More precisely, two branches are distinguished.The first one relates to heuristics, often hybrid quantumclassical algorithms such as variational quantum algorithms [3,8] and in particular QAOA [6].Essentially, these algorithms require the optimization problem to be formulated as a QUBO (Quadratic Unconstrained Binary Optimization) and can be implemented on current noisy quantum computers because the quantum part can be made rather small.The second branch relates to exact algorithms.Unlike the previous algorithms, it is impossible to implement them today but theoretical speed-ups have been proved for several types of problems and algorithms [13,16].
The most emblematic algorithm of this branch is Grover search [9], which achieves a quadratic speed-up when searching for a specific element in an unsorted table, where the complexity is computed as the number of queries of the table and done by an oracle.The authors of [5] use Grover search as a subroutine for a hybrid quantum-classical algorithm that finds with high probability the minimum of an unsorted table, leading to the algorithm known as Quantum Minimum Finding (QMF).Later, the authors of [2] combine QMF with dynamic programming to address NPhard optimization problems.They apply their algorithm to vertex ordering problems, the Traveling Salesman Problem (TSP), and the Minimum Set Cover problem, among others.All these problems satisfy a specific property which implies that they can be solved by classical dynamic programming in O * (  ), where O * is the usual asymptotic notation that ignores the polynomial factors, and  is usually not smaller than 2. The hybrid algorithm from [2] reduces the complexity to O * (  quant ) for  quant < .For instance, the TSP is solved by Held and Karp dynamic programming [10] in O * (2  ), and by the hybrid algorithm of [2] in O * (1.728  ).Subsequently to the work of [2], other NP-hard problems have been tackled with the idea of combining Grover search (or QMF) and classical dynamic programming.This has led to quantum speed-ups for the Steiner Tree problem [11], the graph coloring problem [15], and the subset sum problem [1].
The purpose of this work is to adapt the seminal idea of [2] to NP-hard scheduling problems [17] that satisfy the following property: for a given set of jobs  , the optimal solution for  is the best concatenation of optimal solutions for  and  \  among all  ⊂  such that | | = | |/2 (modulo an additive term that arises in the concatenation).This adaptation requires to introduce a pseudo-polynomial term in the state space of the dynamic programming as well as the aforementioned additive term.We thus obtain an extension of the Dynamic Programming Across the Subsets (DPAS) that many scheduling problems satisfy [17].Herein, we focus on single-machine scheduling problems and show that our bounded-error hybrid quantum-classical algorithm improves the best-known classical exponential complexities, where in some cases a pseudo-polynomial factor   appears.We illustrate it with three examples: minimizing the total weighted completion time with deadlines, minimizing the total weighted completion time with precedence constraints, and minimizing the total weighted tardiness.We summarize in Table 1 Table 1: Comparison of complexities between our hybrid algorithm Quantum-DDPAS and the best-known classical algorithm The rest of the paper is structured as follows.We detail in Section 2 the required property and give examples of single-machine scheduling problems that satisfy it.Then, we describe in Section 3 the hybrid algorithm that solves the problems of interest, recalling basic notions of quantum complexity.Appendix A recalls wellknown bounds useful to derive the complexities of the algorithm while Appendix B provides a detailed proof of the correctness of our main algorithm.

DYNAMIC PROGRAMMING FOR SCHEDULING
Our problems of interest are scheduling problems where solutions are described by permutations of jobs in [] := {1, . . ., } for  ∈ N, and that satisfy a certain property discussed below (see Property 3).This essentially consists of single-machine scheduling problems with constraints.
Let P be the nominal problem we want to solve.We introduce next a family of problems related to P that will be instrumental in deriving the dynamic programming recursion.Let  be a set of nonnegative integers containing 0. We define the family of problems indexed by  ⊆ [] and  ∈  : where Π(, ) ⊆   is the set of feasible permutations of  according to potential constraints and  (., , ) is the objective function.We note OPT[, ] the optimal value of  (, ).With these notations, the nominal problem P can be cast as follows:

Dynamic Programming Across the Subsets
We suppose in what follows that P can be solved by DPAS (Dynamic Programming Across the Subsets).It means that the family of problems must satisfy the following DPAS property.
Property 1 (DPAS).Let  0 ∈  .Problem  ([],  0 ) can be solved by DPAS if there exists a function ℎ : 2  × [] ×  → R, computable in polynomial time, such that the following holds: Notice the presence of the additional parameter  0 in the above definitions, which is typically absent in the scheduling literature.In particular,  0 is a constant throughout the whole recursion (2) and does not impact the resulting computational complexity.The use of that extra parameter defined in Equation ( 1) and in Property 1 shall be necessary later when applying our hybrid algorithm.

□
In this paper, we consider a family of problems that not only satisfy Property 1, but also the Dichotomic DPAS property below.Property 3 (Dichotomic DPAS).Let  0 ∈  .Problem  ( [],  0 ) can be solved by Dichotomic DPAS if there exist three functions  1 : 2  × 2  ×  →  ,  2 : 2  × 2  ×  →  and  : 2  × 2  ×  → R, computable in polynomial time, such that, for all  ⊆ [] of even cardinality: Notice that if OPT[, ] for  ⊆ [] and  ∈  is infeasible, then by convention OPT[, ] = +∞.Furthermore, differently from the previous recurrence (2), recurrence (3) now calls OPT[ ′ ,  ′ ] for  ′ that may be different than  0 .This has an impact when deriving the computational complexity of the algorithm in the next lemma.Proof.We compute Equation (3) We show that 2   () → 0. Let us first consider the sub-sequence ((2  ))  ∈N .For  = 2  , a lower bound of () is the sum of the two last terms: , where  is a constant.Moreover, the sequence (()) ∈N is increasing.Thus,  dominates  → 2  asymptotically, namely  () =  (2  ). □ Notice that solving P using only Dichotomic DPAS is worse than using only DPAS.However, we describe in the next section a hybrid algorithm we call Quantum Dichotomic DPAS (Q-DDPAS) that improves the complexity of solving P by combining DPAS and Dichotomic DPAS with Grover search.Before introducing this algorithm, we give some examples of single-machine scheduling problems that satisfy the Dichotomic DPAS property and then can be solved with Q-DDPAS.

Scheduling Examples
Let us begin with the scheduling problem with deadline constraints and minimization of the total weighted completion time.
Example 5 (Minimizing the total weighted completion time with deadlines).For each job  ∈ [], we are given a weight   , a processing time   , and a deadline d .We note  ( ) =  ∈    and  = ⟦0,  ( [])⟧.For each  ⊆ [] and  ∈  , we consider the problem  (, ) where where   is the completion time of job , and for  ∈ Π(, ): (, ) represents the problem of finding the best feasible solution for jobs in  supposing that starting time is t, and not 0 as usual.Our problem of interest is P =  ([], 0) , often referred to as 1| d |      in the scheduling literature.It can be solved by DPAS.Indeed, Equation where the computation of ℎ is polynomial (linear).This family of problems also satisfies the Dichotomic DPAS property.Indeed, Equation (3) is valid for the following functions: We present another problem that satisfies the Dichotomic DPAS property, which is the scheduling problem with minimization of the total weighted tardiness.
where   is the completion time of job .Our problem of interest is P =  ([], 0) , often referred to as 1| |      in the scheduling literature.It can be solved by DPAS.Indeed, Equation (2) is valid for: where the computation of ℎ is polynomial (quadratic).This family of problems also satisfies the Dichotomic DPAS.Indeed, Equation (3) is valid for the following functions : where the computation of  is polynomial (quadratic).

QUANTUM DICHOTOMIC DPAS ALGORITHM
In this section, we introduce a hybrid bounded-error algorithm called Quantum Dichotomic DPAS (Q-DDPAS) that solves scheduling problems satisfying the Dichotomic DPAS property described in the last section.It is an adaptation of the algorithm in [2].We describe the quantum part of our algorithm in the gate-based quantum computing model, namely, as a quantum circuit decomposed into single and two-qubit quantum gates.The computational time of such a quantum circuit is quantified by the number of these elementary quantum operations [12].Henceforth, we assume to have random access to quantum memory (QRAM) [7].Notice that this is a strong assumption because QRAM is not available on current universal quantum hardware and is not expected to be so in the near future.

Preliminaries
We begin with some notions of complexity for quantum circuits and some notations for the description of Q-DDPAS.
Definition 8. Let us consider a family of quantum circuits (Q  ) ∈N of complexity O ( ()), meaning that Q  is a circuit that applies on  qubits and contains  () universal quantum gates, where Observation 9 (Complexity of qantum circuits).Let  1 and  2 be two quantum circuits, with complexity O ( 1 ()) and O ( 2 ()), respectively.The complexity of the composition The tensor product  1 ⊗  2 has the same complexity.
Observation 10 (Classical algorithm into qantum circuit).Any classical algorithm A can be described as a quantum circuit  A .The complexity of  A is equal to the complexity of A.
We define two useful sets for the description of our algorithm, both indexed by a subset and a parameter, (, ).Essentially, the first set Λ(, ) contains all the possible balanced bi-partitions of  and the associated parameter values of  1 and  2 .The second set Ω(, ) contains the optimal solutions for each bi-partition in Λ(, ).
Let us introduce the quantum circuits that constitute the building blocks of our algorithm, and let us provide for each of them their complexity.The two first circuits  Λ and  Ω amount to put into uniform superposition the elements of Λ, respectively Ω. Definition 12 (Circuit  Λ ).For  ⊆ [] such that | | is even, and for  ∈  , we define  Λ as follows: Notice that we index the objects that represent sets by , and the objects that represent parameters in  by .
Property 13 (Complexity of  Λ ).The complexity of  Λ is polynomial in the size of the input.
Proof.First, let us prove that the construction of the superposition of subsets of  of size | |/2 is polynomial.Let  ⊆ [] of size  (we suppose  to be even).Let us prove that the construction of a quantum superposition of balanced bi-partition (both subsets are of size  2 ) of  can be done in polynomial time.We know that there are ] the quantum circuit corresponding to the Quantum Minimum Finding algorithm [5] that computes with high probability the minimum value of  and the corresponding minimizer: Property 19 (Complexity of  QMF [5]).The complexity of the Quantum Minimum Finding algorithm is O √  •   () , where  is the size of the domain of  and O (  ()) is the complexity of the circuit   .Thus, the complexity of according to Observation 10.
Let us denote the indexes related to the quantum circuit   as To clarify the computations detailed next, we index the corresponding QMF operator as  QMF [   ].We omit the index  because this is an auxiliary register that does not appear in the output of  QMF [  ].

Description of the Algorithm
We describe the Quantum Dichotomic DPAS (Q-DDPAS) algorithm as an adaptation of [2] for scheduling problems satisfying the Dichotomic DPAS property.Without loss of generality, we assume that 4 divides .(a) Apply quantum circuit , where the tuples indexing the different registers are decomposed as follows: and where and where (b) Measure register of indexes  6 2 to find the optimal value OPT[[], 0].The main idea of this algorithm is as follows.First, we compute classically by DPAS the optimal values of all subproblems scheduling with   4 jobs.Second, we call recursively two times QMF to find optimal values of subproblems scheduling with /2 jobs and eventually with  jobs (corresponding to the initial problem).
Theorem 20.The bounded-error Q-DDPAS algorithm solves The proof of Theorem 20 relies on the two lemmas introduced next.However, before stating and proving these lemmas, we observe that the complexity of Q-DDPAS can be further reduced by performing a third call to Dichotomic DPAS recurrence (3) as suggested in [2].For the sake of clarity, we will prove Observation 21 only after having proved Theorem 20.We now introduce the two lemmas necessary to prove Theorem 20.
Lemma 22.The optimal value of P is stored in the register of indexes  6  2 by Q-DDPAS with high probability.
Proof.We provide next a sketch of the proof, referring to Appendix B for the details of the computations.The main idea is to compute the first terms by classical DPAS, and then apply recursively twice Equation (3), which is solved by QMF: We now give some intuition on the effect of the quantum circuit   =  recur  ini and start by explaining the effect of  ini defined in (4).First, the application of  Λ superposes all elements of Λ([], 0) in the registers of indexes  2 (partition  ) and  4 (partition [] \  ).This essentially amounts to superpose all the  /2 bipartitions of [] where each partition is of size /2 (parameters  included).Then, we apply  Ω on register of index  2 , resp. 4 .This superposes all elements of Ω(, ) (for a  of size /2 and  ∈  previously described in registers of indexes  2 , resp. 4 ).This essentially amounts to superpose all the /2 /4 bi-partitions of [𝑛] where each partition is of size /2, parameters  included, and the optimal value associated already stored in the QRAM.
Let us explain the effect of  recur defined in (5).The application of  QMF [  ] on a register encoding ( , ) and the superposition of elements of Ω(, ) stores (with high probability) in an output register OPT[, ] according to the Dichotomic DPAS Property 3. Thus,  QMF [  ] on register of index  2 , resp. 4 , superposes all OPT[, ] in  3 , resp. 5 .In other words, the circuit

𝑟
] that appears in  recur1 superposes (with high probability) all optimal values of Equation (3) for  of size /2.Now that the optimal values are known for sets of size /2 (before, we only knew optimal values for sets of size /4), we apply one more time  Second, let us compute the complexity of the quantum part (using Property 10).
• The complexity of  ini is polynomial in .Indeed,  Λ is polynomial in  (Property 13).Moreover,  Ω is also polynomial in : the classical part stored in the QRAM all OPT[, ] for  of size /4 and  ∈  (Property 15).7)).
• the quantum part applies three levels of recurrence of QMF, finding the minimum over functions with a domain of size = O * (2 0.789 ) (see Equation ( 9)).The quantum part and the classical part have the same complexity, thus the total complexity of Q-DDPAS is the same, namely O * (2 0.789 ) = O * (1.728  ) .□ We summarized in Table 1 the complexities of solving the scheduling problems studied in Section 2 with Q-DDPAS and compare them with the complexities of the best-known current classical algorithms.Q-DDPAS improves the complexity of the exponent but sometimes at the cost of a pseudo-polynomial factor (   for problems 1| d |      and 1||      ).

CONCLUSION
This paper extends the hybrid algorithm of [2] to scheduling problems that satisfy the Dichotomic DPAS property.Such problems, which are often solved in O * (2  ) by classical DPAS, are solved by our bounded-error algorithm in O * (| | • 1.754  ), where | | is meant to be at most pseudo-polynomial in the size of the problem.We detail the application of the resulting hybrid algorithm on three single-machine scheduling problems (1| d |      , 1||      and 1| |      ), showing a reduction of the exponent compared to the best-known classical complexity.We notice that a pseudo-polynomial factor appears in the complexity of two out of three problems.Future works will seek to extend these results to other scheduling problems for which the Dichotomic DPAS holds, such as the 3-machine flowshop scheduling problem [14].

A NOTATIONS AND UPPER BOUNDS
In what follows, we use the notation We also note the binary entropy of  ∈]0, 1[ the quantity  () = −( log 2 () + (1−) log 2 (1−)) .We give some useful upper bounds of binomial coefficients [2]:  (), 0) According to definition of  and the Dichotomic DPAS Property 3, the results stored in register of indexes  6 2 is OPT[[], 0].Notice that optimal permutation  * [[], 0] can be rebuilt with registers of indexes  3 1 ,  5 1 and  6 1 , and with the access to the results of the classical part in the QRAM.

𝑡 2 (
, , ) =  +  ( ) (, , ) = 0 We end with the example of the scheduling problem with precedence constraints and minimization of the total weighted completion time.Example 7 (Minimizing the total weighted completion time with precedence constraints).We are given, for each job  ∈ [] a processing time   and a weight   , and a set of precedence constraints  = {(, ) :  ≺  } that contains all pairs of jobs (, ) such that  precedes .We note  ( ) =  ∈    .Let  = {0}.Here, the family of problems under consideration is indexed only by the chosen subset of [].Thus, for each  ⊆ [], we consider the problem  (, 0) where Π(, 0) = { ∈   |  respects } , and for  ∈ Π(, 0):
Classical part: Compute by classical DPAS the values OPT[, ] for all  of size /4 and for all  ∈  .Store the results in the QRAM.• Quantum part: -For each  of size /2 and  ∈  , compute OPT[, ] through Equation (3) combined with QMF.-Compute OPT[[], 0] with Equation (3) combined with QMF.

4 =
QMF [  ] on these new registers: it outputs OPT[[], 0] on the register of index  6 2 with high probability.□ Lemma 23.The complexity of Q-DDPAS is O * (| | • 1.754  ).Proof.Let us compute the complexity of this algorithm.First, we compute the complexity of the classical part.The proof of Lemma 2 shows that solving all OPT[, ] for all  of size /4 and for all  ∈  is done by DPAS in time | | () O * (2 0.811 ) (see Equation (6)), the complexity of the classical part is O * (| | • 2 0.811 ) .

•
The complexity of  recur is O * √︃ , and so is the complexity of the tensor product.The circuit  recur1 has the same complexity because of the composition with   that is polynomial.The circuit  recur finds the minimum of a function with a domain of size  /2 described by the corresponding quantum circuit  recur1 above.Thus, its complexity is O * Next, we compute  recur  ini |ini⟩ and show that OPT[[], 0] is stored in register of indexes 62 .First, we compute  ini |ini⟩.