In this paper we study dynamic averaging load balancing on general graphs. We consider infinite time and dynamic processes, where in every step new load items are assigned to randomly chosen nodes. A matching is chosen, and the load is averaged over the edges of that matching. We analyze the discrete case where load items are indivisible, moreover our results also carry over to the continuous case where load items can be split arbitrarily. For the choice of the matchings we consider three different models, random matchings of linear size, random matchings containing only single edges, and deterministic sequences of matchings covering the whole graph. We bound the discrepancy, which is defined as the difference between the maximum and the minimum load. Our results cover a broad range of graph classes and, to the best of our knowledge, our analysis is the first result for discrete and dynamic averaging load balancing processes. As our main technical contribution we develop a drift result that allows us to apply techniques based on the effective resistance in an electrical network to the setting of dynamic load balancing.

Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen intense study towards near-term applications on quantum hardware. A crucial parameter for VQAs is the depth of the variational ansatz used - the smaller the depth, the more amenable the ansatz is to near-term quantum hardware in that it gives the circuit a chance to be fully executed before the system decoheres. This potential for depth reduction has made VQAs a staple of Noisy Intermediate-Scale Quantum (NISQ)-era research. In this work, we show that approximating the optimal depth for a given VQA ansatz is intractable. Formally, we show that for any constant ϵ>0, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor N¹⁻ϵ, for N denoting the encoding size of the VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum generalization of NP.) We then show that this hardness persists even in the "simpler" setting of QAOAs. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems. To achieve these results, we bypass the need for a PCP theorem for QCMA by appealing to the disperser-based NP-hardness of approximation construction of [Umans, FOCS 1999].

We study the fundamental scheduling problem 1 | rj | wjUj: schedule a set of n jobs with weights, processing times, release dates, and due dates on a single machine, such that each job starts after its release date and we maximize the weighted number of jobs that complete execution before their due date. Problem 1 | rj | wjUj generalizes both Knapsack and Partition, and the simplified setting without release dates was studied by Hermelin et al. [Annals of Operations Research, 2021] from a parameterized complexity viewpoint. Our main contribution is a thorough complexity analysis of 1 | rj | wjUj in terms of four key problem parameters: the number p# of processing times, the number w# of weights, the number d# of due dates, and the number r# of release dates of the jobs. 1 | rj | wjUj is known to be weakly para-NP-hard even if w# + d# + r# is constant, and Heeger and Hermelin [ESA, 2024] recently showed (weak) W[1]-hardness parameterized by p# or w# even if r# is constant. Algorithmically, we show that 1 | rj | wjUj is fixed-parameter tractable parameterized by p# combined with any two of the remaining three parameters w#, d#, and r#. We further provide pseudo-polynomial XP-time algorithms for parameter r# and d#. To complement these algorithms, we show that 1 | rj | wjUj is (strongly) W[1]-hard when parameterized by d# +r# even if w# is constant. Our results provide a nearly complete picture of the complexity of 1 | rj | wjUj for p#, w#, d#, and r# as parameters, and extend those of Hermelin et al. [Annals of Operations Research, 2021] for the problem 1 || wjUj without release dates.