Collaborative usage of resources is becoming increasingly popular in various fields. One common example are coworking spaces — office rooms with work places that can be rented by individuals on hourly basis. We consider the problem of assigning all booking requests for a day to equivalent office rooms with different but fixed opening times and fixed interchangeable closing times. The closing times are flexible due to daily maintenance, e.g. cleaning, which must be done in all rooms in an arbitrary order. This problem is related to the known Interval Scheduling Prob-lem with Machine Availabilities (ISMA), where each machine has a contiguous availability interval, and each job presents a specific time interval which has to be scheduled. According to our cowork-ing scheduling application, we extend ISMA to Flexible Multithread ISMA (FlexMISMA) by introducing machine capacities that model the number of work places per room and by allowing to permute the end times of machines’ availability periods. In this paper, we determine a tight classification of necessary conditions for the existence of a polynomial time algorithm for FlexMISMA, assuming P ≠ NP . More specifically, we develop a network flow model and present polynomial time algorithms for instances (i) with two machines, and (ii) with arbitrarily many machines of capacity one each. In the same time, we prove that increasing the machine capacity to two renders FlexMISMA NP -hard for arbitrarily many machines. Furthermore, we complement result (i) by showing that the problem is NP -hard already for instances with three machines as a special case of the Vertex-Disjoint

The minimum color-degree perfectb-matching problem (Col-BM) is a new extension of the perfectb-matching problem to edge-colored graphs. The objective of Col-BM is to minimize the maximum number of differently colored edges in a perfectb-matching that are incident to the same node. We show that Col-BM isNP-hard on bipartite graphs by a reduction from (3,B2)-Sat, and conclude that there exists no(2 - epsilon)-approximation algorithm unlessP=NP. However, we identify a class of two-colored complete bipartite graphs on which we can solve Col-BM in polynomial time. Furthermore, we use dynamic programming to devise polynomial-time algorithms solving Col-BM with a fixed number of colors on series-parallel graphs and simple graphs with bounded treewidth.

In the known Interval Scheduling Problem with Machine Availabilities (ISMA), each machine has a contiguous availability interval and each job has a specific time interval which has to be scheduled. The objective is to schedule all jobs such that the machines’ availability intervals are respected or to decide that there exists no such schedule. We extend ISMA by introducing machine capacities to model parallel processing of multiple jobs per machine: the Multi-thread Interval Scheduling with Machine Availabilities (MISMA). In machine scheduling, maintenance plays a crucial role in guaranteeing an efficient operation. The time slots for maintenance at the end of a processing period are often predetermined by staff schedules before the slots are assigned to specific machines. This motivates a variant of MISMA where the end times of the machines’ availability intervals can be permuted, the Flexible Multithread ISMA (FlexMISMA). In this paper, we determine a tight classification of conditions that are required for obtaining a polynomial time algorithm for both MISMA and FlexMISMA. More specifically, we show that FlexMISMA is at least as hard as MISMA. For FlexMISMA, we present polynomial time algorithms for instances (i) with two machines, and (ii) with constantly many parallel jobs at each point in time which both also solve MISMA; and (iii) with arbitrarily many machines of capacity one each in which case MISMA is known to be NP-hard. However, we prove that increasing the capacity to two renders FlexMISMA also NP-hard for arbitrarily many machines. Furthermore, we complement result (i) by showing that both problems are NP-hard already for instances with three machines as a special case of the Vertex-Disjoint Paths problem.