## 1 Introduction  
The mutual exclusion problem is a fundamental process synchronization problem in concurrent systems [1; 2; 3]. It is the problem of controlling the system in such a way that no two processes execute their critical sections (CSs) at a time. Generalizations of mutual exclusion have been studied extensively, e.g., \(k\)-mutual exclusion [4; 5; 6; 7; 8; 9], mutual inclusion [10] and \(l\)-mutual inclusion [11]. The \(k\)-mutual exclusion problem is controlling the system in such a way that at most \(k\) processes can execute their CSs at a time. The mutual inclusion problem is the complement of the mutual exclusion problem; unlike mutual exclusion, where at most one process is in the CS, mutual inclusion places at least one process in the CS. In a similar way, the \(l\)-mutual inclusion problem is the complement of the \(k\)-mutual exclusion problem; unlike \(k\)-mutual exclusion, where at most \(k\) processes are in the CSs, \(l\)-mutual inclusion places at least \(l\) processes in the CSs. These generalizations are unified to a framework \"the critical section problem\" in [12]. Informally, the global \((l,k)\)-CS problem is defined as follows. For each \(0\leq l<k\leq n\), the global \((l,k)\)-CS problem has at least \(l\) and at most \(k\) processes in the CSs in the entire network. This paper discusses the generalized local CS problem, which is a new version of CS problems. When the numbers \(l_{i}\) and \(k_{i}\) are given for each process \(P_{i}\), it is the problem of controlling the system in such a way that the number of processes that can execute their CSs at a time is at least \(l_{i}\) and at most \(k_{i}\) among its neighbors and itself. In this case, we call this problem \"the generalized local \((l_{i},k_{i})\)-critical section problem\". Note that the local \((l,k)\)-CS problem assumes that the values of \(l\) and \(k\) are shared among all processes in the network, whereas the generalized local CS problem assumes that the values of \(l_{i}\) and \(k_{i}\) are set for each process \(P_{i}\). These are the generalizations of local mutual exclusion [13; 14; 15; 16; 17], local \(k\)-mutual exclusion [18] and local mutual inclusion [11]. If every process has \((0,1)\), then the problem is the local mutual exclusion problem. If every process has \((0,k)\), then the problem is the local \(k\)-mutual exclusion. If every process has \((1,|N_{i}|+1)\), then the problem is the local mutual inclusion problem, where \(N_{i}\) is the set of \(P_{i}\)'s neighboring processes. The global CS problem is a special case of the local CS problem when the network topology is complete. However, to the best of our knowledge, our algorithm in this paper is the first solution for the generalized local \((l_{i},k_{i})\)-CS problem. The generalized local \((l_{i},k_{i})\)-CS problem is interesting not only theoretically, but also practically, because it is useful for fault-tolerance and load balancing of distributed systems. For example, we can consider the following future applications. * One application is a formulation of the dynamic invocation of servers for the load balancing. The minimum number of servers that are always invoked for quick responses to requests for \(P_{i}\) is \(l_{i}\). The number of servers is dynamically changed by the system load. However, the total number of servers is limited by available resources like bandwidth for \(P_{i}\), and the number is \(k_{i}\). * Another is fault-tolerance services if each process in the CS provides a service for the network. Because every process has direct access to at least \(l_{i}\) servers, it guarantees fault-tolerant services. However, because providing services involve a significant cost, the number of servers should be limited at most \(k_{i}\) for each process. * The other is that each process in the CS provides service \(A\), and other processes provide service \(B\) for the network. Then, every process in the network has direct access to at least \(l_{i}\) servers of \(A\) and has direct access to at least \(|N_{i}|+1-k_{i}\) servers of \(B\).