Abstract
In the economic activities, the central bank has an important role to cover payments of banks, when they are short of funds to clear their debts. For this purpose, the central bank timely puts funds so that the economic activities go smooth. Since payments in this mechanism are processed sequentially, the total amount of funds put by the central bank critically depends on the order of the payments. Then an interest goes to the amount to prepare if the order of the payments can be controlled by the central bank, or if it is determined under the worst case scenario. This motivates us to introduce a brand-new problem, which we call the settlement fund circulation problem. The problems are formulated as follows: Let G=(V,A) be a directed multigraph with a vertex set V and an arc set A. Each arc a∈A is endowed debt d(a)≥0, and the debts are settled sequentially under a sequence π of arcs. Each vertex v∈V is put fund in the amount of pπ(v)≥0 under the sequence. The minimum/maximum settlement fund circulation problem (Min-SFC/Max-SFC) in a given graph G with debts d:A→R+∪{0} asks to find a bijection π:A→{1,2,…,|A|} that minimizes/maximizes the total funds ∑v∈Vpπ(v). In this paper, we show that both Min-SFC and Max-SFC are NP-hard; in particular, Min-SFC is (I) strongly NP-hard even if G is (i) a multigraph with |V|=2 or (ii) a simple graph with treewidth at most two, and is (II) (not necessarily strongly) NP-hard for simple trees of diameter four, while it is solvable in polynomial time for stars. Also, we identify several polynomial time solvable cases for both problems.
Highlights
It is one of the reasons that debts among banks are cleared in a special system, called interbank settlement system, in which the central bank supports cash management of the banks
We describe our problem by a digraph whose nodes are banks and arcs are loan relationship from one bank to another together with debts as arc weights
In the subsequent two sections (Sections 3 and 4), we discuss about minimum settlement fund circulation problem (MinSFC), which is our main interest in the context of analyzing settlements of debts
Summary
If the debtor fails to prepare cash for the payment until the deadline, it will go bankrupt. Such bankruptcy should be avoided when it could cause significant damage to the economy, and it is true for the case of banks since their debts are highly interconnected each other and bankruptcy of a bank may cause chain reaction of bankruptcy. Cash held by the central bank is used as the fund for the payments. Note that we assume each debt has to be cleared independently and “sequentially”, that is, it is not allowed to cancel out payments; A pays 30 directly to C, and the rest 20 to B, for example.
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