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\in A is endowed debt d(a)\ge 0, and the debts are settled sequentially under a sequence \pi of arcs. Each vertex v\in V is put fund in the amount of p_{\pi}(v)\ge 0 under the sequence. The minimum/maximum settlement fund circulation problem (Min-SFC/Max-SFC) in a given graph G with debts d: A\rightarrow \mathbb{R}_{+}\cup \{0\} asks to find a bijection \pi:A\to \{1,2,\dots,|A|\} that minimizes/maximizes the total funds \sum _{v\in V}p_{\pi }(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.

Abstract Given an undirected multigraphG= (V,E) and a requirement functionrλ: ( ) →Z+(where ( ) is the set of all pairs of vertices andZ+is the set of nonnegative integers), we consider the problem of augmentingGby the smallest number of new edges so that the local edge‐connectivity and vertex‐connectivity between every pairx,y∈Vbecome at leastrλ(x,y) and two, respectively. In this paper, we show that the problem can be solved inO(n3(m+n) log(n2/(m+n))) time, wherenandmare the numbers of vertices and pairs of adjacent vertices inG, respectively. This time complexity can be improved toO((nm+n2logn) logn), in the case of the uniform requirementrλ(x,y)= 𝓁 for allx,y∈V. Furthermore, for the generalrλ, we show that the augmentation problem that preserves the simplicity of the resulting graph can be solved in polynomial time for any fixed 𝓁*= max{rλ(x,y) |x,y∈V}. © 2001 John Wiley & Sons, Inc.