2023-03-23T21:30:48Zhttps://eprints.lib.hokudai.ac.jp/dspace-oai/requestoai:eprints.lib.hokudai.ac.jp:2115/824342022-11-17T02:08:08Zhdl_2115_20038hdl_2115_123Settlement fund circulation problemHayakawa, HitoshiIshii, ToshimasaOno, HirotakaUno, YushiFund settlementAlgorithmDigraphScheduling330In 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 is an element of A is endowed debt d(a) >= 0, and the debts are settled sequentially under a sequence pi of arcs. Each vertex V is an element of V is put fund in the amount of p(pi)(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+ U {0} asks to find a bijection pi : A -> {1,2, ..., vertical bar A vertical bar} that minimizes/maximizes the total funds Sigma(v is an element of 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 vertical bar V vertical bar = 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. (C) 2019 Elsevier B.V. All rights reserved.ElsevierJournal Articleapplication/pdfhttp://hdl.handle.net/2115/82434https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/82434/1/Settlement%20fund%20circulation%20problem.pdf0166-218XDiscrete applied mathematics265861032019-07-31enginfo:doi/10.1016/j.dam.2019.03.017©2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/author