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 brandnew 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 (MinSFC/MaxSFC)
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 MinSFC and MaxSFC are NPhard;
in particular, MinSFC is
(I) strongly NPhard 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) NPhard 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.
BibTeX  Entry
@InProceedings{hayakawa_et_al:LIPIcs:2017:8235,
author = {Hitoshi Hayakawa and Toshimasa Ishii and Hirotaka Ono and Yushi Uno},
title = {{Settlement Fund Circulation Problem}},
booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)},
pages = {46:146:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770545},
ISSN = {18688969},
year = {2017},
volume = {92},
editor = {Yoshio Okamoto and Takeshi Tokuyama},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/8235},
URN = {urn:nbn:de:0030drops82351},
doi = {10.4230/LIPIcs.ISAAC.2017.46},
annote = {Keywords: Fund settlement, Algorithm, Digraph, Scheduling}
}
Keywords: 

Fund settlement, Algorithm, Digraph, Scheduling 
Collection: 

28th International Symposium on Algorithms and Computation (ISAAC 2017) 
Issue Date: 

2017 
Date of publication: 

07.12.2017 