Feedback arc set problem and NP-hardness of minimum recurrent configuration problem of Chip-firing game on directed graphs
نویسندگان
چکیده
In this paper we present further studies of recurrent configurations of Chip-firing games on Eulerian directed graphs (simple digraphs), a class on the way from undirected graphs to general directed graphs. A computational problem that arises naturally from this model is to find the minimum number of chips of a recurrent configuration, which we call the minimum recurrent configuration (MINREC) problem. We point out a close relationship between MINREC and the minimum feedback arc set (MINFAS) problem on Eulerian directed graphs, and prove that both problems are NP-hard.
منابع مشابه
The halting problem for chip-firing on finite directed graphs
We consider a chip-firing game on finite directed graphs and give an answer to a question posed by Bjorner, Lovasz, and Shor in 1991: given an initial configuration of chips, when does it stabilize? The approach they took to address this halting problem involves computing a period vector p with the property that toppling the vertices according to p results in the original configuration, and the...
متن کاملOn the Hardness of Approximating Some NP-optimization Problems Related to Minimum Linear Ordering Problem
We study hardness of approximating several minimaximal and maximinimal NP-optimization problems related to the minimum linear ordering problem (MINLOP). MINLOP is to find a minimum weight acyclic tournament in a given arc-weighted complete digraph. MINLOP is APX-hard but its unweighted version is polynomial time solvable. We prove that, MIN-MAX-SUBDAG problem, which is a generalization of MINLO...
متن کاملCombinatorial algorithms for feedback problems in directed graphs
Given a weighted directed graph G = (V,A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A ⊆ A such that the directed graph (V,A \ A) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containing at least one vertex for each directed cycle. Both problems are NP-complete. We present simple co...
متن کاملOn Making a Distinguished Vertex Minimum Degree by Vertex Deletion
For directed and undirected graphs, we study the problem to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-ha...
متن کاملMinimum Feedback Sets andMulti - Cuts in Directed Graphs
This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (fvs) problem, and the weighted feedback edge set problem (fes). In the fvs (resp. fes) problem, one is given a directed graph with weights on the ver-tices (resp. edges), and is asked to nd a subset of vertices (resp. edges) with minimum total weight that int...
متن کامل