On the complexity of the chip-firing reachability problem

Abstract

In this paper, we study the complexity of the chip-firing reachability problem. We show that for Eulerian digraphs, the reachability problem can be decided in strongly polynomial time, even if the digraph has multiple edges. We also show a special case when the reachability problem can be decided in polynomial time for general digraphs: if the target distribution is recurrent restricted to each strongly connected component. As a further positive result, we show that the chip-firing reachability problem is in co-NP for general digraphs. We also show that the chip-firing halting problem is in co-NP for Eulerian digraphs.

Algorithmic aspects of rotor-routing and the notion of linear equivalence

“…However, as the following proposition shows, if the target configuration is recurrent, the reachability problem is decidable in polynomial time. This result is an analogue of a recent result for the chip-firing game [8]. The proof is also a complete analogue.…”

“…As (y, ̺ 2 ) ∼ (x 1 , ̺ 1 ), we have (x 1 , ̺ 1 ) ∼ (x 2 , ̺ 2 ) if and only if (y, ̺ 2 ) ∼ (x 2 , ̺ 2 ), which by Lemma 3.5 is equivalent to y ∼ x 2 . This can be checked in polynomial time using Gaussian elimination, then solving a system of linear congruence equations (see also [8,Proposition 8]).…”

Rotor-routing reachability is easy, chip-firing reachability is hard

“…In this paper, we study the complexity of the reachability problem of chipfiring and rotor-routing. Previously, the chip-firing reachability problem was shown to be in co − NP [8], and in the special case of polynomial period length (which includes for example Eulerian digraphs), it was shown to be in P [8,12]. Here we show that in general the chip-firing reachability problem is hard: if it were solvable in polynomial time, then the polynomial hierarchy would collapse to NP.…”

Chip-firing based methods in the Riemann–Roch theory of directed graphs

“…Recurrent chip-distributions form an important subset of non-terminating chipdistributions. They are studied in the literature (see [11,14,12] and also [15] in a slightly different model). Here we mention some of their properties needed for the rest of the paper.…”