Comments on ECCC Report TR06-133: The Resolution Width Problem is EXPTIME-Complete

The main argument of the report TR06-133 is in error. The paper claims to prove the result of the title by reduction from the (∃, k)-pebble game, shown to be EXPT IME-complete by Kolaitis and Panttaja. This note shows that the principal lemma is incorrect by providing a simple counterexample. 1 The counter-example The main theorem of the paper depends on the following claim, stated as Lemma 5.3: “If A and B are coloured graphs, and k ≥ 3, then the Spoiler has a winning strategy for the (∃, k)-pebble game on A and B if and only if the Prover has a winning strategy for the k + 2-width game on Σ(A,B).” The counter-example that follows shows that this claim is incorrect. We provide two graphs A and B, for which the Prover has a winning strategy for the 5-width game on Σ(A,B), but on the other hand, the Spoiler does not have a winning strategy for the (∃, 3)-pebble game on A and B. The graphs A and B are easy to describe. The graph A is the complete graph K5, and the graph B is the complete graph K4. The colours of the nodes in the graphs play no role (we can think of all the nodes as being coloured the same colour), so we shall ignore them in the remainder of this note. Research supported by the Natural Sciences and Engineering Research Council of Canada. Research supported by NSERC grant -9700239. 1 Electronic Colloquium on Computational Complexity, Report No. 3 (2009)