Edge Matching with Inequalities, Triangles, Unknown Shape, and Two Players

Abstract

We analyze the computational complexity of several new variants of edge-matching puzzles. First we analyze inequality (instead of equality) constraints between adjacent tiles, proving the problem NP-complete for strict inequalities but polynomial-time solvable for nonstrict inequalities. Second we analyze three types of triangular edge matching, of which one is polynomial-time solvable and the other two are NP-complete; all three are #P-complete. Third we analyze the case where no target shape is specified and we merely want to place the (square) tiles so that edges match exactly; this problem is NP-complete. Fourth we consider four 2-player games based on 1×n edge matching, all four of which are PSPACE-complete. Most of our NP-hardness reductions are parsimonious, newly proving #P and ASP-completeness for, e.g., 1 × n edge matching. Along the way, we prove #P-and ASP-completeness of planar 3-regular directed Hamiltonicity; we provide linear-time algorithms to find antidirected and forbidden-transition Eulerian paths; and we characterize the complexity of new partizan variants of the Geography game on graphs.

Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography

“…Both the edge variant [17] and short version [27] of Undirected Geography are also shown to PSPACE-complete. Various extensions to Undirected Geography has been analyzed [38,5,25].…”

“…We will use the notation standard in combinatorial game theory for Grundy values: * k for k, except that * is shorthand for * 1 and 0 is shorthand for * 0. 5 By grouping all positions with non-zero Grundy values into a class called "Fuzzy", impartial game positions can be partitioned into two outcome classes, characterizing winnability. (1) N ("Fuzzy") -with positive Grundy values; the current (next) player always has a winning strategy.…”