We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on $4$ vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus\mathrm{P}$-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; G\"obel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most $3$, as well as a full classification for the problem of counting list homomorphisms, modulo $2$.
翻译:我们研究从输入图G$到固定图H$的同质体数的等值计算问题。 Faben和Jerrum[ToC'15]在图形$H$上引入了一个明确的标准,并推测,如果满足,问题在多元时间中是可以解决的,否则,问题就在于复杂的等级$opl\mathrm{P}对等问题。我们核查所有图表的假定值$H$,其中排除了4美元顶点的完整图的最小值。此外,我们排除了美元+/mathrm{P}在图形上存在一个额外时间算法,假设如果满足,这个问题在多元时间中是可以解决的,否则,问题就在于这个复杂类别。我们的证据提出了一种新颖的方法,从固定图$H$的全球定义的子结构中得出硬性。使用这个方法,我们吸收了所有先前在解析图(FaCT和Jerrum [ToC'15] 上的所有进展。此外,我们排除了美元“G\bel”和“Richstal dal ligal ligal ligal cal deal ex cas degal degal degal cas pas ex) ex a ex a ex ex ex excience, ex, ex ex expalbilvealmentaldald ex a ex a ex, ex a ex.