We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of $k$-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an $n^{f(t,s)}$-time algorithm to compute modulo $2^t$ the number of subgraph occurrences of patterns that are $s$ vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo $2^t$. Complementing our algorithm, we also give a simple and self-contained proof that counting $k$-matchings modulo odd integers $q$ is Mod_q-W[1]-complete and prove that counting $k$-paths modulo $2$ is Parity-W[1]-complete, answering an open question by Bj\"orklund, Dell, and Husfeldt (ICALP 2015).
翻译:我们系统地调查计算子图案模版固定整数的复杂性。 例如, 已知美元对齐数的等值可以通过简单减少决定因素来在多式时间中确定。 我们将这一数值概括为 $@f( t, s) 美元- 美元- 时间算法, 用来计算 modulo 2 美元- 美元- 美元- 美元- 美元[ 1- 完整, 并证明 美元- 方位 2 美元的计数是 Pity- W1- 完整的, 回答 Bj\\ orklund、 Dell 和 Husfeldt ( CRIal- 2015) 的公开问题 。