Computational complexity is traditionally measured with respect to input size. For graphs, this is typically the number of vertices (or edges) of the graph. However, for large graphs even explicitly representing the graph could be prohibitively expensive. Instead, graphs with enough structure could admit more succinct representations. A number of previous works have considered various succinct representations of graphs, such as small circuits [Galperin, Wigderson '83]. We initiate the study of the computational complexity of problems on factored graphs: graphs that are given as a formula of products and union on smaller graphs. For any graph problem, we define a parameterized version by the number of operations used to construct the graph. For different graph problems, we show that the corresponding parameterized problems have a wide range of complexities that are also quite different from most parameterized problems. We give a natural example of a parameterized problem that is unconditionally not fixed parameter tractable (FPT). On the other hand, we show that subgraph counting is FPT. Finally, we show that reachability for factored graphs is FPT if and only if $\mathbf{NL}$ is in some fixed polynomial time.

翻译：计算复杂性传统上是相对于输入规模来度量的。对于图而言，这通常是图的顶点（或边）的数量。然而，对于大型图而言，即使是显式地表示图也可能代价过高。相反，具有足够结构的图可能允许更简洁的表示。先前的一些工作已经考虑了各种图的简洁表示，例如小型电路[Galperin, Wigderson '83]。我们开创了对因子图上问题计算复杂性的研究：因子图是以较小图的乘积与并运算公式形式给出的图。对于任何图问题，我们通过用于构造图的运算数量来定义一个参数化版本。对于不同的图问题，我们证明了相应的参数化问题具有广泛的复杂性范围，且与大多数参数化问题有显著差异。我们给出了一个参数化问题的自然示例，该问题无条件地不属于固定参数可处理(FPT)类。另一方面，我们证明了子图计数问题是FPT的。最后，我们证明了因子图的可达性问题属于FPT当且仅当$\mathbf{NL}$包含于某个固定的多项式时间内。