The Computational Complexity of Factored Graphs

TL;DR

The paper introduces factored graphs, a succinct representation of graphs built from smaller components via graph operations, and analyzes problems under a parameterized framework with parameter $k$ (the number of leaves). It proves an XP-hardness/XP-completeness dichotomy: LFMIS on factored graphs is $ ext{XP}$-complete while counting small cliques is fixed-parameter tractable in $k$; reachability on factored graphs is $ ext{XNL}$-complete and tightly connected to a central open problem about NL vs. polynomial-time containment. The work develops technical reductions and factored constructions that simulate Turing machine computations, count subgraphs through dimension-based decompositions and inclusion–exclusion, and encode configuration graphs via tensor/Cartesian product-based grids with locality properties. The results illustrate that succinct representations can both hinder and help algorithmic performance depending on the problem, and they connect centralized complexity questions to parameterized questions on compressed inputs. The paper thus advances understanding of how structural succinctness affects graph problems and lays groundwork for further exploration of factored representations and their algorithmic consequences.

Abstract

While graphs and abstract data structures can be large and complex, practical instances are often regular or highly structured. If the instance has sufficient structure, we might hope to compress the object into a more succinct representation. An efficient algorithm (with respect to the compressed input size) could then lead to more efficient computations than algorithms taking the explicit, uncompressed object as input. This leads to a natural question: when does knowing the input instance has a more succinct representation make computation easier? We initiate the study of the computational complexity of problems on factored graphs: graphs that are given as a formula of products and unions on smaller graphs. For any graph problem, we define a parameterized version that takes factored graphs as input, parameterized by the number of (smaller) ordinary graphs used to construct the factored graph. In this setting, we characterize the parameterized complexity of several natural graph problems, exhibiting a variety of complexities. We show that a decision version of lexicographically first maximal independent set is $\mathbf{XP}$-complete, and therefore unconditionally not fixed-parameter tractable ($\mathbf{FPT}$). On the other hand, we show that clique counting is $\mathbf{FPT}$. Finally, we show that reachability is $\mathbf{XNL}$-complete. Moreover, $\mathbf{XNL}$ is contained in $\mathbf{FPT}$ if and only if $\mathbf{NL}$ is contained in some fixed polynomial time.

Figures (13)

• Figure 2: Overview of the Factorization of $G$. Supernodes are enclosed within dotted circles.
• Figure 3: Tree structure of the factored graph $G = ((A \times B) \cup C ) \, \square \, (D \times (E \cup F))$.
• Figure 5: Example partition for $\pi$ where $b = 3, k = 3$. The black edges belong to $\pi_0$, red edges belong to $\pi_1$, and pink edges belong to $\pi_2$.
• Figure 6: Graph Factors $A$, $B$, and $C$.
• Figure 7: Grids $G_V, G_H, G_R,$ and $G_L$ (from left to right).

Theorems & Definitions (83)

• Theorem 1.1: $\xp$-completeness of LFMIS
• Theorem 1.2: Counting Clique Subgraphs is in $\fpt$
• Theorem 1.3: $\xnl$-completeness of Reachability
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7: Factored Graph Tree Structure