Complexity lower bounds for succinct binary structures of bounded clique-width with restrictions

TL;DR

A Rice-like complexity lower bound is presented for any MSO-definable problem on binary structures succinctly encoded by circuits, as long as $\psi$ is non-trivial on structures satisfying bounded clique-width.

Abstract

We present a Rice-like complexity lower bound for any MSO-definable problem on binary structures succinctly encoded by circuits. This work extends the framework recently developed as a counterpoint to Courcelle's theorem for graphs encoded by circuits, in two interplaying directions: (1) by allowing multiple binary relations, and (2) by restricting the interpretation of new symbols. Depending on the pair of an MSO problem $ψ$ and an MSO restriction $χ$, the problem is proven to be NP-hard or coNP-hard or P-hard, as long as $ψ$ is non-trivial on structures satisfying $χ$ with bounded clique-width. Indeed, there are P-complete problems (for logspace reductions) in our extended context. Finally, we strengthen a previous result on the necessity to parameterize the notion of non-triviality, hence supporting the choice of clique-width.

Figures (1)

• Figure 1: Clique-decomposition (left) of a graph (right). Color $1$ is in black and color $2$ is in red. We denote $f_{2\to 1}$ the function such that $f_{2\to 1}(2) = 1$ and $f_{2\to 1}(i) = i$ for any $i \ne 1$. The width of this decomposition is $2$. There is only one arrow label, so we do not mention it in the $\mathsf{join}_M$ operations, such that we can consider $M \subseteq \{\mathsf{left},\mathsf{right}\} \times C^2$, and neither in the $\mathsf{constant}^{\circ}$ operation where we use $\circ$ to mention the loop with the only considered arrow label.

Theorems & Definitions (48)

• Theorem 1: gglgopt25
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1
• Remark 2
• Definition 1
• Remark 3
• Lemma 5: libkin2004elements
• Remark 4