Dichotomies for \#CSP on graphs that forbid a clique as a minor

TL;DR

This work delivers a comprehensive set of dichotomies for $\#CSP$ (and bounded-degree variants) on graph classes forbidding a fixed clique minor, revealing tractable vs. $\#P$-hard regimes determined by the signature set and the minor-closed class. It advances a unified framework that combines holographic transformations, gadget-based reductions, and matchgate methods, enabling polynomial-time algorithms on minor-free/planar-like classes and hardness proofs via structured gadget-interpolation. A key technical advance is the introduction of path gadgets and ga3-width-guided tree decompositions, which allow efficient evaluation while preserving planarity and bounded intersections. The results bridge signature-centric dichotomies with graph-class restrictions, offering a systematic template for analyzing $\#CSP$ across a broad landscape of minor-free graphs and suggesting directions for extending to sym-Hollant and more general minor-closed classes.

Abstract

We prove complexity dichotomies for \#CSP problems (not necessarily symmetric) with Boolean domain and complex range on several typical minor-closed graph classes. These dichotomies give a complete characterization of the complexity of \#CSP on graph classes that forbid a complete graph as a minor. In particular, we also demonstrate that, whether the maximum degree of vertices is bounded may influence the complexity on specific minor-closed graph classes, and this phenomenon has never been observed in the previous related studies. Furthermore, our proofs integrate the properties of each graph class with the techniques from counting complexity, and develop a systematic approach for analyzing the complexity of \#CSP on these graph classes.

Figures (7)

• Figure 1: The shallow vortex grid of order 4, denoted as $H_4$
• Figure 2: The star gadget. Each edge incident to the vertex of degree $k$, represented by a square, is also incident to a vertex of degree 2 represented by a circle.
• Figure 3: An example of the procedure of the algorithm. (a) The graph induced by $\beta(t)\cup\beta(d)$ where $t$ is a leaf and a child of $d$. $\beta(t)$ is enclosed by the red ellipse while $\beta(d)$ is enclosed by the blue ellipse. (b) The construction of the gadget $GG_{\le t}$, where the red edges represent the dangling edges in $D_{\le t}$. (c)The path gadget for $x$, where the path is colored in red and 3 vertices of degree 2 on the path are omitted. (d) Visualization of how $G_{\le t}$ is replaced by a single vertex in $G_{\le d}$. The introduced vertex is colored in red as well as the remaining parts of the path gadgets.
• Figure 4: An example of the procedure of the algorithm. (a) The graph induced by $\beta(d)\cup\beta(t)\cup\beta(l_1)\cup\beta(l_2)$. $\beta(d),\beta(t),\beta(l_1),\beta(l_2)$ are enclosed by the respective orange, green, red, and blue ellipses. Edges in $B_t$ are coloured in orange, while vertices in $A_t$ and edges in $P_t$ are coloured in green.(b) The figure of $GG_{\le t}^\sigma$. Edges belonging to $B_{l_1}$ are coloured in red while those belonging to $B_{l_2}$ are coloured in blue.(c) The figure of $HH_{\le t}^\sigma$. The introduced $v_l$ is coloured in violet. (d) The figure of $GH_{\le t}^\sigma$. The gadget $LL_{\le l}^\sigma$ is enclosed by the black ellipse and the path gadgets are coloured in violet. (e) Visualization of the fact that $B_{l_1}\cup B_{l_2}\subseteq B_t\cup P_t\cup E_{X_l}$.
• Figure 5: A ring blowup of $K_4$. $K_4$ is coloured in black.

Theorems & Definitions (82)

• Theorem 1: robertson2004graph
• Definition 2: Shallow vortex gridthilikos2022killing
• Theorem 3: thilikos2022killing
• Definition 4: $\text{\#CSP}$ and $\text{Holant}$
• Theorem 5
• Definition 6: Tree decompositionROBERTSON1986309
• Lemma 7
• Lemma 8
• Definition 9
• Remark 10