On Algorithmic Meta-Theorems for Solution Discovery: Tractability and Barriers

TL;DR

This paper develops algorithmic meta-theorems for solution discovery on graphs, where one starts from an infeasible configuration and seeks a short transformation to a valid final configuration that satisfies a given property expressible in FO, MSO$_1$, or MSO$_2$. It proves that MSO$_2$-Discovery is in XP with respect to treewidth, and MSO$_1$-Discovery is in FPT with respect to neighborhood diversity, by leveraging dynamic programming over tree or clique decompositions and flow-based reductions. The work also delineates tight hardness boundaries: FO-Discovery is W[1]-hard when parameterized by modulators to stars or paths and by twin cover, and MSO$_1$-Discovery is W[1]-hard for the bandwidth parameter, highlighting gaps in tractability for structural-parameter regimes beyond cliquewidth. Overall, the results offer near-complete fixed-parameter tractability landscapes for solution-discovery problems definable in MSO- and FO-logics, bridging Courcelle-type meta-theorems with non-monotone transformation constraints.

Abstract

Solution discovery asks whether a given (infeasible) starting configuration to a problem can be transformed into a feasible solution using a limited number of transformation steps. This paper investigates meta-theorems for solution discovery for graph problems definable in monadic second-order logic (MSO$_1$ and MSO$_2$) and first-order logic (FO) where the transformation step is to slide a token to an adjacent vertex, focusing on parameterized complexity and structural graph parameters that do not involve the transformation budget $b$. We present both positive and negative results. On the algorithmic side, we prove that MSO$_2$-Discovery is in XP when parameterized by treewidth and that MSO$_1$-Discovery is fixed-parameter tractable when parameterized by neighborhood diversity. On the hardness side, we establish that FO-Discovery is W[1]-hard when parameterized by modulator to stars, modulator to paths, as well as twin cover, numbers. Additionally, we prove that MSO$_1$-Discovery is W[1]-hard when parameterized by bandwidth. These results complement the straightforward observation that solution discovery for the studied problems is fixed-parameter tractable when the budget $b$ is included in the parameter (in particular, parameterized by cliquewidth$+b$, where the cliquewidth of a graph is at most any of the studied parameters), and provide a near-complete (fixed-parameter tractability) meta-theorems investigation for solution discovery problems for MSO- and FO-definable graph problems and structural parameters larger than cliquewidth.

Figures (10)

• Figure 1: Our MSO$_1$ and results: beige boxes indicate tractability of MSO$_1$-Discovery with respect to the corresponding parameter, gray boxes indicate hardness of -Discovery, and white boxes indicate open problems. For parameter bandwidth, we have a hardness result only for MSO$_1$-Discovery; the complexity of -Discovery remains an open problem. An arrow between two boxes indicates the existence of a function in the parameter above that is a lower bound on the parameter below.
• Figure 2: A star vertex-gadget for vertex $v \in V_3$ with $\iota(v)=x$, for an instance with $\kappa=4$, the vertex root nodes for $u, w \in V_3$, and the vertex block node $b_3$.
• Figure 3: A star edge-gadget representing $e = uv$ where $u \in V_i$ and $v \in V_j$ with $\iota(u) = x$ and $\iota(v) = y$, the edge root nodes for $uw$ and $ab$ (edges between $V_i$ and $V_j$), and the edge block node $b_{i,j}$. The edge-gadget for $uv$ is formed of $2n$ leaf nodes, $n$ corresponding to $u$ and $n$ corresponding to $v$.
• Figure 4: A path vertex-gadget for vertex $v \in V_3$ with $\iota(v)=x$, for an instance with $\kappa=4$. The vertex block node $b_3$ is adjacent to all nodes of path vertex-gadgets for all vertices in $V_3$, but it is omitted from the figure for clarity.
• Figure 5: A path edge-gadget for edge $e=uv$ where $u \in V_i$ and $v \in V_j$ with $\iota(u) =x$ and $\iota(v)=y$. It is formed of $2n$ nodes, $n$ corresponding to $u$ and $n$ corresponding to $v$. The edge block node $b_{i,j}$ is adjacent to all nodes of path vertex-gadgets for all edges between $V_i$ and $V_j$, but it is omitted from the figure for clarity.

Theorems & Definitions (58)

• Theorem 2.1
• Theorem 2.1
• Theorem 2.1
• Theorem 2.1
• Theorem 2.1
• Theorem 2.1
• Theorem 4.1: courcelle1990monadic
• Theorem 4.2: courcelle2000linearDBLP:conf/stoc/Korhonen024
• Theorem 4.3
• proof