In this paper, we explore the limits of algorithmic meta-theorems for combinatorial reconfiguration on graphs and prove several intractability results for highly restricted cases, which tightly complement the positive results by Mouawad et al. [IPEC 2014] and Gima et al. [Algorithmica 2024]. In this setting, we study reconfiguration problems on graphs in which the feasible sets are defined by formulas of first-order or monadic second-order logic: for a formula φ(X) with a free set variable X, the problem asks whether two given sets are connected by a token-jumping sequence in which every set satisfies φ on the input graph. Our main contribution is to show that the problem is intractable even for first-order logic and for severely restricted graphs, such as paths and disjoint unions of stars or cliques. Combined with known results, these results settle the parameterized complexity for most of the well-studied structural parameters. We also study the setting where the sets to be reconfigured are small, i.e., their size is part of the parameter, and show that even in this setting the problem is hard for caterpillars, whereas it becomes tractable even for monadic second-order logic when parameterized additionally by shrub-depth.

Lipschitz continuity of algorithms, introduced by Kumabe and Yoshida (FOCS'23), measures the stability of an algorithm against small input perturbations. Algorithms with small Lipschitz continuity are desirable, as they ensure reliable decision-making and reproducible scientific research. Several studies have proposed Lipschitz continuous algorithms for various combinatorial optimization problems, but these algorithms are problem-specific, requiring a separate design for each problem. To address this issue, we provide the first algorithmic meta-theorem in the field of Lipschitz continuous algorithms. Our result can be seen as a Lipschitz continuous analogue of Courcelle’s theorem, which offers Lipschitz continuous algorithms for problems on bounded-treewidth graphs. Specifically, we consider the problem of finding a vertex set in a graph that maximizes or minimizes the total weight, subject to constraints expressed in monadic second-order logic (MSO₂). We show that for any ε > 0, there exists a (1±ε)-approximation algorithm for the problem with a polylogarithmic Lipschitz constant on bounded treewidth graphs. On such graphs, our result outperforms most existing Lipschitz continuous algorithms in terms of approximability and/or Lipschitz continuity. Further, we provide similar results for problems on bounded-clique-width graphs subject to constraints expressed in MSO₁. Additionally, we construct a Lipschitz continuous version of Baker’s decomposition using our meta-theorem as a subroutine.

Given a graph G = (V,E), a set T of vertex pairs, and an integer k, Hitting Geodesic Intervals asks whether there is a set S ⊆ V of size at most k such that for each terminal pair {u,v} ∈ T, the set S intersects at least one shortest u-v path. Aravind and Saxena [WALCOM 2024] introduced this problem and showed several parameterized complexity results. In this paper, we extend the known results in both negative and positive directions and present sharp complexity contrasts with respect to structural graph parameters. We first show that the problem is NP-complete even on graphs with highly restricted shortest-path structures. More precisely, we show the NP-completeness on graphs obtained by adding a single vertex to a disjoint union of 5-vertex paths. By modifying the proof of this result, we also show the NP-completeness on graphs obtained from a path by adding one vertex and on graphs obtained from a disjoint union of triangles by adding one universal vertex. Furthermore, we show the NP-completeness on graphs of bandwidth 4 and maximum degree 5 by replacing the universal vertex in the last case with a long path. Under standard complexity assumptions, these negative results rule out fixed-parameter algorithms for most of the structural parameters studied in the literature (if the solution size k is not part of the parameter). We next present fixed-parameter algorithms parameterized by k plus modular-width and by k plus vertex integrity. The algorithm for the latter case does indeed solve a more general setting that includes the parameterization by the minimum vertex multiway-cut size of the terminal vertices. We show that this is tight in the sense that the problem parameterized by the minimum vertex multicut size of the terminal pairs is W[2]-complete. We then modify the proof of this intractability result and show that the problem is W[2]-complete parameterized by k even in the setting where T = binom(Q,2) for some Q ⊆ V.

In the Telephone Broadcast problem we are given a graph G = (V,E) with a designated source vertex s ∈ V. Our goal is to transmit a message, which is initially known only to s, to all vertices of the graph by using a process where in each round an informed vertex may transmit the message to one of its uninformed neighbors. The optimization objective is to minimize the number of rounds. Following up on several recent works, we investigate the structurally parameterized complexity of Telephone Broadcast. In particular, we first strengthen existing NP-hardness results by showing that the problem remains NP-complete on graphs of bounded tree-depth and also on cactus graphs which are one vertex deletion away from being path forests. Motivated by this (severe) hardness, we study several other parameterizations of the problem and obtain FPT algorithms parameterized by vertex integrity (generalizing a recent FPT algorithm parameterized by vertex cover by Fomin, Fraigniaud, and Golovach [TCS 2024]) and by distance to clique, as well as FPT approximation algorithms parameterized by clique-cover and cluster vertex deletion. Furthermore, we obtain structural results that relate the length of the optimal broadcast protocol of a graph G with its pathwidth and tree-depth. By presenting a substantial improvement over the best previously known bound for pathwidth (Aminian, Kamali, Seyed-Javadi, and Sumedha [ICALP 2025]) we exponentially improve the approximation ratio achievable in polynomial time on graphs of bounded pathwidth from 𝒪(4^pw) to 𝒪(pw).

Abstract Given a graph G and an integer b, Bandwidth asks whether there exists a bijection $$\pi $$ π from V(G) to $$\{1, \ldots , |V(G)|\}$$ { 1 , … , | V ( G ) | } such that $$\max _{\{u, v \} \in E(G)} | \pi (u) - \pi (v) | \le b$$ max { u , v } ∈ E ( G ) | π ( u ) - π ( v ) | ≤ b . This is a classical NP-complete problem, known to remain NP-complete even on very restricted classes of graphs, such as trees of maximum degree 3 and caterpillars of hair length 3. In the realm of parameterized complexity, these results imply that the problem remains NP-hard on graphs of bounded pathwidth, while it is additionally known to be W[1]-hard when parameterized by the tree-depth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number. In this paper we make progress in understanding the parameterized (in)tractability of Bandwidth. We first show that it is FPT when parameterized by the cluster vertex deletion number cvd plus the clique number $$\omega $$ ω , thus significantly strengthening the previously mentioned result for vertex cover number. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results develop and generalize some of the methods of argumentation of the previous results and narrow some of the complexity gaps.

Given a graph G and an integer b, Bandwidth asks whether there exists a bijection π from V(G) to {1, …, |V(G)|} such that max_{{u, v} ∈ E(G)} | π(u) - π(v) | ≤ b. This is a classical NP-complete problem, known to remain NP-complete even on very restricted classes of graphs, such as trees of maximum degree 3 and caterpillars of hair length 3. In the realm of parameterized complexity, these results imply that the problem remains NP-hard on graphs of bounded pathwidth, while it is additionally known to be W[1]-hard when parameterized by the treedepth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number of the input graph. In this paper, we make progress towards the parameterized (in)tractability of Bandwidth. We first show that it is FPT when parameterized by the cluster vertex deletion number cvd plus the clique number ω of the input graph, thus generalizing the previously mentioned result for vertex cover. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results generalize some of the previous results and narrow some of the complexity gaps.

Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by $\textrm{treewidth} + \ell$, where $\ell$ is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if $\ell$ is not part of the parameter, then the problem is PSPACE-complete even on graphs of bounded bandwidth. In this paper, we present the first algorithmic meta-theorems for the case where $\ell$ is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by $\textrm{treedepth} + k$, where $k$ is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth $3$.

A homomorphism $f$ from a guest graph $G$ to a host graph $H$ is locally bijective, injective or surjective if for every $u\in V(G)$, the restriction of $f$ to the neighbourhood of $u$ is bijective, injective or surjective, respectively. The corresponding decision problems, LBHOM, LIHOM and LSHOM, are well studied both on general graphs and on special graph classes. Apart from complexity results when the problems are parameterized by the treewidth and maximum degree of the guest graph, the three problems still lack a thorough study of their parameterized complexity. This paper fills this gap: we prove a number of new FPT, W[1]-hard and para-NP-complete results by considering a hierarchy of parameters of the guest graph $G$. For our FPT results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.