Local search is a fundamental optimization technique that is both widely used in practice and deeply studied in theory, yet its computational complexity remains poorly understood. The traditional frameworks, PLS and the standard algorithm problem, introduced by Johnson, Papadimitriou, and Yannakakis (1988) fail to capture the methodology of local search algorithms: PLS is concerned with finding a local optimum and not with using local search, while the standard algorithm problem restricts each improvement step to follow a fixed pivoting rule. In this work, we introduce a novel formulation of local search which provides a middle ground between these models. In particular, the task is to output not only a local optimum but also a chain of local improvements leading to it. With this framework, we aim to capture the challenge in designing a good pivoting rule. Especially, when combined with the parameterized complexity paradigm, it enables both strong lower bounds and meaningful tractability results. Unlike previous works that combined parameterized complexity with local search, our framework targets the whole task of finding a local optimum and not only a single improvement step. Focusing on two representative meta-problems - Subset Weight Optimization Problem with the c-swap neighborhood and Weighted Circuit with the flip neighborhood - we establish fixed-parameter tractability results related to the number of distinct weights, while ruling out an analogous result when parameterizing by the distance to the nearest optimum via a new type of reduction.

In the standard Freeze-Tag Problem (FTP), an initially awake robot (the source) is in charge of waking up a swarm of sleeping robots by moving towards them, given that all the awake robots can participate in the awakening process. The goal is to minimize the makespan to wake up all robots assuming they move at unit speed. In this paper we introduce the Freeze-Tag-with-Return Problem (FTRP) variant, where the robots must eventually return to their initial positions. In the Euclidean plane with n sleeping robots lying on the unit disk centered at the initial position of the source, we show a non-trivial relationship between FTP and FTRP by proving that the difference between the optimal makespan of both problems never exceeds 1.959, and is at least 1.732 in the worst-case. We also present several upper and lower bounds on the optimal makespan. In particular, we show that if the sleeping robots are in convex positions, then the optimal makespan is at most 2 + 2√2, which is achieved by some instances. From an algorithmic point-of-view, we present single-exponential algorithms for general distance functions. In metric spaces, these algorithms are asymptotically optimal under the ETH, which we show via an NP-hardness reduction on unweighted graphs.

A relation modification problem gets a logical structure and a natural number k as input and asks whether k modifications of the structure suffice to make it satisfy a predefined property. We provide a complete classification of the classical and parameterized complexity of relation modification problems - the latter w. r. t. the modification budget k - based on the descriptive complexity of the respective target property. We consider different types of logical structures on which modifications are performed: Whereas monadic structures and undirected graphs without self-loops each yield their own complexity landscapes, we find that modifying undirected graphs with self-loops, directed graphs, or arbitrary logical structures is equally hard w. r. t. quantifier patterns. Moreover, we observe that all classes of problems considered in this paper are subject to a strong dichotomy in the sense that they are either very easy to solve (that is, they lie in para-AC^{0↑} or TC^0) or intractable (that is, they contain W[2]-hard or NP-hard problems).

In this work, we follow the current trend on temporal graph realization, where one is given a property P and the goal is to determine whether there is a temporal graph, that is, a graph where the edge set changes over time, with property P. We consider the problems where the given property P is a prescribed matrix for the duration, length, or earliest arrival time of pairwise temporal paths. This means that we are given a matrix D and ask whether there is a temporal graph such that for any ordered pair of vertices (s,t), D_{s,t} equals the duration (length, or earliest arrival time, respectively) of any temporal path from s to t minimizing that specific temporal path metric. For shortest and earliest arrival temporal paths, we are the first to consider these problems as far as we know. We analyze these problems for many settings such as: strict and non-strict paths, periodic and non-periodic temporal graphs, and limited number of labels per edge (limited number of occurrences per edge over time). In contrast to all other path metrics, we show that for the earliest arrival times, we can achieve polynomial-time algorithms in periodic and non-periodic temporal graphs and for strict and and non-strict paths. However, the problem becomes NP-hard when the matrix does not contain a single integer but a set or range of possible allowed values. As we show, the problem can still be solved efficiently in this scenario, when the number of entries with more than one value is small, that is, we develop an FPT-algorithm for the number of such entries. For the setting of fastest paths, we achieve new hardness results that answers an open question by Klobas, Mertzios, Molter, and Spirakis [Theor. Comput. Sci. '25] about the parameterized complexity of the problem with respect to the vertex cover number and significantly improves over a previous hardness result for the feedback vertex set number. When considering shortest paths, we show that the periodic versions are polynomial-time solvable whereas the non-periodic versions become NP-hard.

The minimum cut problem is one of the oldest and most fundamental optimization problems in operations research. In this problem, we are given a connected edge-weighted graph (G,ω) and have to find an edge set A (called edge-cut) of smallest total weight such that the removal of the edges of A disconnects G. The problem thus takes the view of an attacker that wants to destroy the global connectivity of the network. Bienstock and Diaz [SICOMP '93] introduced Global Cut Prevention, a two-player version of the minimum cut problem where a defender aims to protect edges to increase the weight of the minimum cut of the resulting graph. More precisely, the input contains an additional edge cost function c that is independent of the attacker weight ω and the defender aims to protect an edge set of total cost at most d such that every edge-cut consisting of unprotected edges has weight at least a+1. We initiate the study of the parameterized complexity of Global Cut Prevention. Here, we consider the most natural parameters such as the budgets d and a of the players, the vertex cover number and treewidth of the input graph, and combinations of these parameters. We show, for example, that the encoding of the costs and weights of the edges has a considerable influence on the problem complexity: If each edge has unit defender cost and unit attacker weight, then Global Cut Prevention is FPT for the vertex cover number. If the attacker weights are arbitrary and encoded in unary, then the problem is W[1]-hard for the vertex cover number but still admits an XP-algorithm. Finally, if the defender cost and the attacker weight are encoded in binary, then the problem becomes NP-hard even on graphs with a vertex cover of size 2.

We study the computational complexity of determining structural properties of edge-periodic temporal graphs (EPGs). EPGs are time-varying graphs that compactly represent periodic behavior of components of a dynamic network, for example, train schedules on a rail network. In EPGs, for each edge e of the graph, a binary string τ ( e ) determines in which time steps the edge is present, namely e is present in time step t if and only if τ ( e ) contains a 1 at position t mod | τ ( e ) | . Due to this periodicity, EPGs serve as very compact representations of complex periodic systems and can even be exponentially smaller than classic temporal graphs representing one period of the same system, as the latter contain the whole sequence of graphs explicitly. In this paper, we study the computational complexity of fundamental questions of the concept of EPGs such as : Is there a time step or a sliding window of size Δ in which the graph (1) is minor-free; (2) contains a minor; (3) is induced subgraph-free; (4) contains an induced subgraph; with respect to a given minor or subgraph. We give a detailed parameterized analysis for multiple combinations of parameters for the problems stated above including several algorithms. Additionally, we study the parameterized complexity of the short traversal problem in EPGs. In this problem, one asks whether there exists a time step t such that one can reach a vertex b from a vertex a at time step at most t + k for given k .

In the NP-hard Weighted Cluster Deletion problem, the input is an undirected graph G = ( V , E ) and an edge-weight function ω : E → N , and the task is to partition the vertex set V into cliques so that the total weight of edges in the cliques is maximized. Recently, it has been shown that Weighted Cluster Deletion is NP-hard on some graph classes where Cluster Deletion , the special case where every edge has unit weight, can be solved in polynomial time. We study the influence of the value t of the largest edge weight assigned by ω on the problem complexity for such graph classes. Our main results are that Weighted Cluster Deletion is fixed-parameter tractable with respect to t on graph classes whose graphs consist of well-separated clusters that are connected by a sparse periphery. Concrete examples for such classes are split graphs and graphs that are close to cluster graphs. We complement our results by strengthening previous hardness results for Weighted Cluster Deletion . For example, we show that Weighted Cluster Deletion is NP-hard on restricted subclasses of cographs even when every edge has weight 1 or 2.

A periodic temporal graph, in its simplest form, is a graph in which every edge appears exactly once in the first Δ time steps, and then it reappears recurrently every Δ time step, where Δ is a given period length. From a network design perspective, a crucial task is to assign the time-labels on the edges in a way that optimizes some criterion. In this paper we introduce a very natural optimality criterion that captures how the temporal distances of all vertex pairs are “stretched”, compared to their distances in the underlying static graph. Given a static graph G, the task is to assign to each edge one time-label between 1 and Δ such that, in the resulting periodic temporal graph with period Δ, the duration of the fastest temporal path from any vertex u to any other vertex v is at most α times the distance between u and v in G. Here, the value of α measures how much the shortest paths are allowed to be stretched once we assign the periodic time-labels. Our results span three different directions: First, we provide a series of approximation and NP-hardness results. Second, we provide approximation and fixed-parameter algorithms. Among them, we provide a simple algorithm which guarantees an approximation strictly smaller than Δ. Third, we consider a parameterized local search extension of the problem where we are given the temporal labeling of the graph, but we are allowed to change the time-labels of at most k edges.

We study the Temporal Dominating Set problem, in which one asks whether a temporal graph G = ( G 1 , ⋯ , G T ) given as a sequence of snapshot graphs over the same vertex set V has a set S of at most k temporal vertices such that each vertex v of V is dominated by some w ∈ S in the snapshot that contains w . Additionally, we consider Temporal Partial Dominating Set , where one asks whether at least t (and not necessarily all) vertices of V can be dominated by S and a further generalization in which the solution may only contain a bounded number of temporal vertices from each snapshot. We analyze how the complexity of Temporal (Partial) Dominating Set is influenced by the maximum snapshot degree and the structure of the underlying graph, the graph with vertex set V whose edge set is the union of all snapshot edge sets. For example, we obtain a complexity dichotomy for the maximum snapshot degree and show that Temporal Partial Dominating Set is fixed-parameter tractable for tw + Δ , where tw and Δ denote the treewidth and the maximum degree of the underlying graph of G , respectively. We also study which of our results transfer to the well-studied Temporal Vertex Cover problem. For example, we show that Temporal Vertex Cover is also fixed-parameter tractable for tw + Δ which substantially extends the previously known polynomial-time algorithms for the case that the underlying graph is a path or cycle.

A temporal graph 𝒢 = (G,λ) can be represented by an underlying graph G = (V,E) together with a function λ that assigns to each edge e ∈ E the set of time steps during which e is present. The reachability graph of 𝒢 is the directed graph D = (V,A) with (u,v) ∈ A if and only if there is a temporal path from u to v. We study the Reachability Graph Realizability (RGR) problem that asks whether a given directed graph D = (V,A) is the reachability graph of some temporal graph. The question can be asked for undirected or directed temporal graphs, for reachability defined via strict or non-strict temporal paths, and with or without restrictions on λ (simple, proper, or both). Answering an open question posed by Casteigts et al. (TCS 2024), we show that all variants of the problem are NP-complete, except for two variants that become trivial in the directed case. For undirected temporal graphs, we consider the complexity of the problem with respect to the solid graph, that is, the graph containing all edges that could potentially receive a label in any realization. We show that the RGR problem is fixed-parameter tractable for the feedback edge set number of the solid graph. As we show, the latter parameter can presumably not be replaced by smaller parameters like feedback vertex set number or treedepth, since the problem is W[2]-hard for them.

Given an undirected graph G, the problem of deciding whether G admits a simple and proper time-labeling that makes it temporally connected is known to be NP-hard (Göbel et al., 1991). In this article, we relax this problem and ask whether a given degree sequence can be realized as a temporally connected graph. Our main results are a complete characterization of the feasible cases, and a recognition algorithm that runs in 𝒪(n) time for graphical degree sequences (realized as simple temporal graphs) and in 𝒪(n+m) time for multigraphical degree sequences (realized as non-simple temporal graphs, where the number of time labels on an edge corresponds to the multiplicity of the edge in the multigraph). In fact, these algorithms can be made constructive at essentially no cost. Namely, we give a constructive 𝒪(n+m) time algorithm that outputs, for a given (multi)graphical degree sequence 𝐝, a temporally connected graph whose underlying (multi)graph is a realization of 𝐝, if one exists.

A temporal graph is a finite sequence of graphs, called snapshots, over the same vertex set. Many temporal graph problems turn out to be much more difficult than their static counterparts. One such problem is Timeline Vertex Cover (also known as MinTimeline_∞), a temporal analogue to the classical Vertex Cover problem. In this problem, one is given a temporal graph 𝒢 and two integers k and 𝓁, and the goal is to cover each edge of each snapshot by selecting for each vertex at most k activity intervals of length at most 𝓁 each. Here, an edge uv in the ith snapshot is covered, if an activity interval of u or v is active at time i. In this work, we continue the algorithmic study of Timeline Vertex Cover and introduce the Timeline Dominating Set problem where we want to dominate all vertices in each snapshot by the selected activity intervals. We analyze both problems from a classical and parameterized point of view and also consider partial problem versions, where the goal is to cover (dominate) at least t edges (vertices) of the snapshots. With respect to the parameterized complexity, we consider the temporal graph parameters vertex-interval-membership-width (vimw) and interval-membership-width (imw). We show that all considered problems admit FPT-algorithms when parameterized by vimw+k+𝓁. This provides a smaller parameter combination than the ones used for previously known FPT-algorithms for Timeline Vertex Cover. Surprisingly, for imw+k+𝓁, Timeline Dominating Set turns out to be easier than Timeline Vertex Cover, by also admitting an FPT-algorithm, whereas the vertex cover version is NP-hard even if imw+k+𝓁 is constant. We also consider parameterization by combinations of n, the vertex set size, with k or 𝓁 and parameterization by t. Here, we show for example that both partial problems are fixed-parameter tractable for t which significantly improves and generalizes a previous result for a special case of Partial Timeline Vertex Cover with k = 1.

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Abstract An independent set S in a graph G is k-swap-optimal if there is no independent set $$S'$$ S ′ such that $$\varvec{|S'|>|S|}$$ | S ′ | > | S | and $$\varvec{|(S'\setminus S)\cup (S\setminus S')|\le k}$$ | ( S ′ \ S ) ∪ ( S \ S ′ ) | ≤ k . Motivated by applications in data reduction, we study whether we can determine efficiently if a given vertex v is contained in some k-swap-optimal independent set or in all k-swap-optimal independent sets. We show that these problems are NP-hard for constant values of k even on graphs with constant maximum degree. Moreover, we show that the problems are $$\varvec{\Sigma ^{\text {P}}_{2}}$$ Σ 2 P -hard when k is not constant, even on graphs of constant maximum degree. We obtain similar hardness results for determining whether an edge is contained in a k-swap optimal max cut. Finally, we consider a certain type of edge-swap neighborhood for the Longest Path problem. We show that for a given edge we can decide in $$\varvec{f(\Delta +k)\cdot n^{\mathcal {O}(1)}}$$ f ( Δ + k ) · n O ( 1 ) time whether it is in some k-optimal path.

In the periodic temporal graph realization problem introduced by Klobas et al. [SAND '24] one is given a period Δ and an n × n matrix D of desired fastest travel times, and the task is to decide if there is a simple periodic temporal graph with period Δ such that the fastest travel time between any pair of vertices matches the one specified by D . We generalize the problem from simple temporal graphs to temporal graphs where each edge can appear up to ℓ times in each period, for some given integer ℓ . For the resulting problem Multi-Label Periodic TGR , we show that it is fixed-parameter tractable for parameter n and for parameter vc + Δ , where vc is the vertex cover number of the underlying graph. We also show the existence of a polynomial kernel for parameter nu + d max , where nu is the number of non-universal vertices of the underlying graph and d max is the largest entry of D . Furthermore, we show that the problem is NP -hard for each ℓ ≥ 5 , even if the underlying graph is a tree, a case that was known to be solvable in polynomial time if the task is to construct a simple periodic temporal graph, that is, if ℓ = 1 .

We show that the simplest local search heuristics for two natural Euclidean clustering problems are PLS-complete. First, we show that the Hartigan--Wong method for $k$-Means clustering is PLS-complete, even when $k = 2$. Second, we show the same result for the Flip heuristic for Max Cut, even when the edge weights are given by the (squared) Euclidean distances between the points in some set $\mathcal{X} \subseteq \mathbb{R}^d$; a problem which is equivalent to Min Sum 2-Clustering.

A temporal graph is a graph whose edges only appear at certain points in time. Reachability in these graphs is defined in terms of paths that traverse the edges in chronological order (temporal paths). This form of reachability is neither symmetric nor transitive, the latter having important consequences on the computational complexity of even basic questions, such as computing temporal connected components. In this paper, we introduce several parameters that capture how far a temporal graph $\mathcal{G}$ is from being transitive, namely, \emph{vertex-deletion distance to transitivity} and \emph{arc-modification distance to transitivity}, both being applied to the reachability graph of $\mathcal{G}$. We illustrate the impact of these parameters on the temporal connected component problem, obtaining several tractability results in terms of fixed-parameter tractability and polynomial kernels. Significantly, these results are obtained without restrictions of the underlying graph, the snapshots, or the lifetime of the input graph. As such, our results isolate the impact of non-transitivity and confirm the key role that it plays in the hardness of temporal graph problems.

In the NP-hard Π-Network Sparsification problem, we are given an edge-weighted graph G, a collection 𝒞 of c subsets of V(G), called communities, and two numbers 𝓁 and b, and the question is whether there exists a spanning subgraph G' of G with at most 𝓁 edges of total weight at most b such that G'[C] fulfills Π for each community C ∈ 𝒞. We study the fine-grained and parameterized complexity of two special cases of this problem: Connectivity NWS where Π is the connectivity property and Stars NWS, where Π is the property of having a spanning star. First, we provide a tight 2^Ω(n²+c)-time running time lower bound based on the ETH for both problems, where n is the number of vertices in G even if all communities have size at most 4, G is a clique, and every edge has unit weight. For the connectivity property, the unit weight case with G being a clique is the well-studied problem of computing a hypergraph support with a minimum number of edges. We then study the complexity of both problems parameterized by the feedback edge number t of the solution graph G'. For Stars NWS, we present an XP-algorithm for t answering an open question by Korach and Stern [Discret. Appl. Math. '08] who asked for the existence of polynomial-time algorithms for t = 0. In contrast, we show for Connectivity NWS that known polynomial-time algorithms for t = 0 [Korach and Stern, Math. Program. '03; Klemz et al., SWAT '14] cannot be extended to larger values of t by showing NP-hardness for t = 1.

We introduce a cops and robbers game with one cop and one robber on a special type of time-varying graphs (TVGs), namely edge-periodic graphs. These are TVGs in which, for each edge e, a binary string τ(e) is given such that the edge e is present in time step t if and only if τ(e) contains a 1 at position tmod|τ(e)|. This periodicity allows for a compact representation of infinite TVGs. We prove that even for very simple underlying graphs, i.e., directed and undirected cycles, the problem of deciding whether a cop-winning strategy exists is NP-hard and W[1]-hard parameterized by the number of vertices. Furthermore, we show that this decision problem can be solved on general edge-periodic graphs in PSPACE. Finally, we present tight bounds on the minimum length of a directed or undirected cycle that guarantees the cycle to be robber-winning.

In the NP-hard Max c-Cut problem, one is given an undirected edge-weighted graph G and wants to color the vertices of G with c colors such that the total weight of edges with distinctly colored endpoints is maximal. The case with c=2 is the famous Max Cut problem. To deal with the NP-hardness of this problem, we study parameterized local search algorithms. More precisely, we study LS-Max c-Cut where we are additionally given a vertex coloring f and an integer k and the task is to find a better coloring f' that differs from f in at most k entries, if such a coloring exists; otherwise, f is k-optimal. We show that LS-Max c-Cut presumably cannot be solved in g(k) · nᴼ⁽¹⁾ time even on bipartite graphs, for all c ≥ 2. We then show an algorithm for LS-Max c-Cut with running time O((3eΔ)ᵏ · c · k³ · Δ · n), where Δ is the maximum degree of the input graph. Finally, we evaluate the practical performance of this algorithm in a hill-climbing approach as a post-processing for state-of-the-art heuristics for Max c-Cut. We show that using parameterized local search, the results of this heuristic can be further improved on a set of standard benchmark instances.

Abstract In the NP-hard Colored (s,t)-Cut problem, the input is a graph G = (V,E) together with an edge-coloring ℓ : E → C, two vertices s and t, and a number k. The question is whether there is a set $S\subseteq C$ S ⊆ C of at most k colors such that deleting every edge with a color from S destroys all paths between s and t in G. We continue the study of the parameterized complexity of Colored (s,t)-Cut. First, we consider parameters related to the structure of G. For example, we study parameterization by the number ξi of edge deletions that are needed to transform G into a graph with maximum degree i. We show that Colored (s,t)-Cut is W[2]-hard when parameterized by ξ3, but fixed-parameter tractable when parameterized by ξ2. Second, we consider parameters related to the coloring ℓ. We show fixed-parameter tractability for three parameters that are potentially smaller than the total number of colors |C| and provide a linear-size problem kernel for a parameter related to the number of edges with rare edge colors.

In Bayesian Network Structure Learning (BNSL), we are given a variable set and parent scores for each variable and aim to compute a DAG, called Bayesian network, that maximizes the sum of parent scores, possibly under some structural constraints. Even very restricted special cases of BNSL are computationally hard, and, thus, in practice heuristics such as local search are used. In a typical local search algorithm, we are given some BNSL solution and ask whether there is a better solution within some pre-defined neighborhood of the solution. We study ordering-based local search, where a solution is described via a topological ordering of the variables. We show that given such a topological ordering, we can compute an optimal DAG whose ordering is within inversion distance r in subexponential FPT time; the parameter r allows to balance between solution quality and running time of the local search algorithm. This running time bound can be achieved for BNSL without any structural constraints and for all structural constraints that can be expressed via a sum of weights that are associated with each parent set. We show that for other modification operations on the variable orderings, algorithms with an FPT time for r are unlikely. We also outline the limits of ordering-based local search by showing that it cannot be used for common structural constraints on the moralized graph of the network.

A Bayesian network is a directed acyclic graph that represents statistical dependencies between variables of a joint probability distribution. A fundamental task in data science is to learn a Bayesian network from observed data. Polytree Learning is the problem of learning an optimal Bayesian network that fulfills the additional property that its underlying undirected graph is a forest. In this work, we revisit the complexity of Polytree Learning. We show that Polytree Learning can be solved in single-exponential FPT time for the number of variables. Moreover, we consider the influence of d, the number of variables that might receive a nonempty parent set in the final DAG on the complexity of Polytree Learning. We show that Polytree Learning is presumably not fixed-parameter tractable for d, unlike Bayesian network learning which is fixed-parameter tractable for d. Finally, we show that if d and the maximum parent set size are bounded, then we can obtain efficient algorithms.

We consider the cops and robbers game variant consisting of one cop and one robber on time-varying graphs (TVG). The considered TVGs are edge periodic graphs, i.e., for each edge, a binary string s_e determines in which time step the edge is present, namely the edge e is present in time step t if and only if the string s_e contains a 1 at position t mod |s_e|. This periodicity allows for a compact representation of an infinite TVG. We prove that even for very simple underlying graphs, i.e., directed and undirected cycles the problem whether a cop-winning strategy exists is NP-hard and W[1]-hard parameterized by the number of vertices. Our second main result are matching lower bounds for the ratio between the length of the underlying cycle and the least common multiple (lcm) of the lengths of binary strings describing edge-periodicies over which the graph is robber-winning. Our third main result improves the previously known EXPTIME upper bound for Periodic Cop & Robber on general edge periodic graphs to PSPACE-membership.

We consider the (parameterized) complexity of a cop and robber game on periodic, temporal graphs and a problem on periodic sequences to which these games relate intimately. In particular, we show that it is NP-hard to decide (a) whether there is some common index at which all given periodic, binary sequences are 0, and (b) whether a single cop can catch a single robber on an edge-periodic temporal graph. We further present results for various parameterizations of both problems and show that hardness not only applies in general, but also for highly limited instances. As one main result we show that even if the graph has a size-2 vertex cover and is acyclic in each time step, the cop and robber game on periodic, temporal graphs is NP-hard and W[1]-hard when parameterized by the size of the underlying input graph.