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Niels Grüttemeier

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6 papers

1 author row

IJCAI Conference 2023 Conference Paper

Parameterized Local Search for Max c-Cut

Jaroslav Garvardt
Niels Grüttemeier
Christian Komusiewicz
Nils Morawietz

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.

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TCS Journal 2022 Journal Article

Colored cut games

Nils Morawietz
Niels Grüttemeier
Christian Komusiewicz
Frank Sommer

In a graph G = ( V, E ) with an edge coloring ℓ: E → C and two distinguished vertices s and t, a colored ( s, t ) -cut is a set C ˜ ⊆ C such that deleting all edges with some color c ∈ C ˜ from G disconnects s and t. Motivated by applications in the design of robust networks, we introduce colored cut games. In these games, an attacker and a defender choose colors to delete and to protect, respectively, in an alternating fashion. The attacker wants to achieve a colored ( s, t ) -cut and the defender wants to prevent this. First, we show that for an unbounded number of alternations, colored cut games are PSPACE-complete even on subcubic graphs. We then show that, even on subcubic graphs, colored cut games with i alternations are complete for classes in the polynomial hierarchy whose level depends on i. To complete the dichotomy, we show that all colored cut games are polynomial-time solvable on graphs with maximum degree at most 2. Next, we show that all colored cut games admit a polynomial kernel for the parameter k + κ r where k denotes the total attacker budget and, for any constant r, κ r is the number of vertex deletions that are necessary to transform G into a graph where the longest path has length at most r. For κ 1, which is the vertex cover number vc of the input graph, the kernel has size O ( vc 2 k 2 ). Moreover, we introduce an algorithm solving the most basic colored cut game, Colored ( s, t ) -Cut, in 2 vc + k n O ( 1 ) time.

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JAIR Journal 2022 Journal Article

Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis

Niels Grüttemeier
Christian Komusiewicz

We study the problem of learning the structure of an optimal Bayesian network when additional constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the network or its moralized graph are close, in terms of vertex or edge deletions, to a sparse graph class Π. For example, we show that learning an optimal network whose moralized graph has vertex deletion distance at most k from a graph with maximum degree 1 can be computed in polynomial time when k is constant. This extends previous work that gave an algorithm with such a running time for the vertex deletion distance to edgeless graphs. We then show that further extensions or improvements are presumably impossible. For example, we show that learning optimal networks where the network or its moralized graph have maximum degree 2 or connected components of size at most c, c ≥ 3, is NP-hard. Finally, we show that learning an optimal network with at most k edges in the moralized graph presumably has no f(k) · |I|O(1)-time algorithm and that, in contrast, an optimal network with at most k arcs can be computed in 2O(k) · |I|O(1) time where |I| is the total input size.

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AAAI Conference 2021 Conference Paper

Efficient Bayesian Network Structure Learning via Parameterized Local Search on Topological Orderings

Niels Grüttemeier
Christian Komusiewicz
Nils Morawietz

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.

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IJCAI Conference 2021 Conference Paper

On the Parameterized Complexity of Polytree Learning

Niels Grüttemeier
Christian Komusiewicz
Nils Morawietz

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.

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IJCAI Conference 2020 Conference Paper

Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis

Niels Grüttemeier
Christian Komusiewicz

We study the problem of learning the structure of an optimal Bayesian network when additional structural constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the moralized graph can be transformed to a graph from a sparse graph class Π by at most k vertex deletions. We show that for Π being the graphs with maximum degree 1, an optimal network can be computed in polynomial time when k is constant, extending previous work that gave an algorithm with such a running time for Π being the class of edgeless graphs [Korhonen & Parviainen, NIPS 2015]. We then show that further extensions or improvements are presumably impossible. For example, we show that when Π is the set of graphs in which each component has size at most three, then learning an optimal network is NP-hard even if k=0. Finally, we show that learning an optimal network with at most k edges in the moralized graph presumably is not fixed-parameter tractable with respect to k and that, in contrast, computing an optimal network with at most k arcs can be computed is fixed-parameter tractable in k.

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