arXiv subjects

Johannes Tantow

Publications and source records attributed to Johannes Tantow.

3 recordsLinked to original sources

arXiv2026

Solving Stackelberg Vertex Cover on trees using split and join

The Stackelberg Vertex Cover problem is a bilevel optimization problem with two players on a graph $G = (F \cup P, E)$ where each vertex from $F$ has a weight and the first player selects a price for each vertex in $P$. Afterwards, the second player finds a minimum vertex cover $X$ and the first player receives the set price for each vertex from $X \cap P$. The goal is to maximize the revenue of the first player. This problem was recently shown to be NP-complete for bipartite graphs while being solvable in linear time on paths. We present three new algorithms for solving Stackelberg Vertex Cover on certain kinds of trees: (1) a pseudo-polynomial algorithm working on general trees when all weights are integer, i.e., it is FPT with the maximum weight as a parameter; (2) a strongly polynomial algorithm for trees having the property that the least common ancestor of any two vertices from $P$ is again in $P$ (this case includes paths); and (3) an FPT-algorithm for trees, where the parameter is the maximum number $P$-vertices $v_i$ that an $F$-vertex $u$ can reach while using no other $P$-vertices. These algorithms are based on a lemma that allows us to split instances at a vertex $u$ into multiple sub-instances, which follows from LP duality and integrality of the vertex cover LP on bipartite graphs. The lemma requires that the minimum vertex covers of the sub-instances agree on $u$ (either all include $u$ or all don't). For this we introduce the concept of commitments. Finally, we show that the Stackelberg Vertex Cover problem with commitments is weakly NP-complete.

cs.DS↗

arXiv2026

PLS-complete problems with lexicographic cost functions: Max-$k$-SAT and Abelian Permutation Orbit Minimization

How hard is it to find a local optimum? If we are given a graph and want to find a locally maximal cut--meaning that the number of edges in the cut can't be improved by moving a single vertex from one side to the other--then just iterating improving steps finds a local maximum in $ |E|$ steps. If, on the other hand, the edges are weighted, this problem becomes hard for the class PLS (Polynomial Local Search). We are interested in optimization problems with lexicographic costs. For Max-Cut this would mean that the edges $e_1,\dots, e_m$ have costs $c(e_i) = 2^i$. For such a cost function finding a global Max-Cut is easy. In contrast, we show that it is PLS-complete to find an assignment for a 4-CNF formula that is locally maximal (when the clauses have lexicographic weights); and also for a 3-CNF when we allow switching two variables at a time. We use these results to answer a question in Scheder and Tantow, who showed that finding a lexicographic local minimum of a string $s \in \{0,1\}^n$ under the action of a list of given permutations $π_1, \dots, π_k \in S_{n}$ is PLS-complete. They ask whether the problem stays PLS-complete when the $π_1,\dots,π_k$ commute, i.e., generate an Abelian subgroup $G$ of $S_n$. We show that it does, and in fact stays PLS-complete even (1) when every element in $G$ has order two or (2) when $G$ is cyclic. Additionally, we use it to further investigate the complexity of computing pure $α$-Nash equilibria in congestion games. Using lexicographic 4-SAT, we obtain a simple proof of the PLS-completeness originally shown by Skopalik and Vöcking that can be extended to exponential and polynomial delay functions with positive coefficients. The number of strategies per player and players per resource is bounded. However, the degree of the polynomials is not bounded by a constant.

cs.CC↗

arXiv2025

PLS-completeness of string permutations

Bitstrings can be permuted via permutations and compared via the lexicographic order. In this paper we study the complexity of finding a minimum of a bitstring via given permutations. As a global optima is known to be NP-complete, we study the local optima via the class PLS and show hardness for PLS. Additionally, we show that even for one permutation the global optimization is NP-complete and give a formula that has these permutation as symmetries. This answers an open question inspired from Kolodziejczyk and Thapen and stated at the SAT and interactions seminar in Dagstuhl.

cs.CC↗