Michael Lampis

Michael Lampis, Valia Mitsou, Edouard Nemery et al.

In this paper we study the Spanning Tree Congestion problem, where we are given a graph $G=(V,E)$ and are asked to find a spanning tree $T$ of minimum maximum congestion. Here, the congestion of an edge $e\in T$ is the number of edges $uv\in E$ such that the (unique) path from $u$ to $v$ in $T$ traverses $e$. We consider this well-studied NP-hard problem from the point of view of (structural) parameterized complexity and obtain the following results. We resolve a natural open problem by showing that Spanning Tree Congestion is not FPT parameterized by treewidth (under standard assumptions). More strongly, we present a generic reduction which applies to (almost) any parameter of the form ``vertex-deletion distance to class $\mathcal{C}$'', thus obtaining W[1]-hardness for parameters more restricted than treewidth, including tree-depth plus feedback vertex set, or incomparable to treewidth, such as twin cover. Via a slight tweak of the same reduction we also show that the problem is NP-complete on interval graphs of modular-width $4$. Even though it is known that Spanning Tree Congestion remains NP-hard on instances with only one vertex of unbounded degree, it is currently open whether the problem remains hard on bounded-degree graphs. We resolve this question by showing NP-hardness on graphs of maximum degree 8. Complementing the problem's W[1]-hardness for treewidth...

Michael Lampis

A stable or locally-optimal cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. In this paper we study Minimum Stable Cut, the problem of finding a stable cut of minimum weight. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We begin by showing that the problem remains weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this hardness with a pseudo-polynomial DP algorithm solving the problem in time $(Δ\cdot W)^{O(tw)}n^{O(1)}$, where $tw$ is the treewidth, $Δ$ the maximum degree, and $W$ the maximum weight. On the other hand, bounding $Δ$ is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Minimum Stable Cut by both $tw$ and $Δ$ and obtain an FPT algorithm running in time $2^{O(Δtw)}(n+\log W)^{O(1)}$. Our main result for the weighted problem is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in $(nW)^{o(pw)}$ or $2^{o(Δpw)}(n+\log W)^{O(1)}$, then the ETH is false. Complementing this, we show that we can, however, obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions, that is, solutions where no single vertex can unilaterally increase the weight of its incident cut edges by more than a factor of $(1+\varepsilon)$. Motivated by these mostly negative results, we consider Unweighted Minimum Stable Cut. Here our results already imply a much faster exact algorithm running in time $Δ^{O(tw)}n^{O(1)}$. We show that this is also probably essentially optimal: an algorithm running in $n^{o(pw)}$ would contradict the ETH.

Michael Lampis, Manolis Vasilakis

Capacitated Vertex Cover is the hard-capacitated variant of Vertex Cover: given a graph, a capacity for every vertex, and an integer $k$, the task is to select at most $k$ vertices that cover all edges and assign each edge to one of its chosen endpoints so that no chosen vertex receives more incident edges than its capacity. This problem is a classical benchmark in parameterized complexity, as it was among the first natural problems shown to be W[1]-hard when parameterized by treewidth. We revisit its exact complexity from a fine-grained parameterized perspective and obtain a much sharper picture for several standard parameters. For the natural parameter $k$, we prove under the Exponential Time Hypothesis (ETH) that no algorithm with running time $k^{o(k)} n^{\mathcal{O}(1)}$ exists. In particular, this shows that the known algorithms with running time $k^{\mathcal{O}(\mathrm{tw})} n^{\mathcal{O}(1)}$ are essentially optimal. We then turn to more general structural parameters. For vertex cover number $\mathrm{vc}$, we give evidence against a $2^{\mathcal{O}(\mathrm{vc}^{2-\varepsilon})} n^{\mathcal{O}(1)}$ algorithm, as such an improvement would imply corresponding progress for a broader class of integer-programming-type problems. We complement this barrier with a nearly matching upper bound for vertex integrity $\mathrm{vi}$, improving the previously known double-exponential dependence to an algorithm with running time $\mathrm{vi}^{\mathcal{O}(\mathrm{vi}^{2})} n^{\mathcal{O}(1)}$ using $N$-fold integer programming. For treewidth, we show that the standard dynamic programming algorithm with running time $n^{\mathcal{O}(\mathrm{tw})}$ is essentially optimal under the ETH, even if one parameterizes by tree-depth. Turning to clique-width, we prove that Capacitated Vertex Cover remains NP-hard already on graphs of linear clique-width $6$...

Michael Lampis, Yiren Lu

We revisit the complexity of deciding whether a graphical game admits a pure Nash equilibrium (PNE) parameterized by standard measures of the input graph, such as treewidth. The natural dynamic programming algorithm for this problem has parameter dependence $α^{(Δ+1)\text{tw}}$ where $α$ is the maximum number of strategies available to each player, each player's utility depends on at most $Δ$ other players, and the input graph has width $\text{tw}$. Our first contribution is to point out that an algorithm by Thomas and van Leeuwen [Algorithmica 2015] claiming to improve this dependence to $α^{O(\text{tw})}$ is flawed and, more strongly, such an algorithm would imply that FPT=W[1]. We then set out to pinpoint the fine-grained complexity of this problem with respect to standard parameters and show that the natural DP algorithm is not optimal, as the problem can be solved with dependence $α^{\lfloor \frac{2Δ}{3} + 1 \rfloor \text{tw}}$, $α^{\lfloor \fracΔ{2} + 1 \rfloor \text{pw}}$, and $α^{\text{ctw}}$, where $\text{pw}, \text{ctw}$ are the pathwidth and cutwidth of the input respectively. Our main algorithmic tool is a tightening of the relationship between the width of a graph $G$, its maximum degree, and the width of $G^2$, which may be of independent interest. Complementing these results, we show that our algorithms for pathwidth and cutwidth are likely to be optimal, as improving them is equivalent to falsifying the pw-SETH.

Michael Lampis, Stefan Mengel, Valia Mitsou

We propose reductions to quantified Boolean formulas (QBF) as a new approach to showing fixed-parameter linear algorithms for problems parameterized by treewidth. We demonstrate the feasibility of this approach by giving new algorithms for several well-known problems from artificial intelligence that are in general complete for the second level of the polynomial hierarchy. By reduction from QBF we show that all resulting algorithms are essentially optimal in their dependence on the treewidth. Most of the problems that we consider were already known to be fixed-parameter linear by using Courcelle's Theorem or dynamic programming, but we argue that our approach has clear advantages over these techniques: on the one hand, in contrast to Courcelle's Theorem, we get concrete and tight guarantees for the runtime dependence on the treewidth. On the other hand, we avoid tedious dynamic programming and, after showing some normalization results for CNF-formulas, our upper bounds often boil down to a few lines.