Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism

In this paper we design {\sf FPT}-algorithms for two parameterized problems. The first is \textsc{List Digraph Homomorphism}: given two digraphs $G$ and $H$ and a list of allowed vertices of $H$ for every vertex of $G$, the question is whether there exists a homomorphism from $G$ to $H$ respecting the list constraints. The second problem is a variant of \textsc{Multiway Cut}, namely \textsc{Min-Max Multiway Cut}: given a graph $G$, a non-negative integer $\ell$, and a set $T$ of $r$ terminals, the question is whether we can partition the vertices of $G$ into $r$ parts such that (a) each part contains one terminal and (b) there are at most $\ell$ edges with only one endpoint in this part. We parameterize \textsc{List Digraph Homomorphism} by the number $w$ of edges of $G$ that are mapped to non-loop edges of $H$ and we give a time $2^{O(\ell\cdot\log h+\ell^2\cdot \log \ell)}\cdot n^{4}\cdot \log n$ algorithm, where $h$ is the order of the host graph $H$. We also prove that \textsc{Min-Max Multiway Cut} can be solved in time $2^{O((\ell r)^2\log \ell r)}\cdot n^{4}\cdot \log n$. Our approach introduces a general problem, called {\sc List Allocation}, whose expressive power permits the design of parameterized reductions of both aforementioned problems to it. Then our results are based on an {\sf FPT}-algorithm for the {\sc List Allocation} problem that is designed using a suitable adaptation of the {\em randomized contractions} technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk, and Pilipczuk, FOCS 2012]).Comment: An extended abstract of this work will appear in the Proceedings of the 10th International Symposium on Parameterized and Exact Computation (IPEC), Patras, Greece, September 201

On the complexity of computing the kk-restricted edge-connectivity of a graph

The \emph{$k$-restricted edge-connectivity} of a graph $G$, denoted by $\lambda_k(G)$, is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least $k$ vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the complexity of computing $\lambda_k(G)$. Very recently, in the parameterized complexity community the notion of \emph{good edge separation} of a graph has been defined, which happens to be essentially the same as the $k$-restricted edge-connectivity. Motivated by the relevance of this invariant from both combinatorial and algorithmic points of view, in this article we initiate a systematic study of its computational complexity, with special emphasis on its parameterized complexity for several choices of the parameters. We provide a number of NP-hardness and W[1]-hardness results, as well as FPT-algorithms.Comment: 16 pages, 4 figure

Reducing CMSO model checking to highly connected graphs

Given a Counting Monadic Second Order (CMSO) sentence psi, the CMSO[psi] problem is defined as follows. The input to CMSO[psi] is a graph G, and the objective is to determine whether G |= psi. Our main theorem states that for every CMSO sentence psi, if CMSO[psi] is solvable in polynomial time on "globally highly connected graphs", then CMSO[psi] is solvable in polynomial time (on general graphs). We demonstrate the utility of our theorem in the design of parameterized algorithms. Specifically we show that technical problem-specific ingredients of a powerful method for designing parameterized algorithms, recursive understanding, can be replaced by a black-box invocation of our main theorem. We also show that our theorem can be easily deployed to show fixed parameterized tractability of a wide range of problems, where the input is a graph G and the task is to find a connected induced subgraph of G such that "few" vertices in this subgraph have neighbors outside the subgraph, and additionally the subgraph has a CMSO-definable property

Clustering to Given Connectivities

We define a general variant of the graph clustering problem where the criterion of density for the clusters is (high) connectivity. In Clustering to Given Connectivities, we are given an n-vertex graph G, an integer k, and a sequence Lambda= of positive integers and we ask whether it is possible to remove at most k edges from G such that the resulting connected components are exactly t and their corresponding edge connectivities are lower-bounded by the numbers in Lambda. We prove that this problem, parameterized by k, is fixed parameter tractable, i.e., can be solved by an f(k)* n^{O(1)}-step algorithm, for some function f that depends only on the parameter k. Our algorithm uses the recursive understanding technique that is especially adapted so to deal with the fact that we do not impose any restriction to the connectivity demands in Lambda

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Parameterized Enumeration of Neighbour Strings and Kemeny Aggregations

In this thesis, we consider approaches to enumeration problems in the parameterized complexity setting. We obtain competitive parameterized algorithms to enumerate all, as well as several of, the solutions for two related problems Neighbour String and Kemeny Rank Aggregation. In both problems, the goal is to find a solution that is as close as possible to a set of inputs (strings and total orders, respectively) according to some distance measure. We also introduce a notion of enumerative kernels for which there is a bijection between solutions to the original instance and solutions to the kernel, and provide such a kernel for Kemeny Rank Aggregation, improving a previous kernel for the problem. We demonstrate how several of the algorithms and notions discussed in this thesis are extensible to a group of parameterized problems, improving published results for some other problems