Parameterized Directed kk-Chinese Postman Problem and kk Arc-Disjoint Cycles Problem on Euler Digraphs

In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given a connected weighted digraph $G$ and asked to find $k$ non-empty closed directed walks covering all arcs of $G$ such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128) asked for the parameterized complexity of $k$-DCPP when $k$ is the parameter. We prove that the $k$-DCPP is fixed-parameter tractable. We also consider a related problem of finding $k$ arc-disjoint directed cycles in an Euler digraph, parameterized by $k$. Slivkins (ESA 2003) showed that this problem is W[1]-hard for general digraphs. Generalizing another result by Slivkins, we prove that the problem is fixed-parameter tractable for Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler digraphs remains W[1]-hard even for Euler digraphs

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

Editing to Eulerian Graphs

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let $ea$, $ed$ and $vd$ denote the operations edge addition, edge deletion and vertex deletion respectively. For any $S\subseteq \{ea,ed,vd\}$, we define Connected Degree Parity Editing$(S)$ (CDPE($S$)) to be the problem that takes as input a graph $G$, an integer $k$ and a function $\delta\colon V(G)\rightarrow\{0,1\}$, and asks whether $G$ can be modified into a connected graph $H$ with $d_{H}(v)\equiv\delta(v)~(\bmod~2)$ for each $v\in V(H)$, using at most $k$ operations from $S$. We prove that 1. if $S=\{ea\}$ or $S=\{ea,ed\}$, then CDPE($S$) can be solved in polynomial time; 2. if $\{vd\} \subseteq S\subseteq \{ea,ed,vd\}$, then CDPE($S$) is NP-complete and W[1]-hard when parameterized by $k$, even if $\delta\equiv 0$. Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE($S$) problem for all $S\subseteq \{ea,ed,vd\}$. We obtain the same classification for a natural variant of the CDPE($S$) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value. As an important implication of our results, we obtain polynomial-time algorithms for the Eulerian Editing problem and its directed variant.Comment: 33 pages. An extended abstract of this paper will appear in the proceedings of FSTTCS 201

Determinantal Sieving

We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial $P(X)$ on a set of variables $X=\{x_1,\ldots,x_n\}$ and a linear matroid $M=(X,\mathcal{I})$ of rank $k$, both over a field $\mathbb{F}$ of characteristic 2, in $2^k$ evaluations we can sieve for those terms in the monomial expansion of $P$ which are multilinear and whose support is a basis for $M$. Alternatively, using $2^k$ evaluations of $P$ we can sieve for those monomials whose odd support spans $M$. Applying this framework, we improve on a range of algebraic FPT algorithms, such as: 1. Solving $q$-Matroid Intersection in time $O^*(2^{(q-2)k})$ and $q$-Matroid Parity in time $O^*(2^{qk})$, improving on $O^*(4^{qk})$ (Brand and Pratt, ICALP 2021) 2. $T$-Cycle, Colourful $(s,t)$-Path, Colourful $(S,T)$-Linkage in undirected graphs, and the more general Rank $k$ $(S,T)$-Linkage problem, all in $O^*(2^k)$ time, improving on $O^*(2^{k+|S|})$ respectively $O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))})$ (Fomin et al., SODA 2023) 3. Many instances of the Diverse X paradigm, finding a collection of $r$ solutions to a problem with a minimum mutual distance of $d$ in time $O^*(2^{r(r-1)d/2})$, improving solutions for $k$-Distinct Branchings from time $2^{O(k \log k)}$ to $O^*(2^k)$ (Bang-Jensen et al., ESA 2021), and for Diverse Perfect Matchings from $O^*(2^{2^{O(rd)}})$ to $O^*(2^{r^2d/2})$ (Fomin et al., STACS 2021) All matroids are assumed to be represented over a field of characteristic 2. Over general fields, we achieve similar results at the cost of using exponential space by working over the exterior algebra. For a class of arithmetic circuits we call strongly monotone, this is even achieved without any loss of running time. However, the odd support sieving result appears to be specific to working over characteristic 2

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Linear Orderings of Sparse Graphs

The Linear Ordering problem consists in finding a total ordering of the vertices of a directed graph such that the number of backward arcs, i.e., arcs whose heads precede their tails in the ordering, is minimized. A minimum set of backward arcs corresponds to an optimal solution to the equivalent Feedback Arc Set problem and forms a minimum Cycle Cover. Linear Ordering and Feedback Arc Set are classic NP-hard optimization problems and have a wide range of applications. Whereas both problems have been studied intensively on dense graphs and tournaments, not much is known about their structure and properties on sparser graphs. There are also only few approximative algorithms that give performance guarantees especially for graphs with bounded vertex degree. This thesis fills this gap in multiple respects: We establish necessary conditions for a linear ordering (and thereby also for a feedback arc set) to be optimal, which provide new and fine-grained insights into the combinatorial structure of the problem. From these, we derive a framework for polynomial-time algorithms that construct linear orderings which adhere to one or more of these conditions. The analysis of the linear orderings produced by these algorithms is especially tailored to graphs with bounded vertex degrees of three and four and improves on previously known upper bounds. Furthermore, the set of necessary conditions is used to implement exact and fast algorithms for the Linear Ordering problem on sparse graphs. In an experimental evaluation, we finally show that the property-enforcing algorithms produce linear orderings that are very close to the optimum and that the exact representative delivers solutions in a timely manner also in practice. As an additional benefit, our results can be applied to the Acyclic Subgraph problem, which is the complementary problem to Feedback Arc Set, and provide insights into the dual problem of Feedback Arc Set, the Arc-Disjoint Cycles problem

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..