An O∗(1.0821n)O^*(1.0821^n)-Time Algorithm for Computing Maximum Independent Set in Graphs with Bounded Degree 3

We give an $O^*(1.0821^n)$-time, polynomial space algorithm for computing Maximum Independent Set in graphs with bounded degree 3. This improves all the previous running time bounds known for the problem

On the Parameterized Complexity of Biclique Cover and Partition

Given a bipartite graph G, we consider the decision problem called BicliqueCover for a fixed positive integer parameter k where we are asked whether the edges of G can be covered with at most k complete bipartite subgraphs (a.k.a. bicliques). In the BicliquePartition problem, we have the additional constraint that each edge should appear in exactly one of the k bicliques. These problems are both known to be NP-complete but fixed parameter tractable. However, the known FPT algorithms have a running time that is doubly exponential in k, and the best known kernel for both problems is exponential in k. We build on this kernel and improve the running time for BicliquePartition to O*(2^{2k^2+k*log(k)+k}) by exploiting a linear algebraic view on this problem. On the other hand, we show that no such improvement is possible for BicliqueCover unless the Exponential Time Hypothesis (ETH) is false by proving a doubly exponential lower bound on the running time. We achieve this by giving a reduction from 3SAT on n variables to an instance of BicliqueCover with k=O(log(n)). As a further consequence of this reduction, we show that there is no subexponential kernel for BicliqueCover unless P=NP. Finally, we point out the significance of the exponential kernel mentioned above for the design of polynomial-time approximation algorithms for the optimization versions of both problems. That is, we show that it is possible to obtain approximation factors of n/log(n) for both problems, whereas the previous best approximation factor was n/sqrt(log(n))

Fixed-Parameter Tractability of the Weighted Edge Clique Partition Problem

We develop an FPT algorithm and a bi-kernel for the Weighted Edge Clique Partition (WECP) problem, where a graph with $n$ vertices and integer edge weights is given together with an integer $k$, and the aim is to find $k$ cliques, such that every edge appears in exactly as many cliques as its weight. The problem has been previously only studied in the unweighted version called Edge Clique Partition (ECP), where the edges need to be partitioned into $k$ cliques. It was shown that ECP admits a kernel with~$k^2$ vertices [Mujuni and Rosamond, 2008], but this kernel does not extend to WECP. The previously fastest algorithm known for ECP has a runtime of $2^{\mathcal{O}(k^2)}n^{O(1)}$ [Issac, 2019]. For WECP we develop a bi-kernel with $4^k$ vertices, and an algorithm with runtime $2^{\mathcal{O}(k^{3/2}w^{1/2}\log(k/w))}n^{O(1)}$, where $w$ is the maximum edge weight. The latter in particular improves the runtime for ECP to~$2^{\mathcal{O}(k^{3/2}\log k)}n^{O(1)}$

Efficient Constructions for the Gy\H{o}ri-Lov\'{a}sz Theorem on Almost Chordal Graphs

In the 1970s, Gy\H{o}ri and Lov\'{a}sz showed that for a $k$-connected $n$-vertex graph, a given set of terminal vertices $t_1, \dots, t_k$ and natural numbers $n_1, \dots, n_k$ satisfying $\sum_{i=1}^{k} n_i = n$, a connected vertex partition $S_1, \dots, S_k$ satisfying $t_i \in S_i$ and $|S_i| = n_i$ exists. However, polynomial algorithms to actually compute such partitions are known so far only for $k \leq 4$. This motivates us to take a new approach and constrain this problem to particular graph classes instead of restricting the values of $k$. More precisely, we consider $k$-connected chordal graphs and a broader class of graphs related to them. For the first, we give an algorithm with $O(n^2)$ running time that solves the problem exactly, and for the second, an algorithm with $O(n^4)$ running time that deviates on at most one vertex from the given required vertex partition sizes

A Primal-Dual Algorithm for Multicommodity Flows and Multicuts in Treewidth-2 Graphs

We study the problem of multicommodity flow and multicut in treewidth-2 graphs and prove bounds on the multiflow-multicut gap. In particular, we give a primal-dual algorithm for computing multicommodity flow and multicut in treewidth-2 graphs and prove the following approximate max-flow min-cut theorem: given a treewidth-2 graph, there exists a multicommodity flow of value f with congestion 4, and a multicut of capacity c such that c ? 20 f. This implies a multiflow-multicut gap of 80 and improves upon the previous best known bounds for such graphs. Our algorithm runs in polynomial time when all the edges have capacity one. Our algorithm is completely combinatorial and builds upon the primal-dual algorithm of Garg, Vazirani and Yannakakis for multicut in trees and the augmenting paths framework of Ford and Fulkerson

Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs

w_k. In particular for the balanced version, i.e. w? = w? == w_k, this gives a partition with 1/3w_i ? w(T_i) ? 3w_i

Balanced Crown Decomposition for Connectivity Constraints

We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems. In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a 3-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component

Algorithms and Bounds for Very Strong Rainbow Coloring

A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (\src(G)) of the graph. When proving upper bounds on \src(G), it is natural to prove that a coloring exists where, for \emph{every} shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call \emph{very strong rainbow connection number} (\vsrc(G)). In this paper, we give upper bounds on \vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on \src(G) for these classes, showing that the study of \vsrc(G) enables meaningful progress on bounding \src(G). Then we study the complexity of the problem to compute \vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that \vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for \src(G). We also observe that deciding whether \vsrc(G) = k is fixed-parameter tractable in $k$ and the treewidth of $G$. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor $n^{1-\varepsilon}$, unless P$=$NP

Solving Directed Feedback Vertex Set by Iterative Reduction to Vertex Cover

In the Directed Feedback Vertex Set (DFVS) problem, one is given a directed graph G = (V,E) and wants to find a minimum cardinality set S ? V such that G-S is acyclic. DFVS is a fundamental problem in computer science and finds applications in areas such as deadlock detection. The problem was the subject of the 2022 PACE coding challenge. We develop a novel exact algorithm for the problem that is tailored to perform well on instances that are mostly bi-directed. For such instances, we adapt techniques from the well-researched vertex cover problem. Our core idea is an iterative reduction to vertex cover. To this end, we also develop a new reduction rule that reduces the number of not bi-directed edges. With the resulting algorithm, we were able to win third place in the exact track of the PACE challenge. We perform computational experiments and compare the running time to other exact algorithms, in particular to the winning algorithm in PACE. Our experiments show that we outpace the other algorithms on instances that have a low density of uni-directed edges