Parameterized Algorithms for Maximum Cut with Connectivity Constraints

We study two variants of Maximum Cut, which we call Connected Maximum Cut and Maximum Minimal Cut, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas Maximum Cut on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-width, and twin-cover number. Finally, we obtain FPT algorithms with respect to the solution size

Random Perfect Graphs

We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number $\alpha(P_n)$ and clique number $\omega(P_n)$ is close to a concentrated distribution $L(n)$ which plays an important role in our generation method. We also prove that the probability that $P_n$ contains any given graph $H$ as an induced subgraph is asymptotically $0$ or $\frac12$ or $1$. Further we show that almost all perfect graphs are $2$-clique-colourable, improving a result of Bacs\'o et al from 2004; they are almost all Hamiltonian; they almost all have connectivity $\kappa(P_n)$ equal to their minimum degree; they are almost all in class one (edge-colourable using $\Delta$ colours, where $\Delta$ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon $W_P(x, y) = \frac12(\mathbb{1}[x \le 1/2] + \mathbb{1}[y \le 1/2])$

Fixed-Parameter Tractability of Multicut in Directed Acyclic Graphs

The Multicut problem, given a graph G, a set of terminal pairs $\mathcal{T}=\{(s_i,t_i)\ |\ 1\leq i\leq r\}$, and an integer $p$, asks whether one can find a cutset consisting of at most $p$ nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, $t_i$ is not reachable from $s_i$ for each $1\leq i\leq r$. The fixed-parameter tractability of Multicut in undirected graphs, parameterized by the size of the cutset only, has been recently proved by Marx and Razgon [SIAM J. Comput., 43 (2014), pp. 355--388] and, independently, by Bousquet, Daligault, and Thomassé [Proceedings of STOC, ACM, 2011, pp. 459--468], after resisting attacks as a long-standing open problem. In this paper we prove that Multicut is fixed-parameter tractable on directed acyclic graphs when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains $W[1]$-hard

Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization

A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [IPEC 2018], we introduce a natural generalization of this problem, which we call d-Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d-Cut on (2d+2)-regular graphs and a polynomial algorithm for graphs of maximum degree at most d+2. The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [IPEC 2018], is a polynomial kernel for d-Cut for every positive integer d, parameterized by the distance to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time O^*(2^n)

GRASP/VND Optimization Algorithms for Hard Combinatorial Problems

Two hard combinatorial problems are addressed in this thesis. The first one is known as the ”Max CutClique”, a combinatorial problem introduced by P. Martins in 2012. Given a simple graph, the goal is to find a clique C such that the number of links shared between C and its complement C C is maximum. In a first contribution, a GRASP/VND methodology is proposed to tackle the problem. In a second one, the N P-Completeness of the problem is mathematically proved. Finally, a further generalization with weighted links is formally presented with a mathematical programming formulation, and the previous GRASP is adapted to the new problem. The second problem under study is a celebrated optimization problem coming from network reliability analysis. We assume a graph G with perfect nodes and imperfect links, that fail independently with identical probability ρ ∈ [0,1]. The reliability RG(ρ), is the probability that the resulting subgraph has some spanning tree. Given a number of nodes and links, p and q, the goal is to find the (p,q)-graph that has the maximum reliability RG(ρ), uniformly in the compact set ρ ∈ [0,1]. In a first contribution, we exploit properties shared by all uniformly most-reliable graphs such as maximum connectivity and maximum Kirchhoff number, in order to build a novel GRASP/VND methodology. Our proposal finds the globally optimum solution under small cases, and it returns novel candidates of uniformly most-reliable graphs, such as Kantor-Mobius and Heawood graphs. We also offer a literature review, ¨ and a mathematical proof that the bipartite graph K4,4 is uniformly most-reliable. Finally, an abstract mathematical model of Stochastic Binary Systems (SBS) is also studied. It is a further generalization of network reliability models, where failures are modelled by a general logical function. A geometrical approximation of a logical function is offered, as well as a novel method to find reliability bounds for general SBS. This bounding method combines an algebraic duality, Markov inequality and Hahn-Banach separation theorem between convex and compact sets

Crown Reductions and Decompositions: Theoretical Results and Practical Methods

Two kernelization schemes for the vertex cover problem, an NP-hard problem in graph theory, are compared. The first, crown reduction, is based on the identification of a graph structure called a crown and is relatively new while the second, LP-kernelization has been used for some time. A proof of the crown reduction algorithm is presented, the algorithm is implemented and theorems are proven concerning its performance. Experiments are conducted comparing the performance of crown reduction and LP- kernelization on real world biological graphs. Next, theorems are presented that provide a logical connection between the crown structure and LP-kernelization. Finally, an algorithm is developed for decomposing a graph into two subgraphs: one that is a crown and one that is crown free