A 1.751.75 LP approximation for the Tree Augmentation Problem

In the Tree Augmentation Problem (TAP) the goal is to augment a tree $T$ by a minimum size edge set $F$ from a given edge set $E$ such that $T \cup F$ is $2$-edge-connected. The best approximation ratio known for TAP is $1.5$. In the more general Weighted TAP problem, $F$ should be of minimum weight. Weighted TAP admits several $2$-approximation algorithms w.r.t. to the standard cut LP-relaxation, but for all of them the performance ratio of $2$ is tight even for TAP. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than $2$ is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than $2$, which is a long time open problem even for TAP. In this paper we introduce such an LP-relaxation and give an algorithm that computes a feasible solution for TAP of size at most $1.75$ times the optimal LP value. This gives some hope to break the ratio $2$ for the weighted case. Our algorithm computes some initial edge set by solving a partial system of constraints that form the integral edge-cover polytope, and then applies local search on $3$-leaf subtrees to exchange some of the edges and to add additional edges. Thus we do not need to solve the LP, and the algorithm runs roughly in time required to find a minimum weight edge-cover in a general graph.Comment: arXiv admin note: substantial text overlap with arXiv:1507.0279

Half-integrality, LP-branching and FPT Algorithms

A recent trend in parameterized algorithms is the application of polytope tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). However, although interesting results have been achieved, the methods require the underlying polytope to have very restrictive properties (half-integrality and persistence), which are known only for few problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a relaxation of the search space from $\{0,1\}^V$ to $\{0,1/2,1\}^V$ such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such relaxations, we provide a much broader class of half-integral polytopes with the required properties, unifying and extending previously known cases. In addition to the insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an $O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover with label set $\Sigma$ and an $O^*(4^k)$-time algorithm for Group Feedback Vertex Set, including the setting where the group is only given by oracle access. All these significantly improve on previous results. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012). Additionally, we propose a network flow-based approach to solve some cases of the relaxation problem. This gives the first linear-time FPT algorithm to edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA paper

Non-Uniform Robust Network Design in Planar Graphs

Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on \textbf{bulk-robustness}, a model recently introduced~\cite{bulk} for addressing the need to model non-uniform failure patterns in systems. We significantly extend the techniques used in~\cite{bulk} to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.Comment: 17 pages, 2 figure

Linear-Time FPT Algorithms via Network Flow

In the area of parameterized complexity, to cope with NP-Hard problems, we introduce a parameter k besides the input size n, and we aim to design algorithms (called FPT algorithms) that run in O(f(k)n^d) time for some function f(k) and constant d. Though FPT algorithms have been successfully designed for many problems, typically they are not sufficiently fast because of huge f(k) and d. In this paper, we give FPT algorithms with small f(k) and d for many important problems including Odd Cycle Transversal and Almost 2-SAT. More specifically, we can choose f(k) as a single exponential (4^k) and d as one, that is, linear in the input size. To the best of our knowledge, our algorithms achieve linear time complexity for the first time for these problems. To obtain our algorithms for these problems, we consider a large class of integer programs, called BIP2. Then we show that, in linear time, we can reduce BIP2 to Vertex Cover Above LP preserving the parameter k, and we can compute an optimal LP solution for Vertex Cover Above LP using network flow. Then, we perform an exhaustive search by fixing half-integral values in the optimal LP solution for Vertex Cover Above LP. A bottleneck here is that we need to recompute an LP optimal solution after branching. To address this issue, we exploit network flow to update the optimal LP solution in linear time.Comment: 20 page

Stronger ILPs for the Graph Genus Problem

The minimum genus of a graph is an important question in graph theory and a key ingredient in several graph algorithms. However, its computation is NP-hard and turns out to be hard even in practice. Only recently, the first non-trivial approach - based on SAT and ILP (integer linear programming) models - has been presented, but it is unable to successfully tackle graphs of genus larger than 1 in practice. Herein, we show how to improve the ILP formulation. The crucial ingredients are two-fold. First, we show that instead of modeling rotation schemes explicitly, it suffices to optimize over partitions of the (bidirected) arc set A of the graph. Second, we exploit the cycle structure of the graph, explicitly mapping short closed walks on A to faces in the embedding. Besides the theoretical advantages of our models, we show their practical strength by a thorough experimental evaluation. Contrary to the previous approach, we are able to quickly solve many instances of genus > 1