Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most $k$ of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a \BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed $k$. It is known that this implies a polynomial-time compression algorithm that turns OCT instances into equivalent instances of size at most \BigOh(4^k), a so-called kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in $k$, has turned into one of the main open questions in the study of kernelization. This work provides the first (randomized) polynomial kernelization for OCT. We introduce a novel kernelization approach based on matroid theory, where we encode all relevant information about a problem instance into a matroid with a representation of size polynomial in $k$. For OCT, the matroid is built to allow us to simulate the computation of the iterative compression step of the algorithm of Reed, Smith, and Vetta, applied (for only one round) to an approximate odd cycle transversal which it is aiming to shrink to size $k$. The process is randomized with one-sided error exponentially small in $k$, where the result can contain false positives but no false negatives, and the size guarantee is cubic in the size of the approximate solution. Combined with an \BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a reduction of the instance to size \BigOh(k^{4.5}), implying a randomized polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape

Edge Bipartization Faster Than 2^k

In the Edge Bipartization problem one is given an undirected graph $G$ and an integer $k$, and the question is whether $k$ edges can be deleted from $G$ so that it becomes bipartite. In 2006, Guo et al. [J. Comput. Syst. Sci., 72(8):1386-1396, 2006] proposed an algorithm solving this problem in time $O(2^k m^2)$; today, this algorithm is a textbook example of an application of the iterative compression technique. Despite extensive progress in the understanding of the parameterized complexity of graph separation problems in the recent years, no significant improvement upon this result has been yet reported. We present an algorithm for Edge Bipartization that works in time $O(1.977^k nm)$, which is the first algorithm with the running time dependence on the parameter better than $2^k$. To this end, we combine the general iterative compression strategy of Guo et al. [J. Comput. Syst. Sci., 72(8):1386-1396, 2006], the technique proposed by Wahlstrom [SODA 2014, 1762-1781] of using a polynomial-time solvable relaxation in the form of a Valued Constraint Satisfaction Problem to guide a bounded-depth branching algorithm, and an involved Measure & Conquer analysis of the recursion tree

Fixed-Parameter Algorithms in Analysis of Heuristics for Extracting Networks in Linear Programs

We consider the problem of extracting a maximum-size reflected network in a linear program. This problem has been studied before and a state-of-the-art SGA heuristic with two variations have been proposed. In this paper we apply a new approach to evaluate the quality of SGA\@. In particular, we solve majority of the instances in the testbed to optimality using a new fixed-parameter algorithm, i.e., an algorithm whose runtime is polynomial in the input size but exponential in terms of an additional parameter associated with the given problem. This analysis allows us to conclude that the the existing SGA heuristic, in fact, produces solutions of a very high quality and often reaches the optimal objective values. However, SGA contain two components which leave some space for improvement: building of a spanning tree and searching for an independent set in a graph. In the hope of obtaining even better heuristic, we tried to replace both of these components with some equivalent algorithms. We tried to use a fixed-parameter algorithm instead of a greedy one for searching of an independent set. But even the exact solution of this subproblem improved the whole heuristic insignificantly. Hence, the crucial part of SGA is building of a spanning tree. We tried three different algorithms, and it appears that the Depth-First search is clearly superior to the other ones in building of the spanning tree for SGA. Thereby, by application of fixed-parameter algorithms, we managed to check that the existing SGA heuristic is of a high quality and selected the component which required an improvement. This allowed us to intensify the research in a proper direction which yielded a superior variation of SGA

Treewidth reduction for constrained separation and bipartization problems

We present a method for reducing the treewidth of a graph while preserving all the minimal $s-t$ separators. This technique turns out to be very useful for establishing the fixed-parameter tractability of constrained separation and bipartization problems. To demonstrate the power of this technique, we prove the fixed-parameter tractability of a number of well-known separation and bipartization problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set). These results answer a number of open questions in the area of parameterized complexity.Comment: STACS final version of our result. For the complete description of the result please see version

05301 Abstracts Collection -- Exact Algorithms and Fixed-Parameter Tractability

From 24.07.05 to 29.07.05, the Dagstuhl Seminar 05301 ``Exact Algorithms and Fixed-Parameter Tractability\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. This is a collection of abstracts of the presentations given during the seminar

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

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