Directed Acyclic Subgraph Problem Parameterized above the Poljak-Turzik Bound

An oriented graph is a directed graph without directed 2-cycles. Poljak and Turzik (1986) proved that every connected oriented graph G on n vertices and m arcs contains an acyclic subgraph with at least m/2+(n-1)/4 arcs. Raman and Saurabh (2006) gave another proof of this result and left it as an open question to establish the parameterized complexity of the following problem: does G have an acyclic subgraph with least m/2 + (n-1)/4 + k arcs, where k is the parameter? We answer this question by showing that the problem can be solved by an algorithm of runtime (12k)!n^{O(1)}. Thus, the problem is fixed-parameter tractable. We also prove that there is a polynomial time algorithm that either establishes that the input instance of the problem is a Yes-instance or reduces the input instance to an equivalent one of size O(k^2)

Beyond Max-Cut: \lambda-Extendible Properties Parameterized Above the Poljak-Turz\'{i}k Bound

Poljak and Turz\'ik (Discrete Math. 1986) introduced the notion of \lambda-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0<\lambda<1 and \lambda-extendible property \Pi, any connected graph G on n vertices and m edges contains a subgraph H \in {\Pi} with at least \lambda m+ (1-\lambda)/2 (n-1) edges. The property of being bipartite is 1/2-extendible, and thus this bound generalizes the Edwards-Erd\H{o}s bound for Max-Cut. We define a variant, namely strong \lambda-extendibility, to which the bound applies. For a strongly \lambda-extendible graph property \Pi, we define the parameterized Above Poljak- Turz\'ik (APT) (\Pi) problem as follows: Given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H \in {\Pi} and H has at least \lambda m + (1-\lambda)/2 (n - 1) + k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turz\'ik bound. We consider properties {\Pi} for which APT (\Pi) is fixed- parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, APT (\Pi) is FPT for all 0<\lambda<1. Our results hold for properties of oriented graphs and graphs with edge labels. Our results generalize the result of Crowston et al. (ICALP 2012) on Max-Cut parameterized above the Edwards-Erd\H{o}s bound, and yield FPT algorithms for several graph problems parameterized above lower bounds, e.g., Max q-Colorable Subgraph problem. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem is FPT, thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).Comment: 23 pages, no figur

Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a $d$-degenerate graph $G$ and an integer $k$, outputs an independent set $Y$, such that for every independent set $X$ in $G$ of size at most $k$, the probability that $X$ is a subset of $Y$ is at least $\left({(d+1)k \choose k} \cdot k(d+1)\right)^{-1}$.The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph $G$, a set $T = \{\{s_1, t_1\}, \{s_2, t_2\}, \ldots, \{s_\ell, t_\ell\}\}$ of terminal pairs and an integer $k$, returns an induced subgraph $G^\star$ of $G$ that maintains all the inclusion minimal multicuts of $G$ of size at most $k$, and does not contain any $(k+2)$-vertex connected set of size $2^{{\cal O}(k)}$. In particular, $G^\star$ excludes a clique of size $2^{{\cal O}(k)}$ as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for Stable $s$-$t$ Separator, Stable Odd Cycle Transversal and Stable Multicut on general graphs, and for Stable Directed Feedback Vertex Set on $d$-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our algorithms can be derandomized at the cost of a small overhead in the running time.Comment: 35 page

09511 Abstracts Collection -- Parameterized complexity and approximation algorithms

From 14. 12. 2009 to 17. 12. 2009., the Dagstuhl Seminar 09511 ``Parameterized complexity and approximation algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Directed Multicut is W[1]-hard, Even for Four Terminal Pairs

We prove that Multicut in directed graphs, parameterized by the size of the cutset, is W[1]-hard and hence unlikely to be fixed-parameter tractable even if restricted to instances with only four terminal pairs. This negative result almost completely resolves one of the central open problems in the area of parameterized complexity of graph separation problems, posted originally by Marx and Razgon [SIAM J. Comput. 43(2):355-388 (2014)], leaving only the case of three terminal pairs open. Our gadget methodology allows us also to prove W[1]-hardness of the Steiner Orientation problem parameterized by the number of terminal pairs, resolving an open problem of Cygan, Kortsarz, and Nutov [SIAM J. Discrete Math. 27(3):1503-1513 (2013)].Comment: v2: Added almost tight ETH lower bound