An efficient algorithm for finding maximum cycle packings in reducible flow graphs

Reducible flow graphs occur naturally in connection with flow-charts of computer programs and are used extensively for code optimization and global data flow analysis. In this paper we present an O(n2m log (n 2/m)) algorithm for finding a maximum cycle packing in any weighted reducible flow graph with n vertices and m arcs. © Springer-Verlag 2004.postprin

Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3-SAT

In directed graphs, we investigate the problems of finding: 1) a minimum feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2) a feedback vertex set inducing an acyclic graph (also called the Vertex 2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS). We show that these problems are strongly related to (variants of) Monotone 3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in positive form. As a consequence, we deduce several NP-completeness results on restricted versions of these problems. In particular, we define the 2-Choice version of an optimization problem to be its restriction where the optimum value is known to be either D or D+1 for some integer D, and the problem is reduced to decide which of D or D+1 is the optimum value. We show that the 2-Choice versions of MFVS, Acyclic MFVS, Min Ones Monotone 3-SAT and Min Ones Monotone NAE 3-SAT are NP-complete. The two latter problems are the variants of Monotone 3-SAT and respectively Monotone NAE 3-SAT requiring that the truth assignment minimize the number of variables set to true. Finally, we propose two classes of directed graphs for which Acyclic FVS is polynomially solvable, namely flow reducible graphs (for which MFVS is already known to be polynomially solvable) and C1P-digraphs (defined by an adjacency matrix with the Consecutive Ones Property)

Markov-Chain-Based Heuristics for the Feedback Vertex Set Problem for Digraphs

A feedback vertex set (FVS) of an undirected or directed graph G=(V, A) is a set F such that G-F is acyclic. The minimum feedback vertex set problem asks for a FVS of G of minimum cardinality whereas the weighted minimum feedback vertex set problem consists of determining a FVS F of minimum weight w(F) given a real-valued weight function w. Both problems are NP-hard [Karp72]. Nethertheless, they have been found to have applications in many fields. So one is naturally interested in approximation algorithms. While most of the existing approximation algorithms for feedback vertex set problems rely on local properties of G only, this thesis explores strategies that use global information about G in order to determine good solutions. The pioneering work in this direction has been initiated by Speckenmeyer [Speckenmeyer89]. He demonstrated the use of Markov chains for determining low cardinality FVSs. Based on his ideas, new approximation algorithms are developed for both the unweighted and the weighted minimum feedback vertex set problem for digraphs. According to the experimental results presented in this thesis, these new algorithms outperform all other existing approximation algorithms. An additional contribution, not related to Markov chains, is the identification of a new class of digraphs G=(V, A) which permit the determination of an optimum FVS in time O(|V|^4). This class strictly encompasses the completely contractible graphs [Levy/Low88]

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Topological Interference Management With Decoded Message Passing

The topological interference management (TIM) problem studies partially-connected interference networks with no channel state information except for the network topology (i.e., connectivity graph) at the transmitters. In this paper, we consider a similar problem in the uplink cellular networks, while message passing is enabled at the receivers (e.g., base stations), so that the decoded messages can be routed to other receivers via backhaul links to help further improve network performance. For this TIM problem with decoded message passing (TIM-MP), we model the interference pattern by conflict digraphs, connect orthogonal access to the acyclic set coloring on conflict digraphs, and show that one-to-one interference alignment boils down to orthogonal access because of message passing. With the aid of polyhedral combinatorics, we identify the structural properties of certain classes of network topologies where orthogonal access achieves the optimal degrees-of-freedom (DoF) region in the information-theoretic sense. The relation to the conventional index coding with simultaneous decoding is also investigated by formulating a generalized index coding problem with successive decoding as a result of decoded message passing. The properties of reducibility and criticality are also studied, by which we are able to prove the linear optimality of orthogonal access in terms of symmetric DoF for the networks up to four users with all possible network topologies (218 instances). Practical issues of the tradeoff between the overhead of message passing and the achievable symmetric DoF are also discussed, in the hope of facilitating efficient backhaul utilization

On the minimum number of inversions to make a digraph kk-(arc-)strong

The {\it inversion} of a set $X$ of vertices in a digraph $D$ consists of reversing the direction of all arcs of $D\langle X\rangle$. We study $sinv'_k(D)$ (resp. $sinv_k(D)$) which is the minimum number of inversions needed to transform $D$ into a $k$-arc-strong (resp. $k$-strong) digraph and sinv'_k(n) = \max\{sinv'_k(D) \mid D~\mbox{is a 2k$-edge-connected digraph of order$n}\}. We show : $(i): \frac{1}{2} \log (n - k+1) \leq sinv'_k(n) \leq \log n + 4k -3$ ; $(ii):$ for any fixed positive integers $k$ and $t$, deciding whether a given oriented graph $\vec{G}$ satisfies $sinv'_k(\vec{G}) \leq t$ (resp. $sinv_k(\vec{G}) \leq t$) is NP-complete ; $(iii):$ if $T$ is a tournament of order at least $2k+1$, then $sinv'_k(T) \leq sinv_k(T) \leq 2k$, and $sinv'_k(T) \leq \frac{4}{3}k+o(k)$; $(iv):\frac{1}{2}\log(2k+1) \leq sinv'_k(T) \leq sinv_k(T)$ for some tournament $T$ of order $2k+1$; $(v):$ if $T$ is a tournament of order at least $19k-2$ (resp. $11k-2$), then $sinv'_k(T) \leq sinv_k(T) \leq 1$ (resp. $sinv_k(T) \leq 3$); $(vi):$ for every $\epsilon>0$, there exists $C$ such that $sinv'_k(T) \leq sinv_k(T) \leq C$ for every tournament $T$ on at least $2k+1 + \epsilon k$ vertices

Measure-Driven Algorithm Design and Analysis: A New Approach for Solving NP-hard Problems

NP-hard problems have numerous applications in various fields such as networks, computer systems, circuit design, etc. However, no efficient algorithms have been found for NP-hard problems. It has been commonly believed that no efficient algorithms for NP-hard problems exist, i.e., that P6=NP. Recently, it has been observed that there are parameters much smaller than input sizes in many instances of NP-hard problems in the real world. In the last twenty years, researchers have been interested in developing efficient algorithms, i.e., fixed-parameter tractable algorithms, for those instances with small parameters. Fixed-parameter tractable algorithms can practically find exact solutions to problem instances with small parameters, though those problems are considered intractable in traditional computational theory. In this dissertation, we propose a new approach of algorithm design and analysis: discovering better measures for problems. In particular we use two measures instead of the traditional single measure?input size to design algorithms and analyze their time complexity. For several classical NP-hard problems, we present improved algorithms designed and analyzed with this new approach, First we show that the new approach is extremely powerful for designing fixedparameter tractable algorithms by presenting improved fixed-parameter tractable algorithms for the 3D-matching and 3D-packing problems, the multiway cut problem, the feedback vertex set problems on both directed and undirected graph and the max-leaf problems on both directed and undirected graphs. Most of our algorithms are practical for problem instances with small parameters. Moreover, we show that this new approach is also good for designing exact algorithms (with no parameters) for NP-hard problems by presenting an improved exact algorithm for the well-known satisfiability problem. Our results demonstrate the power of this new approach to algorithm design and analysis for NP-hard problems. In the end, we discuss possible future directions on this new approach and other approaches to algorithm design and analysis