Guy Even

Approximating minimum feedback sets and multicuts in directed graphs

Abstract. This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set (FES) problem. In the {FVS} (resp. FES) problem, one is given a directed graph with weights (each of which is at least one) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NP-hard problems and have many applications. We also consider a generalization of these problems: subset-fvs and subset-fes, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NP-hard even when |X|=2 . We present approximation algorithms for the subset-fvs and subset-fes problems. The first algorithm we present achieves an approximation factor of O(log2|X|) . The second algorithm achieves an approximation factor of O(min{log τ* log log τ*, log n log log n)} , where τ* is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the subset-fes and subset-fvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1+ɛ) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.

Divide-and-conquer approximation algorithms via spreading metrics

We present a novel divide-and-conquer paradigm for approximating NP-hard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divide-and-conquer approach is applicable. Second, a fractional spreading metric is computable in polynomial time. The spreading metric assigns lengths to either edges or vertices of the input graph, such that all subgraphs for which the optimization problem is nontrivial have large diameters. In addition, the spreading metric provides a lower bound, <inline-equation><f> <g>t</g></f> </inline-equation>, on the cost of solving the optimization problem. We present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is <italic>O</italic>(min{log <inline-equation><f> <g>t</g>,</f> </inline-equation>log log <inline-equation><f> <g>t</g></f> </inline-equation>, log <italic>k</italic> log log <italic>k</italic>}) where <italic>k</italic> denotes the number of “interesting” vertices in the problem instance, and is at most the number of vertices. We present seven problems that can be formulated to fit the paradigm. For all these problems our algorithm improves previous results. The problems are: (1) linear arrangement; (2) embedding a graph in a <italic>d</italic>-dimensional mesh; (3) interval graph completion; (4) minimizing storage-time product; (5) subset feedback sets in directed graphs and multicuts in circular networks; (6) symmetric multicuts in directed networks; (7) balanced partitions and <italic>p</italic>-separators (for small values of <italic>p</italic>) in directed graphs.

Conflict-Free Colorings of Simple Geometric Regions with Applications to Frequency Assignment in Cellular Networks

Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call minimum conflict-free coloring (min-CF-coloring). In its general form, the input of the min-CF-coloring problem is a set system

Hitting sets when the VC-dimension is small

(X,{\cal S})

Approximating Minimum Feedback Sets and Multi-Cuts in Directed Graphs

, where each

Delay-optimized implementation of IEEE floating-point addition

S \in {\cal S}

Fast Approximate Graph Partitioning Algorithms

is a subset of X. The output is a coloring

Observability of Boolean networks: A graph-theoretic approach

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Approximations of general independent distributions

of the sets in

Lower bounds for sampling algorithms for estimating the average

{\cal S}