Piotr Berman

A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem

A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. The problem considered is that of finding a minimum feedback vertex set given a weighted and undirected graph. We present a simple and efficient approximation algorithm with performance ratio of at most 2, improving previous best bounds for either weighted or unweighted cases of the problem. Any further improvement on this bound, matching the best constant factor known for the vertex cover problem, is deemed challenging. The approximation principle, underlying the algorithm, is based on a generalized form of the classical local ratio theorem, originally developed for approximation of the vertex cover problem, and a more flexible style of its application.

Improved approximations for the Steiner tree problem

For a set S contained in a metric space, a Steiner tree of S is a tree that connects the points in S. Finding a minimum cost Steiner tree is an NP-hard problem in euclidean and rectilinear metrics as well as in graphs. We give an approximation algorithm and show that the worst-case ratio of the cost of our solutions to the optimal cost is better than previously known ratios in graphs, and in rectilinear metric on the plane. Our method offers a trade-off between the running time and the ratio; on one hand it always allows to improve the ratio, on the other it allows to obtain previously known ratios with much greater efficiency. We use properties of optimal rectilinear Steiner trees to obtain significantly better ratio and running time in rectilinear metric.

Power efficient monitoring management in sensor networks

Optimizing the energy consumption in wireless sensor networks has recently become the most important performance objective. We assume the sensor network model in which sensors can interchange idle and active modes. Given monitoring regions, battery life and energy consumption rate for each sensor, we formulate the problem of maximizing sensor network lifetime, i.e., time during which the monitored area is (partially or fully) covered. Our contributions include (1) an efficient data structure to represent the monitored area with at most n/sup 2/ points guaranteeing the full coverage which is superior to the previously used approach based on grid points, (2) efficient provably good centralized algorithms for sensor monitoring schedule maximizing the total lifetime including (1+ln(1-q)/sup -1/)-approximation algorithm for the case when a q-portion of the monitored area is required to cover, e.g., for the 90% area coverage our schedule guarantees to be at most 3.3 times shorter than the optimum, (4) a family of efficient distributed protocols with trade-off between communication and monitoring power consumption, (5) extensive experimental study of the proposed algorithms showing significant advantage in quality, scalability and flexibility.

Bidding Protocols for Deploying Mobile Sensors

Constructing a sensor network with a mix of mobile and static sensors can achieve a balance between sensor coverage and sensor cost. In this paper, we design two bidding protocols to guide the movement of mobile sensors in such sensor networks to increase the coverage to a desirable level. In the protocols, static sensors detect coverage holes locally by using Voronoi diagrams and bid mobile sensors to move. Mobile sensors accept the highest bids and heal the largest holes. Simulation results show that our protocols achieve suitable trade-off between coverage and sensor cost

On Some Tighter Inapproximability Results (Extended Abstract)

We give a number of improved inapproximability results, including the best up to date explicit approximation thresholds for bounded occurence satisfiability problems like MAX-2SAT and E2-LIN-2, and the bounded degree graph problems, like MIS, Node Cover, and MAX CUT. We prove also for the first time inapproximability of the problem of Sorting by Reversals and display an explicit approximation threshold.

Fast Sorting by Reversal

Analysis of genomes evolving by inversions leads to a combinatorial problem of sorting by reversals studied in detail recently. Following a series of work recently, Hannenhalli and Pevzner developed the first polynomial algorithm for the problem of sorting signed permutations by reversals and proposed an O(n4) implementation of the algorithm. In this paper we exploit a few combinatorial properties of the cycle graph of a permutation and propose an O(n2α(n)) implementation of the algorithm where α is the inverse Ackerman function. Besides making this algorithm practical, our technique improves implementations of the other rearrangement distance problems.

On the complexity of approximating the independent set problem

We show that for some positive constant c it is not feasible to approximate Independent Set (for graphs of n nodes) within a factor of n c , provided Maximum 2-Satisfiability does not have a randomized polynomial time approximation scheme. We also study reductions preserving the quality of approximations and exhibit complete problems.

A d /2 approximation for maximum weight independent set in d -claw free graphs

A graph is d-claw free if no node has d distinct independent neighbors. In the most usual applications, the nodes of this graph form a family of sets with fewer than d elements, and the edges indicate overlapping pairs of sets. Consequently, an independent set in the graph is a packing in our family of sets. In this paper we consider the following problem. Given is a d-claw free graph G = (V, E, w) where w : V → IR+. We describe an algorithm with running time polynomial in |V|d that finds an independent set A such that w(A*)/w(A) ≤ d/2, where A* is an independent that maximizes w(A*). The previous best polynomial time approximation algorithm obtained w(A*)/w(A) ≤ 2d/3.

On-line navigation in a room

We consider the problem of navigating through an unknown environment in which the obstacles are disjoint oriented rectangles enclosed in an n x n square room. The task of navigating algorithm is to reach the center of the room starting from one of the corners. While there always exists a path of length n, the best previously known navigating algorithm finds paths of length n201nn . We give an efficient deterministic algorithm which finds a path of length O(n ln n); this algorithm uses tactile information only. Moreover, we prove that any deterministic algorithm can be forced to traverse a distance of Ω(n ln n), even if it uses visual information.

8/7-approximation algorithm for (1,2)-TSP

We design a polynomial time 8/7-approximation algorithm for the Traveling Salesman Problem in which all distances are either one or two. This improves over the best known approximation factor for that problem. As a direct application we get a 7/6-approximation algorithm for the Maximum Path Cover Problem, similarly improving upon the best known approximation factor for that problem. The result depends on a new method of consecutive path cover improvements and on a new analysis of certain related color alternating paths. This method could be of independent interest.