Gruia Călinescu

An improved approximation algorithm for multiway cut

Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due to Dahlhaus, Johnson, Papadimitriou, Seymour, and Yannakakis gave a performance guarantee of 2(1?1k). In this paper, we present a new linear programming relaxation for Multiway Cut and a new approximation algorithm based on it. The algorithm breaks the threshold of 2 for approximating Multiway Cut, achieving a performance ratio of at most 1.5?1k. This improves the previous result for every value of k. In particular, for k=3 we get a ratio of 76<43.

Selecting forwarding neighbors in wireless ad hoc networks

Broadcasting is a fundamental operation which is frequent in wireless ad hoc networks. A simple broadcasting mechanism, known as flooding, is to let every node retransmit the message to all its 1-hop neighbors when receiving the first copy of the message. Despite its simplicity, flooding is very inefficient and can result in high redundancy, contention, and collision. One approach to reducing the redundancy is to let each node forward the message only to a small subset of 1-hop neighbors that cover all of the nodes 2-hop neighbors. In this paper we propose two practical heuristics for selecting the minimum number of forwarding neighbors: an O(nlog n) time algorithm that selects at most 6 times more forwarding neighbors than the optimum, and an O(nlog 2n) time algorithm with an improved approximation ratio of 3, where n is the number of 1- and 2-hop neighbors. The best previously known algorithm, due to Bronnimann and Goodrich [2], guarantees O(1) approximation in O(n3 log n) time.

Selecting forwarding neighbors in wireless Ad Hoc networks

Broadcasting is a fundamental operation which is frequent in wireless ad hoc networks. A simple broadcasting mechanism, known as flooding, is to let every node retransmit the message to all its 1-hop neighbors when receiving the first copy of the message. Despite its simplicity, flooding is very inefficient and can result in high redundancy, contention, and collision. One approach to reducing the redundancy is to let each node forward the message only to a small subset of 1-hop neighbors that cover all of the nodes 2-hop neighbors. In this paper, we propose two practical heuristics for selecting the minimum number of forwarding neighbors: an &Ogr;(n log n) time algorithm that selects at most 6 times more forwarding neighbors than the optimum, and an &Ogr;(n2) time algorithm with an improved approximation ratio of 3, where n is the number of 1- and 2-hop neighbors. The best previously known algorithm, due to Bronnimann and Goodrich [2], guarantees &Ogr;(1) approximation in &Ogr;(n3 log n) time.

A better approximation algorithm for finding planar subgraphs

The MAXIMUM PLANAR SUBGRAPH problem?given a graphG, find a largest planar subgraph ofG?has applications in circuit layout, facility layout, and graph drawing. No previous polynomial-time approximation algorithm for this NP-Complete problem was known to achieve a performance ratio larger than 1/3, which is achieved simply by producing a spanning tree ofG. We present the first approximation algorithm for MAXIMUM PLANAR SUBGRAPH with higher performance ratio (4/9 instead of 1/3). We also apply our algorithm to find large outerplanar subgraphs. Last, we show that both MAXIMUM PLANAR SUBGRAPH and its complement, the problem of removing as few edges as possible to leave a planar subgraph, are Max SNP-Hard.

Splittable traffic partition in WDM/SONET rings to minimize SONET ADMs

SONET (Synchronous Optical NETworks) add-drop multiplexers (ADMs) are the dominant cost factor in the WDM(Wavelength Division Multiplexing)/SONET rings. The number of SONET ADMs required by a set of traffic streams is determined by the routing and wavelength assignment of the traffic streams. Previous works took as input the traffic streams with routings given a priori and developed various heuristics for wavelength assignment to minimize the SONET ADM costs. However, little was known about the performance guarantees of these heuristics. This paper contributes mainly in two aspects. First, in addition to the traffic streams with pre-specified routing, this paper also studies minimizing the ADM requirement by traffic streams without given routings, a problem which is shown to be NP-hard. Several heuristics for integrated routing and wavelength assignment are proposed to minimize the SONET ADM costs. Second, the approximation ratios of those heuristics for wavelength assignment only and those heuristics for integrated routing and wavelength assignment are analyzed. The new Preprocessed Iterative Matching heuristic has the best approximation ratio: at most 3/2.

Separating points by axis-parallel lines

We study the problem of separating n points in the plane, no two of which have the same x- or y-coordinate, using a minimum number of vertical and horizontal lines avoiding the points, so that each cell of the subdivision contains at most one point. Extending previous NP-hardness results due to Freimer et al. we prove that this problem and some variants of it are APX-hard. We give a 2-approximation algorithm for this problem, and a d-approximation algorithm for the d-dimensional variant, in which the points are to be separated using axis-parallel hyperplanes. To this end, we reduce the point separation problem to the rectangle stabbing problem studied by Gaur et al. Their approximation algorithm uses LP-rounding. We present an alternative LP-rounding procedure which also works for the rectangle stabbing problem. We show that the integrality ratio of the LP is exactly 2.

Alphabet-Independent and Scaled Dictionary Matching

The rapidly growing need for analysis of digitized images in multimedia systems has led to a variety of interesting problems in multidimensional pattern matching. One of the problems is that of scaled matching, finding all appearances of a pattern in a text in all sizes. Another important problem is dictionary matching, a quick search through a dictionary of preprocessed patterns in order to find all dictionary patterns that appear in the input text. In this paper we provide a simple algorithm for two-dimensional scaled matching. Our algorithm is the first linear-time alphabet-independent scaled matching algorithm. Its running time is O(|T|), where |T| is the text size, and is independent of |?|, the size of the alphabet. The main idea behind our algorithm is to identify and exploit a scaling-invariant property of patterns. Our technique generalizes to produce the first known algorithm for scaled dictionary matching. We can find all appearances of all dictionary patterns that appear in the input text in any discrete scale. The time bounds of our algorithm are equal to the currently known exact (no scaling) two-dimensional dictionary matching algorithms.

Multicuts in Unweighted Graphs with Bounded Degree and Bounded Tree-Width

The Multicut problem is defined as follows: given a graph G and a collection of pairs of distinct vertices (s i; t i) of G, find a small- est set of edges of G whose removal disconnects each s i from the corre- sponding t i. Our main result is a polynomial-time approximation scheme for Multicut in unweighted graphs with bounded degree and bounded tree-width: for any ∈ > 0, we presented a polynomial-time algorithm with performance ratio at most 1 + ∈. In the particular case when the input is a bounded-degree tree, we have a linear-time implementation of the algorithm. We also provided some hardness results. We proved that Multicut is still NP-hard for binary trees and that, unless P = NP, no polynomial-time approximation scheme exists if we drop any of the the three conditions: unweighted, bounded-degree, bounded-tree-width. Some of these results extend to the vertex version of Multicut.

Relay Nodes in Wireless Sensor Networks

Motivated by application to wireless sensor networks, we study the following problem. We are given a set Sof wireless sensor nodes, given as a set of points in the two-dimensional plane, and real numbers 0 < r≤ R. We must place a minimum-size set Qof wireless relay nodes in the two dimensional plane to connect S, where connectivity is explained formally next. The nodes of Scan communicate to nodes within distance r, and the relay nodes of Qcan communicate within distance R. Once the nodes of Qare placed, they together with Sinducean undirected graph G= (V,E) defined as follows: V= S? Q, and

1.61-approximation for min-power strong connectivity with two power levels

E = \{ uv | u,v \in Q \mbox{ and } ||u,v|| \leq R \} \cup \{ xu | x \in S \mbox{ and } u \in (Q \cup S) \mbox{ and } ||u,x|| \leq r\}