Results

**11 - 18**of**18**### Clique Clustering yields a PTAS for max-Coloring Interval Graphs

, 2011

"... We are given an interval graph G = (V,E) where each interval I ∈ V has a weight wI ∈ R +. The goal is to color the intervals V with an arbitrary number of color classes C1,C2,...,Ck such that ∑ k i=1 maxI∈Ci wI is minimized. This problem, called max-coloring interval graphs, contains the classical p ..."

Abstract
- Add to MetaCart

(Show Context)
We are given an interval graph G = (V,E) where each interval I ∈ V has a weight wI ∈ R +. The goal is to color the intervals V with an arbitrary number of color classes C1,C2,...,Ck such that ∑ k i=1 maxI∈Ci wI is minimized. This problem, called max-coloring interval graphs, contains the classical problem of coloring interval graphs as a special case for uniform weights, and it arises in many practical scenarios such as memory management. Pemmaraju, Raman, and Varadarajan showed that max-coloring interval graphs is NP-hard (SODA’04) and presented a 2-approximation algorithm. Closingagap which hasbeen openfor years, wesettle theapproximationcomplexity ofthisproblembygivingapolynomial-timeapproximation scheme (PTAS), that is, we show that there is an (1+ǫ)-approximation algorithm for any ǫ> 0. Besides using standard preprocessing techniques such as geometric rounding and shifting, our main building block is a general technique for trading the overlap structure of an interval graph for accuracy, which we call clique clustering. 1

### Latency Constrained Aggregation in Chain Networks Admits a PTAS

, 2010

"... This paper studies the aggregation of messages in networks that consist of a chain of nodes, and each message is time-constrained such that it needs to be aggregated during a given time interval, called its due interval. The objective is to minimize the maximum sending cost of any node, which is for ..."

Abstract
- Add to MetaCart

(Show Context)
This paper studies the aggregation of messages in networks that consist of a chain of nodes, and each message is time-constrained such that it needs to be aggregated during a given time interval, called its due interval. The objective is to minimize the maximum sending cost of any node, which is for example a concern in wireless sensor networks, where it is crucial to distribute the energy consumption as equally as possible. First, we settle the complexity of this problem by proving its NP-hardness, even for the case of unit length due intervals. Second, we give a QPTAS, which we extend to a PTAS for the special case that the lengths of the due intervals are constants. This is in particular interesting, since we prove that this problem becomes APX-hard if we consider tree networks instead of chain networks, even for the case of unit length due intervals. Specifically, we show that it cannot be approximated within 4/3 − ǫ for any ǫ> 0, unless P=NP.

### Minimal Routing Cooperation using Route Weight for MANET

"... Now days people moves to flexible and wireless network facilities hence Mobile ad-hoc network and sensor networks are growing very fast. Ad-hoc network can performed great task using multi-hop communication in such environment where dedicated infrastructure is hard to established, where node are mov ..."

Abstract
- Add to MetaCart

(Show Context)
Now days people moves to flexible and wireless network facilities hence Mobile ad-hoc network and sensor networks are growing very fast. Ad-hoc network can performed great task using multi-hop communication in such environment where dedicated infrastructure is hard to established, where node are movable and topology changes rapidly. Such type of network have to suffer with several constraints for example limited energy of nodes, information of the coordinate location of the mobile nodes at any geographical location, and need of real-time or multi-cast communication. Lots of research has been done in this field as importance and challenges in this field for covering various situations. This paper gives detail classification of the ad hoc routing protocols and to survey and proposed a new method for effective route discovery. Proposed scheme is based on total delay of path and remain power of path for selection route between source and destination. Proposed method is expected to be more effective and efficient route discovery from existing. Keywords Ad hoc networks, sensor networks, routing protocols,

### Contents

"... WP5.2 Dynamical aspects of network design and topology control State-of-the-art survey and algorithmic solutions ..."

Abstract
- Add to MetaCart

(Show Context)
WP5.2 Dynamical aspects of network design and topology control State-of-the-art survey and algorithmic solutions

### 1 When In-Network Processing Meets Time: Complexity and Effects of Joint Optimization in Wireless Sensor Networks

"... Abstract—As sensornets are increasingly being deployed in mission-critical applications, it becomes imperative that we consider application QoS requirements in in-network processing (INP). Towards understanding the complexity of joint QoS and INP optimization, we study the problem of jointly optimiz ..."

Abstract
- Add to MetaCart

(Show Context)
Abstract—As sensornets are increasingly being deployed in mission-critical applications, it becomes imperative that we consider application QoS requirements in in-network processing (INP). Towards understanding the complexity of joint QoS and INP optimization, we study the problem of jointly optimizing packet packing (i.e., aggregating shorter packets into longer ones) and the timeliness of data delivery. We identify the conditions under which the problem is strong NP-hard, and we find that the problem complexity heavily depends on aggregation constraints (in particular, maximum packet size and re-aggregation tolerance) instead of network and traffic properties. For cases when the problem is NP-hard, we show that there is no polynomial-time approximation scheme (PTAS); for cases when the problem can be solved in polynomial time, we design polynomial time, offline algorithms for finding the optimal packet packing schemes. To understand the impact of joint QoS and INP optimization on sensornet performance, we design a distributed, online protocol tPack that schedules packet transmissions to maximize the local utility of packet packing at each node. Using a testbed of 130 TelosB motes, we experimentally evaluate the properties of tPack. We find that jointly optimizing data delivery timeliness and packet packing and considering real-world aggregation constraints significantly improve network performance. Our findings shed light on the challenges, benefits, and solutions of joint QoS and INP optimization, and they also suggest open problems for future research. Index Terms—Wireless network, sensor network, real-time, packet packing, in-network processing I.

### A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

"... We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor ap-proximations are known, with the current best ratio of 3. Our main result is a weakly robust polynomial time approximat ..."

Abstract
- Add to MetaCart

(Show Context)
We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor ap-proximations are known, with the current best ratio of 3. Our main result is a weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) that it computes a clique partition, we show that it is guaranteed to be within (1+ε) ratio of the optimum if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDG’s is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an O log ∗ n