[Show abstract][Hide abstract] ABSTRACT: The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. This paper characterizes claw-free cubic graphs whose 1-skeleton graphs of perfect matching polytopes have diameter 1.
Discrete Mathematics Algorithms and Applications 03/2014; 06(02). DOI:10.1142/S1793830914500256
[Show abstract][Hide abstract] ABSTRACT: Consider an acyclic directed network $G$ with sources $S_1, S_2,..., S_l$ and
distinct sinks $R_1, R_2,..., R_l$. For $i=1, 2,..., l$, let $c_i$ denote the
min-cut between $S_i$ and $R_i$. Then, by Menger's theorem, there exists a
group of $c_i$ edge-disjoint paths from $S_i$ to $R_i$, which will be referred
to as a group of Menger's paths from $S_i$ to $R_i$ in this paper. Although
within the same group they are edge-disjoint, the Menger's paths from different
groups may have to merge with each other. It is known that by choosing Menger's
paths appropriately, the number of mergings among different groups of Menger's
paths is always bounded by a constant, which is independent of the size and the
topology of $G$. The tightest such constant for the all the above-mentioned
networks is denoted by $\mathcal{M}(c_1, c_2,..., c_2)$ when all $S_i$'s are
distinct, and by $\mathcal{M}^*(c_1, c_2,..., c_2)$ when all $S_i$'s are in
fact identical. It turns out that $\mathcal{M}$ and $\mathcal{M}^*$ are closely
related to the network encoding complexity for a variety of networks, such as
multicast networks, two-way networks and networks with multiple sessions of
unicast. Using this connection, we compute in this paper some exact values and
bounds in network encoding complexity using a graph theoretical approach.
[Show abstract][Hide abstract] ABSTRACT: Connected dominating set (CDS) has been proposed as the virtual backbone to alleviate the broadcasting storm in wireless sensor network. Most recent research has extensively focused on the construction of connected dominating set. However, the nodes in the CDS need to dominate all its neighbors, and then some nodes cover a large number of neighboring nodes. Therefore, it is desirable to construct a capacitated dominating set, where each node can dominate only a certain number of neighbors. In this paper, we study capacitated dominating set and connected capacitated dominating set, and propose two approximation algorithms with small approximation ratios.
Discrete Mathematics Algorithms and Applications 03/2011; 3(1). DOI:10.1142/S1793830911000961
[Show abstract][Hide abstract] ABSTRACT: In this paper we consider how to collect data from sensors deployed in the Euclidean plane in a time-efficient way. We assume that all sensors could adjust their transmission ranges and aggregate data received from other sensors. We adopt a collision-free transmission model using proper schedules for data transmission. We study the problem of finding the schedule under which data from all sensors could be transmitted to the data sink in the minimal time. We propose an approximation algorithm for this NP-hard problem whose performance ratio is bounded by a constant. This significantly improves the existing approximation algorithm that does not have a constant performance ratio.
[Show abstract][Hide abstract] ABSTRACT: Unit disk graphs are the intersection graphs of equal sized disks in the plane, they are widely used as a mathematical model for wireless ad-hoc networks and some problems in computational geometry. In this paper we first show that Roman dominating set and connected Roman dominating set problems in unit disk graphs are NP-complete, and then present two approximation algorithms for these problems.
Discrete Mathematics Algorithms and Applications 03/2010; 2(1). DOI:10.1142/S1793830910000504
[Show abstract][Hide abstract] ABSTRACT: A wireless sensor network usually consists of a large number of sensor nodes deployed in a field. One of the major communication
operations is to broadcast a message from one node to the rest of the others. In this paper, we adopt the conflict-free communication
model and study how to compute a transmission schedule that determines when and where a node should forward the message so
that all nodes could receive the message in minimum time. We give two approximation algorithms for this NP-hard problem that
have better theoretically guaranteed performances than the existing algorithms. The proposed approach could be applied to
some other similar problems.
Keywordsbroadcast schedule-approximation algorithm-wireless sensor network-unit disk graph
MSC90B18
Frontiers of Mathematics in China 01/2010; 5(1):75-87. DOI:10.1007/s11464-009-0050-4 · 0.50 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Let G be a simple graph containing a perfect matching. G is said to be bipartite matching extendable (BM-extendable) if every matching M whose induced subgraph is a bipartite graph extends to a perfect matching. Extremal graph problems are at the core of graph theory. In this paper, we characterize maximally BM-unextendable graphs, maximally BM-extendable graphs in the class of complete k-partite graphs with k⩾2.
[Show abstract][Hide abstract] ABSTRACT: Minimum m-connected k-dominating set problem is as follows: Given a graph G=(V,E) and two natural numbers m and k, find a subset S⊆V of minimal size such that every vertex in V∖S is adjacent to at least k vertices in S and the induced graph of S is m-connected. In this paper we study this problem with unit disc graphs and small m, which is motivated by the design of fault-tolerant virtual backbone for wireless sensor networks. We propose two approximation
algorithms with constant performance ratios for m≤2. We also discuss how to design approximation algorithms for the problem with arbitrarily large m.
[Show abstract][Hide abstract] ABSTRACT: In wireless sensor networks, a virtual backbone has been proposed as the routing infrastructure to alleviate the broadcasting storm problem and perform some other tasks such as area monitoring. Previous work in this area has mainly focused on how to set up a small virtual backbone for high efficiency, which is modelled as the minimum Connected Dominating Set (CDS) problem. In this paper we consider how to establish a small virtual backbone to balance efficiency and fault tolerance. This problem can be formalized as the minimum m-connected k-tuple dominating set problem, which is a general version of minimum CDS problem with m=1 and k=1. We propose three centralized algorithms with small approximation ratios for small m and improve the current best results for small k.
[Show abstract][Hide abstract] ABSTRACT: A wide range of applications for wireless ad hoc networks are time-critical and impose stringent requirement on the communication latency. This paper studies the problem Minimum-Latency Broadcast Scheduling (MLBS) in wireless ad hoc networks represented by unit-disk graphs. This problem is NP-hard. A trivial lower bound on the minimum broadcast latency is the radius R of the network with respect to the source of the broadcast, which is the maximum distance of all the nodes from the source of the broadcast. The previously best-known approximation algorithm for MLBS produces a broadcast schedule with latency at most 648 R. In this paper, we present three progressively improved approximation algorithms for MLBS. They produce broadcast schedules with latency at most 24 R -23, 16 R -15, and R + O (log R) respectively.
INFOCOM 2007. 26th IEEE International Conference on Computer Communications. IEEE; 06/2007
[Show abstract][Hide abstract] ABSTRACT: Unit disk graphs are the intersection graphs of equal sized disks in the plane, they are widely used as a mathematical model for wireless ad-hoc networks and some problems in computational geometry. In this paper we first show that the Roman domination problem in unit disk graphs is NP-hard, and then present a simple linear time approximation algorithm and a polynomial-time approximation scheme for this problem, respectively.
Computational Science - ICCS 2007, 7th International Conference, Beijing, China, May 27 - 30, 2007, Proceedings, Part III; 01/2007
[Show abstract][Hide abstract] ABSTRACT: Given a wired network of processors, and a source node that needs to broadcast a message to all other processors in the network, the minimum broadcast time problem is to find a scheme that accomplishes the broadcast in a minimum number of time rounds under the constraint that at each time round, no processor can forward the received message to more than one of its neighbors in the network. This NP-hard problem has been extensively studied in literatures. In this paper we focus on a variant of the minimum broadcast time problem: the minimum multicast time problem in wireless sensor networks under collision-free data transmission model. The goal of the problem is to multicast a message from the source node to a set of destination nodes in a minimum number of time rounds. This problem remains NP-hard even in the Euclidean plane and the current best approximation algorithm has performance ratio of 41. In this paper we propose a new algorithm that has performance ratio of 15.
IEEE Wireless Communications and Networking Conference, WCNC 2007, Hong Kong, China, 11-15 March, 2007; 01/2007
[Show abstract][Hide abstract] ABSTRACT: We consider the single machine parallel-batching scheduling problem with precedence relations, release dates and identical processing times to minimize total weighted completion time. We give a polynomial-time approximation scheme (PTAS) for the problem.
Journal of Industrial and Management Optimization 08/2005; 1(3). DOI:10.3934/jimo.2005.1.353 · 0.84 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In wireless sensor networks, virtual backbone has been proposed as the routing infrastructure to alleviate the broadcasting
storm problem and perform some other tasks such as area monitoring. Previous work in this area has mainly focused on how to
set up a small virtual backbone for high efficiency, which is modelled as the minimum Connected Dominating Set (CDS) problem.
In this paper we consider how to establish a small virtual backbone to balance efficiency and fault tolerance. This problem
can be formalized as the minimum m-connected k-dominating set problem, which is a general version of minimum CDS problem with m = 1 and k = 1. In this paper we will propose some approximation algorithms for this problem that beat the current best performance
ratios.