Lap Chi Lau

On achieving optimal throughput with network coding

With the constraints of network topologies and link capacities, achieving the optimal end-to-end throughput in data networks has been known as a fundamental but computationally hard problem. In this paper, we seek efficient solutions to the problem of achieving optimal throughput in data networks, with single or multiple unicast, multicast and broadcast sessions. Although previous approaches lead to solving NP-complete problems, we show the surprising result that, facilitated by the recent advances of network coding, computing the strategies to achieve the optimal end-to-end throughput can be performed in polynomial time. This result holds for one or more communication sessions, as well as in the overlay network model. Supported by empirical studies, we present the surprising observation that in most topologies, applying network coding may not improve the achievable optimal throughput; rather, it facilitates the design of significantly more efficient algorithms to achieve such optimality.

On achieving maximum multicast throughput in undirected networks

The transmission of information within a data network is constrained by the network topology and link capacities. In this paper, we study the fundamental upper bound of information dissemination rates with these constraints in undirected networks, given the unique replicable and encodable properties of information flows. Based on recent advances in network coding and classical modeling techniques in flow networks, we provide a natural linear programming formulation of the maximum multicast rate problem. By applying Lagrangian relaxation on the primal and the dual linear programs (LPs), respectively, we derive a) a necessary and sufficient condition characterizing multicast rate feasibility, and b) an efficient and distributed subgradient algorithm for computing the maximum multicast rate. We also extend our discussions to multiple communication sessions, as well as to overlay and ad hoc network models. Both our theoretical and simulation results conclude that, network coding may not be instrumental to achieve better maximum multicast rates in most cases; rather, it facilitates the design of significantly more efficient algorithms to achieve such optimality.

Survivable network design with degree or order constraints

We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv+3)-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv +3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain’s iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost λ-edge-connected subgraph with at least k vertices, which we call the (k, λ)-subgraph problem. This generalizes some well-studied classical problems such as the k-MST and the minimum cost λ-edgeconnected subgraph problems. We give a polylogarithmic approximation for the (k, 2)-subgraph problem. However, by relating it to the Densest k-Subgraph problem, we provide evidence that the (k, λ)-subgraph problem might be hard to approximate for arbitrary λ.

A Constant Bound on Throughput Improvement of Multicast Network Coding in Undirected Networks

Recent research in network coding shows that, joint consideration of both coding and routing strategies may lead to higher information transmission rates than routing only. A fundamental question in the field of network coding is: how large can the throughput improvement due to network coding be? In this paper, we prove that in undirected networks, the ratio of achievable multicast throughput with network coding to that without network coding is bounded by a constant ratio of 2, i.e., network coding can at most double the throughput. This result holds for any undirected network topology, any link capacity configuration, any multicast group size, and any source information rate. This constant bound 2 represents the tightest bound that has been proved so far in general undirected settings, and is to be contrasted with the unbounded potential of network coding in improving multicast throughput in directed networks.

Survivable Network Design with Degree or Order Constraints

We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds

Improved Cheeger's inequality: analysis of spectral partitioning algorithms through higher order spectral gap

B_v

Additive approximation for bounded degree survivable network design

on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a

Recognizing Powers of Proper Interval, Split, and Chordal Graphs

(2,2B_v+3)

An Approximate Max-Steiner-Tree-Packing Min-Steiner-Cut Theorem

-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex

Bipartite roots of graphs

v