We analyze the performance of spectral clustering for community extraction in stochastic block models. We show that, under mild conditions, spectral clustering applied to the adjacency matrix of the network can consistently recover hidden communities even when the order of the maximum expected degree is as small as $\log n$, with $n$ the number of nodes. This result applies to some popular polynomial time spectral clustering algorithms and is further extended to degree corrected stochastic block models using a spherical $k$-median spectral clustering method. A key component of our analysis is a combinatorial bound on the spectrum of binary random matrices, which is sharper than the conventional matrix Bernstein inequality and may be of independent interest.

/pdf/consistency-of-spectral-clustering-in-stochastic-block-2a7nuytdtl.pdf

Consistency of spectral clustering in stochastic block models

We present a 1.488-approximation algorithm for the metric uncapacitated facility location (UFL) problem. Previously, the best algorithm was due to Byrka (2007). Byrka proposed an algorithm parametrized by @c and used it with @c~1.6774. By either running his algorithm or the algorithm proposed by Jain, Mahdian and Saberi ([email protected]?02), Byrka obtained an algorithm that gives expected approximation ratio 1.5. We show that if @c is randomly selected, the approximation ratio can be improved to 1.488. Our algorithm cuts the gap with the 1.463 approximability lower bound by almost 1/3.

A 1.488 approximation algorithm for the uncapacitated facility location problem

In this paper, we define multicast for an ad hoc network through nodes' mobility as MotionCast and study the delay and capacity tradeoffs for it. Assuming nodes move according to an independently and identically distributed (i.i.d.) pattern and each desires to send packets to k distinctive destinations, we compare the delay and capacity in two transmission protocols: one uses 2-hop relay algorithm without redundancy; the other adopts the scheme of redundant packets transmissions to improve delay while at the expense of the capacity. In addition, we obtain the maximum capacity and the minimum delay under certain constraints. We find that the per-node delay and capacity for the 2-hop algorithm without redundancy are Θ(1/k) and Θ(n log k), respectively; for the 2-hop algorithm with redundancy, they are Ω(1/k√(n log k)) and Θ(√(n log k)), respectively. The capacity of the 2-hop relay algorithm without redundancy is better than the multicast capacity of static networks developed by Li [IEEE/ACM Trans. Netw., vol. 17, no. 3, pp. 950-961, Jun. 2009] as long as k is strictly less than n in an order sense, while when k=Θ(n), mobility does not increase capacity anymore. The ratio between delay and capacity satisfies delay/rate ≥ O(nk log k) for these two protocols, which are both smaller than that of directly extending the fundamental tradeoff for unicast established by Neely and Modiano [IEEE Trans. Inf. Theory, vol. 51, no. 6, pp. 1917-1937, Jun. 2005] to multicast, i.e., delay/rate ≥ O(n k2). More importantly, we have proved that the fundamental delay-capacity tradeoff ratio for multicast is delay/rate ≥ O(n log k), which would guide us to design better routing schemes for multicast.

/pdf/delay-and-capacity-tradeoff-analysis-for-motioncast-4n4vvoy95n.pdf

Delay and capacity tradeoff analysis for motioncast

City Logistics Centers (CLC) are an important part of the modern urban logistics system, and the selection of the location of a CLC has become a key problem in logistics and supply chain management. Integrating the economic, environmental, and social dimensions of sustainable development, this paper presents a new evaluation system for the location selection of a CLC from a sustainability perspective. A fuzzy multi-attribute group decision making (FMAGDM) technique based on a linguistic 2-tuple is used to evaluate potential alternative CLC locations. In this method, the linguistic evaluation values of all the evaluation criteria are transformed into linguistic 2-tuples. A new 2-tuple hybrid ordered weighted averaging (THOWA) operator is presented to aggregate the overall evaluation values of all experts into a collective evaluation value for each alternative, which is then used to rank and select alternative CLC locations. An application example is provided to validate the method developed and to highlight the implementation, practicality, and effectiveness by comparing with the fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method.

Location selection of city logistics centers under sustainability

We present a 1.488 approximation algorithm for the metric uncapacitated facility location (UFL) problem. Previously the best algorithm was due to Byrka [1]. By linearly combining two algorithms A1(γf) for γf ≈ 1.6774 and the (1.11, 1.78)-approximation algorithm A2 proposed by Jain, Mahdian and Saberi [8], Byrka gave a 1.5 approximation algorithm for the UFL problem. We show that if γf is randomly selected from some distribution, the approximation ratio can be improved to 1.488. Our algorithm cuts the gap with the 1.463 approximability lower bound by almost 1/3.

We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of 
$1+\sqrt{3}+\epsilon$, improving upon the decade-old ratio of $3+\epsilon$. Our approach is based on two components, each of which, we believe, is of independent interest. 
First, we show that in order to give an $\alpha$-approximation algorithm for $k$-median, it is sufficient to give a \emph{pseudo-approximation algorithm} that finds an $\alpha$-approximate solution by opening $k+O(1)$ facilities. This is a rather surprising result as there exist instances for which opening $k+1$ facilities may lead to a significant smaller cost than if only $k$ facilities were opened. 
Second, we give such a pseudo-approximation algorithm with $\alpha= 1+\sqrt{3}+\epsilon$. Prior to our work, it was not even known whether opening $k + o(k)$ facilities would help improve the approximation ratio.

Approximating $k$-Median via Pseudo-Approximation

We present a novel approximation algorithm for k-median that achieves an approximation guarantee of 1+√3+e, improving upon the decade-old ratio of 3+e. Our approach is based on two components, each of which, we believe, is of independent interest. First, we show that in order to give an α-approximation algorithm for k-median, it is sufficient to give a pseudo-approximation algorithm that finds an α-approximate solution by opening k+O(1) facilities. This is a rather surprising result as there exist instances for which opening k+1 facilities may lead to a significant smaller cost than if only k facilities were opened.Second, we give such a pseudo-approximation algorithm with α= 1+√3+e. Prior to our work, it was not even known whether opening k + o(k) facilities would help improve the approximation ratio.

Approximating k-median via pseudo-approximation

In this paper, we study the multicast capacity of a large scale random wireless network. We simply consider the extended multihop network, where a number of wireless nodes vi(1 ≤ i ≤ n) are randomly located in a square region with side-length a = √n, by use of Poisson distribution with density 1. All nodes transmit at constant power P, and the power decays along path, with attenuation exponent α > 2. The data rate of a transmission is determined by the SINR as B log(1 + SINR). There are ns randomly and independently chosen multicast sessions. Each multicast has k randomly chosen terminals. We show that, when k ≤ θ1n/(log n)2α+6, and ns ≥ θ2n1/2+β, the capacity that each multicast session can achieve, with high probability, is at least c8√n/ns√k, where θ1, θ2, and c8 are some special constants and β > 0 is any positive real number. Our result generalizes the unicast capacity [3] for random networks using percolation theory.

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Shi Li

Papers

A 1.488 approximation algorithm for the uncapacitated facility location problem

A 1.488 approximation algorithm for the uncapacitated facility location problem

Approximating $k$-Median via Pseudo-Approximation

Approximating k-median via pseudo-approximation

Capacity of large scale wireless networks under Gaussian channel model