Power minimization for OFDM Transmission with Subcarrier-pair based Opportunistic DF Relaying

This paper develops a sum-power minimized resource allocation (RA) algorithm subject to a sum-rate constraint for cooperative orthogonal frequency division modulation (OFDM) transmission with subcarrier-pair based opportunistic decode-and-forward (DF) relaying. The improved DF protocol first proposed in [1] is used with optimized subcarrier pairing. Instrumental to the RA algorithm design is appropriate definition of variables to represent source/relay power allocation, subcarrier pairing and transmission-mode selection elegantly, so that after continuous relaxation, the dual method and the Hungarian algorithm can be used to find an (at least approximately) optimum RA with polynomial complexity. Moreover, the bisection method is used to speed up the search of the optimum Lagrange multiplier for the dual method. Numerical results are shown to illustrate the power-reduction benefit of the improved DF protocol with optimized subcarrier pairing.Comment: 4 pages, accepted by IEEE Communications Letter

Bisection of Bounded Treewidth Graphs by Convolutions

In the Bisection problem, we are given as input an edge-weighted graph G. The task is to find a partition of V(G) into two parts A and B such that ||A| - |B|| <= 1 and the sum of the weights of the edges with one endpoint in A and the other in B is minimized. We show that the complexity of the Bisection problem on trees, and more generally on graphs of bounded treewidth, is intimately linked to the (min, +)-Convolution problem. Here the input consists of two sequences (a[i])^{n-1}_{i = 0} and (b[i])^{n-1}_{i = 0}, the task is to compute the sequence (c[i])^{n-1}_{i = 0}, where c[k] = min_{i=0,...,k}(a[i] + b[k - i]). In particular, we prove that if (min, +)-Convolution can be solved in O(tau(n)) time, then Bisection of graphs of treewidth t can be solved in time O(8^t t^{O(1)} log n * tau(n)), assuming a tree decomposition of width t is provided as input. Plugging in the naive O(n^2) time algorithm for (min, +)-Convolution yields a O(8^t t^{O(1)} n^2 log n) time algorithm for Bisection. This improves over the (dependence on n of the) O(2^t n^3) time algorithm of Jansen et al. [SICOMP 2005] at the cost of a worse dependence on t. "Conversely", we show that if Bisection can be solved in time O(beta(n)) on edge weighted trees, then (min, +)-Convolution can be solved in O(beta(n)) time as well. Thus, obtaining a sub-quadratic algorithm for Bisection on trees is extremely challenging, and could even be impossible. On the other hand, for unweighted graphs of treewidth t, by making use of a recent algorithm for Bounded Difference (min, +)-Convolution of Chan and Lewenstein [STOC 2015], we obtain a sub-quadratic algorithm for Bisection with running time O(8^t t^{O(1)} n^{1.864} log n)

A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection

Given an undirected graph with edge costs and node weights, the minimum bisection problem asks for a partition of the nodes into two parts of equal weight such that the sum of edge costs between the parts is minimized. We give a polynomial time bicriteria approximation scheme for bisection on planar graphs. Specifically, let $W$ be the total weight of all nodes in a planar graph $G$. For any constant $\varepsilon > 0$, our algorithm outputs a bipartition of the nodes such that each part weighs at most $W/2 + \varepsilon$ and the total cost of edges crossing the partition is at most $(1+\varepsilon)$ times the total cost of the optimal bisection. The previously best known approximation for planar minimum bisection, even with unit node weights, was $O(\log n)$. Our algorithm actually solves a more general problem where the input may include a target weight for the smaller side of the bipartition.Comment: To appear in STOC 201

Consistency Thresholds for the Planted Bisection Model

The planted bisection model is a random graph model in which the nodes are divided into two equal-sized communities and then edges are added randomly in a way that depends on the community membership. We establish necessary and sufficient conditions for the asymptotic recoverability of the planted bisection in this model. When the bisection is asymptotically recoverable, we give an efficient algorithm that successfully recovers it. We also show that the planted bisection is recoverable asymptotically if and only if with high probability every node belongs to the same community as the majority of its neighbors. Our algorithm for finding the planted bisection runs in time almost linear in the number of edges. It has three stages: spectral clustering to compute an initial guess, a "replica" stage to get almost every vertex correct, and then some simple local moves to finish the job. An independent work by Abbe, Bandeira, and Hall establishes similar (slightly weaker) results but only in the case of logarithmic average degree.Comment: latest version contains an erratum, addressing an error pointed out by Jan van Waai