Beating ratio 0.5 for weighted oblivious matching problems

ESA 2016 is organized in collaboration with the European Association for Theoretical Computer Science (EATCS) and is a part of ALGO 2016We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem. Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs). (1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). (2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as matching coverage, in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied. Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5.published_or_final_versio

Size-constrained Weighted Ancestors with Applications

The weighted ancestor problem on a rooted node-weighted tree $T$ is a generalization of the classic predecessor problem: construct a data structure for a set of integers that supports fast predecessor queries. Both problems are known to require $\Omega(\log\log n)$ time for queries provided $\mathcal{O}(n\text{ poly} \log n)$ space is available, where $n$ is the input size. The weighted ancestor problem has attracted a lot of attention by the combinatorial pattern matching community due to its direct application to suffix trees. In this formulation of the problem, the nodes are weighted by string depth. This attention has culminated in a data structure for weighted ancestors in suffix trees with $\mathcal{O}(1)$ query time and an $\mathcal{O}(n)$-time construction algorithm [Belazzougui et al., CPM 2021]. In this paper, we consider a different version of the weighted ancestor problem, where the nodes are weighted by any function $\textsf{weight}$ that maps the nodes of $T$ to positive integers, such that $\textsf{weight}(u)\le \textsf{size}(u)$ for any node $u$ and $\textsf{weight}(u_1)\le \textsf{weight}(u_2)$ if node $u_1$ is a descendant of node $u_2$, where $\textsf{size}(u)$ is the number of nodes in the subtree rooted at $u$. In the size-constrained weighted ancestor (SWAQ) problem, for any node $u$ of $T$ and any integer $k$, we are asked to return the lowest ancestor $w$ of $u$ with weight at least $k$. We show that for any rooted tree with $n$ nodes, we can locate node $w$ in $\mathcal{O}(1)$ time after $\mathcal{O}(n)$-time preprocessing. In particular, this implies a data structure for the SWAQ problem in suffix trees with $\mathcal{O}(1)$ query time and $\mathcal{O}(n)$-time preprocessing, when the nodes are weighted by $\textsf{weight}$. We also show several string-processing applications of this result

Optimal Resource Allocation and Relay Selection in Bandwidth Exchange Based Cooperative Forwarding

In this paper, we investigate joint optimal relay selection and resource allocation under bandwidth exchange (BE) enabled incentivized cooperative forwarding in wireless networks. We consider an autonomous network where N nodes transmit data in the uplink to an access point (AP) / base station (BS). We consider the scenario where each node gets an initial amount (equal, optimal based on direct path or arbitrary) of bandwidth, and uses this bandwidth as a flexible incentive for two hop relaying. We focus on alpha-fair network utility maximization (NUM) and outage reduction in this environment. Our contribution is two-fold. First, we propose an incentivized forwarding based resource allocation algorithm which maximizes the global utility while preserving the initial utility of each cooperative node. Second, defining the link weight of each relay pair as the utility gain due to cooperation (over noncooperation), we show that the optimal relay selection in alpha-fair NUM reduces to the maximum weighted matching (MWM) problem in a non-bipartite graph. Numerical results show that the proposed algorithms provide 20- 25% gain in spectral efficiency and 90-98% reduction in outage probability.Comment: 8 pages, 7 figure

On distributed scheduling in wireless networks exploiting broadcast and network coding

In this paper, we consider cross-layer optimization in wireless networks with wireless broadcast advantage, focusing on the problem of distributed scheduling of broadcast links. The wireless broadcast advantage is most useful in multicast scenarios. As such, we include network coding in our design to exploit the throughput gain brought in by network coding for multicasting. We derive a subgradient algorithm for joint rate control, network coding and scheduling, which however requires centralized link scheduling. Under the primary interference model, link scheduling problem is equivalent to a maximum weighted hypergraph matching problem that is NP-complete. To solve the scheduling problem distributedly, locally greedy and randomized approximation algorithms are proposed and shown to have bounded worst-case performance. With random network coding, we obtain a fully distributed cross-layer design. Numerical results show promising throughput gain using the proposed algorithms, and surprisingly, in some cases even with less complexity than cross-layer design without broadcast advantage

Simple Distributed Weighted Matchings

Wattenhofer [WW04] derive a complicated distributed algorithm to compute a weighted matching of an arbitrary weighted graph, that is at most a factor 5 away from the maximum weighted matching of that graph. We show that a variant of the obvious sequential greedy algorithm [Pre99], that computes a weighted matching at most a factor 2 away from the maximum, is easily distributed. This yields the best known distributed approximation algorithm for this problem so far