Approximating max-min linear programs with local algorithms
A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min LPs where the objective is to maximise \(\min_k \...
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| Published in: | arXiv.org |
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| Main Authors: | , , , |
| Format: | Paper |
| Language: | English |
| Published: |
Ithaca
Cornell University Library, arXiv.org
08.10.2007
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| Subjects: | |
| ISSN: | 2331-8422 |
| Online Access: | Get full text |
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| Summary: | A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min LPs where the objective is to maximise \(\min_k \sum_v c_{kv} x_v\) subject to \(\sum_v a_{iv} x_v \le 1\) for each \(i\) and \(x_v \ge 0\) for each \(v\). Here \(c_{kv} \ge 0\), \(a_{iv} \ge 0\), and the support sets \(V_i = \{v : a_{iv} > 0 \}\), \(V_k = \{v : c_{kv}>0 \}\), \(I_v = \{i : a_{iv} > 0 \}\) and \(K_v = \{k : c_{kv} > 0 \}\) have bounded size. In the distributed setting, each agent \(v\) is responsible for choosing the value of \(x_v\), and the communication network is a hypergraph \(\mathcal{H}\) where the sets \(V_k\) and \(V_i\) constitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if \(|V_i|\) and \(|V_k|\) are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in \(\mathcal{H}\). |
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| Bibliography: | SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50 |
| ISSN: | 2331-8422 |
| DOI: | 10.48550/arxiv.0710.1499 |