# hw5.pdf - Fall 2021 CS 583: Approximation Algorithms...

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Fall 2021, CS 583: Approximation AlgorithmsHomework 5Due: 11/30/2021 in GradescopeInstructions and Policy:Each student should write up their own solutions independently.You need to indicate the names of the people you discussed a problem with; ideally youshould discuss with no more than two other people. You may be able to find solutions tothe problems in various papers and books but it would defeat the purpose of learning if copythem. You should cite all sources that you use and write in your own words.Read through all the problems and think about them and how they relate to what we coveredin the lectures. Solve as many problems as you can. Please submit solutions to Problem 1,6 and at least two other problems. Some problems are closely related so it may benefit youto solve them together or view them as parts of an extended problem.Please write clearly and concisely - clarity and brevity will be rewarded.Refer to knownfacts as necessary.Problem 1We have mostly seen edge-weighted network design problems in undirectedgraphs. As we discussed in lecture, directed network design problems are hard to approxi-mate. Undirectednode-weightednetwork design problems fall in between. Here we considernode-weighted Steiner tree problem.The input is a graphG= (V, E) with non-negativenode weightsw:VR+, and a set ofkterminalsSV. The goal is to find a Steiner treeT= (X, F) for the terminals (that is,SX) with minimum node weight. We will assumethat the weight of the terminals is 0 without loss of generality since every terminal has tobe included in the tree.We saw that Steiner tree in edge-weighted graphs admits a sim-ple 2-approximation via the MST heuristic, and improved approximations are also known.In contrast the node-weighted problem is harder. This problem will walk you through anapproach to approximate it via Set Cover type ideas.Describe a reduction from Set Cover onmsets andnelements to node-weighted Steinertree withk=n.Informally argue that anα(k)-approximation for node-weightedSteiner tree implies anα(n)-approximation for Set Cover. This should convince youthat node-weighted Steiner tree does not admit a better than lnkapproximation.Given a set of terminals, a spider is a star-like tree with a center vertexcandklegswhere each leg is ac-tpath for some terminalt(the center itself can be a terminal).Thus, all the leaves are terminals ifk >1. Ifk= 1 then we require that the centerto be a terminal and in this case the spider is simply a path connecting two terminals.See figure. Argue that any minimal Steiner tree (X, F) contains a collection of node-disjoint spiders that together include all but one terminal fromS.1

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Term
Spring
Professor
Bing Liu
Tags
Hypergraph Multiway Cut