{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article1876","name":"Approximating Solution Structure","abstract":"hen it is hard to compute an optimal solution $y in optsol(x)$ to an\r\ninstance $x$ of a problem, one may be willing to settle for an efficient\r\nalgorithm $A$ that computes an approximate solution $A(x)$. The most\r\npopular type of approximation algorithm in Computer Science (and indeed\r\nmany other applications) computes solutions whose value is within some multiplicative factor of the optimal solution value, {em e.g.},\r\n$max(frac{val(A(x))}{optval(x)}, frac{optval(x)}{val(A(x))}) leq\r\nh(|x|)$ for some function $h()$. However, an algorithm might also\r\nproduce a solution whose structure is ``close'' to the structure of an\r\noptimal solution relative to a specified solution-distance function $d$,\r\n{em i.e.}, $d(A(x), y) leq h(|x|)$ for some $y in optsol(x)$. Such\r\nstructure-approximation algorithms have applications within Cognitive\r\nScience and other areas. Though there is an\r\nextensive literature dating back over 30 years on value-approximation,\r\nthere is to our knowledge no work on general techniques for assessing\r\nthe structure-(in)approximability of a given problem.\r\n\r\nIn this talk, we describe a framework for investigating the\r\npolynomial-time and fixed-parameter structure-(in)approximability of\r\ncombinatorial optimization problems relative to metric solution-distance\r\nfunctions, {em e.g.}, Hamming distance. We motivate this framework by\r\n(1) describing a particular application within Cognitive Science and (2)\r\nshowing that value-approximability does not necessarily imply\r\nstructure-approximability (and vice versa). This framework includes\r\ndefinitions of several types of structure approximation algorithms\r\nanalogous to those studied in value-approximation, as well as\r\nstructure-approximation problem classes and a\r\nstructure-approximability-preserving reducibility. We describe a set of techniques for proving the degree of\r\nstructure-(in)approximability of a given problem, and summarize all\r\nknown results derived using these techniques. We also list 11 open\r\nquestions summarizing particularly promising directions for future\r\nresearch within this framework.\r\n\r\nvspace*{0.15in}\r\n\r\noindent\r\n(co-presented with Todd Wareham)\r\nvspace*{0.15in}\r\n\r\njointwork{Hamilton, Matthew; M\"{u}ller, Moritz; van Rooij, Iris; Wareham, Todd}","keywords":["Approximation Algorithms","Solution Structure"],"author":[{"@type":"Person","name":"van Rooij, Iris","givenName":"Iris","familyName":"van Rooij"},{"@type":"Person","name":"Hamilton, Matthew","givenName":"Matthew","familyName":"Hamilton"},{"@type":"Person","name":"M\u00fcller, Moritz","givenName":"Moritz","familyName":"M\u00fcller"},{"@type":"Person","name":"Wareham, Todd","givenName":"Todd","familyName":"Wareham"}],"position":3,"pageStart":1,"pageEnd":24,"dateCreated":"2007-11-28","datePublished":"2007-11-28","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"van Rooij, Iris","givenName":"Iris","familyName":"van Rooij"},{"@type":"Person","name":"Hamilton, Matthew","givenName":"Matthew","familyName":"Hamilton"},{"@type":"Person","name":"M\u00fcller, Moritz","givenName":"Moritz","familyName":"M\u00fcller"},{"@type":"Person","name":"Wareham, Todd","givenName":"Todd","familyName":"Wareham"}],"copyrightYear":"2007","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.07281.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume660","volumeNumber":7281,"name":"Dagstuhl Seminar Proceedings, Volume 7281","dateCreated":"2007-11-28","datePublished":"2007-11-28","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article1876","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume660"}}}