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Approximating Solution Structure

Authors Iris van Rooij, Matthew Hamilton, Moritz Müller, Todd Wareham

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Iris van Rooij
Matthew Hamilton
Moritz Müller
Todd Wareham

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Iris van Rooij, Matthew Hamilton, Moritz Müller, and Todd Wareham. Approximating Solution Structure. In Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs. Dagstuhl Seminar Proceedings, Volume 7281, pp. 1-24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2007)


hen it is hard to compute an optimal solution $y in optsol(x)$ to an instance $x$ of a problem, one may be willing to settle for an efficient algorithm $A$ that computes an approximate solution $A(x)$. The most popular type of approximation algorithm in Computer Science (and indeed many other applications) computes solutions whose value is within some multiplicative factor of the optimal solution value, {em e.g.}, $max(frac{val(A(x))}{optval(x)}, frac{optval(x)}{val(A(x))}) leq h(|x|)$ for some function $h()$. However, an algorithm might also produce a solution whose structure is ``close'' to the structure of an optimal solution relative to a specified solution-distance function $d$, {em i.e.}, $d(A(x), y) leq h(|x|)$ for some $y in optsol(x)$. Such structure-approximation algorithms have applications within Cognitive Science and other areas. Though there is an extensive literature dating back over 30 years on value-approximation, there is to our knowledge no work on general techniques for assessing the structure-(in)approximability of a given problem. In this talk, we describe a framework for investigating the polynomial-time and fixed-parameter structure-(in)approximability of combinatorial optimization problems relative to metric solution-distance functions, {em e.g.}, Hamming distance. We motivate this framework by (1) describing a particular application within Cognitive Science and (2) showing that value-approximability does not necessarily imply structure-approximability (and vice versa). This framework includes definitions of several types of structure approximation algorithms analogous to those studied in value-approximation, as well as structure-approximation problem classes and a structure-approximability-preserving reducibility. We describe a set of techniques for proving the degree of structure-(in)approximability of a given problem, and summarize all known results derived using these techniques. We also list 11 open questions summarizing particularly promising directions for future research within this framework. vspace*{0.15in} oindent (co-presented with Todd Wareham) vspace*{0.15in} jointwork{Hamilton, Matthew; M"{u}ller, Moritz; van Rooij, Iris; Wareham, Todd}
  • Approximation Algorithms
  • Solution Structure


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