View source for Baker's technique

In [[theoretical computer science]], '''Baker's technique''' is a method for designing [[polynomial-time approximation scheme]]s (PTASs) for problems on [[planar graph]]s. It is named after [[Brenda Baker]], who announced it in a 1983 conference and published it in the ''[[Journal of the ACM]]'' in 1994.

The idea for Baker's technique is to break the graph into layers, such that the problem can be solved optimally on each layer, then combine the solutions from each layer in a reasonable way that will result in a feasible solution.  This technique has given PTASs for the following problems: [[subgraph isomorphism]], [[maximum independent set]], [[minimum vertex cover]], [[minimum dominating set]], minimum [[edge dominating set]], maximum triangle matching, and many others.

The [[bidimensionality|bidimensionality theory]] of  [[Erik Demaine]], Fedor Fomin, [[Mohammad Hajiaghayi|Hajiaghayi]], and Dimitrios Thilikos and its offshoot ''simplifying decompositions'' ({{harvtxt|Demaine|Hajiaghayi|Kawarabayashi|2005}},{{harvtxt|Demaine|Hajiaghayi|Kawarabayashi|2011}}) generalizes and greatly expands the applicability of Baker's technique
for a vast set of problems on [[planar graph]]s and more generally  [[graph minor|graphs excluding a fixed minor]], such as bounded genus graphs, as well as to other classes of graphs not closed under taking minors such as the [[1-planar graph]]s.

==Example of technique==
The example that we will use to demonstrate Baker's technique is the maximum weight [[Independent set (graph theory)|independent set]] problem.

===Algorithm===
 INDEPENDENT-SET(<math>G</math>, <math>w</math>, <math>\epsilon</math>)
     Choose an arbitrary vertex <math> r </math>
     <math>k = 1/\epsilon</math>
     find the breadth-first search levels for <math> G </math> rooted at <math> r </math> <math>\pmod k</math>: <math>\{V_0,V_1, \ldots, V_{k-1} \}</math>
 
     '''for''' <math>\ell = 0, \ldots, k-1</math>
         find the components <math>G^\ell_1, G^\ell_2, \ldots,</math> of <math>G</math> after deleting <math>V_\ell</math>
 
     '''for''' <math>i = 1,2, \ldots </math>
        compute <math>S_i^\ell</math>, the maximum-weight independent set of <math>G_i^\ell</math>
 
     <math>S^\ell = \cup_i S_i^\ell</math>
     let <math>S^{\ell^*}</math> be the solution of maximum weight among <math>\{S^0,S^1, \ldots, S^{k-1} \}</math>
 
     '''return''' <math>S^{\ell^*}</math>

Notice that the above algorithm is feasible because each <math>S^\ell </math> is the union of disjoint independent sets.

===Dynamic programming===
[[Dynamic programming]] is used when we compute the maximum-weight independent set for each <math>G_i^\ell</math>. This dynamic program works because each <math>G_i^\ell</math> is a [[K-outerplanar graph|<math>k</math>-outerplanar graph]]. Many NP-complete problems can be solved with dynamic programming on <math>k</math>-outerplanar graphs. Baker's technique can be interpreted as covering the given planar graphs with subgraphs of this type, finding the solution to each subgraph using dynamic programming, and gluing the solutions together.

==References==
{{refbegin|colwidth=30em}}
*{{citation
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 | publisher = IEEE Computer Society
 | title = 24th Annual Symposium on Foundations of Computer Science, Tucson, Arizona, USA, 7–9 November 1983
 | year = 1983}}
*{{citation
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 | volume = 317
 | year = 1988}}
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*{{citation
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* {{cite web
 | author = Albin James Maliakal
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 | year = 2024
 | access-date = 2024-10-08
 }}
{{refend}}

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[[Category:1983 in computing]]
[[Category:Planar graphs]]
[[Category:Approximation algorithms]]