Pseudo-Boolean function: Difference between revisions

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{{Short description|Generalization of binary functions}}

In [[mathematics]] and [[optimization]], a '''pseudo-Boolean function''' is a [[function (mathematics)|function]] of the form

:<math>f: \mathbf{B}^n \to \R,</math>

where {{math|1='''B''' = {{mset|0, 1}}}} is a ''[[Boolean ___domain]]'' and {{mvar|n}} is a nonnegative [[integer]] called the [[arity]] of the function. A [[Boolean function]] is then a special case, where the codomainvalues of <math>f</math>isare also restricted to 0 or 1.

==Representations==

Any pseudo-Boolean function can be written uniquely as a [[multi-linear]] polynomial:<ref>{{Citecite journal |titlelast1=Hammer |first1=P.L. |last2=Rosenberg |first2=I. |last3=Rudeanu |first3=S.|date=1963 |title=On the determination of the minima of pseudo-Boolean functions|last1 = Hammer|first1 language=ro P.L.|date = 1963|journal = Studii și cercetări matematice|last2 = Rosenberg|first2 = I.|last3 = Rudeanu|first3 = S.|issue =14 14|pages = 359–364|language = ro|issn = 0039-4068}}</ref><ref>{{Citecite book |titlelast1=Hammer |first1=Peter L. |last2=Rudeanu |first2=Sergiu |year=1968 |title=Boolean Methods in Operations Research and Related Areas|last1 = Hammer|first1 = Peter L.|publisher = Springer|year = 1968|isbn = 978-3-642-85825-3|last2 = Rudeanu|first2 = Sergiu}}</ref>

:<math>f(\boldsymbol{x}) = a + \sum_i a_ix_i + \sum_{i<j}a_{ij}x_ix_j + \sum_{i<j<k}a_{ijk}x_ix_jx_k + \ldots</math>

The '''degree''' of the pseudo-Boolean function is simply the degree of the [[polynomial]] in this representation.

In many settings (e.g., in [[Analysis of Boolean functions|Fourier analysis of pseudo-Boolean functions]]), a pseudo-Boolean function is viewed as a function <math>f</math> that maps <math>\{-1,1\}^n</math> to <math>\mathbb{R}</math>. Again in this case we can uniquely write <math>f</math> as a multi-linear polynomial:

<math> f(x)= \sum_{I\subseteq [n]}\hat{f}(I)\prod_{i\in I}x_i, </math> where <math> \hat{f}(I) </math> are Fourier coefficients of <math>f</math> and <math>[n]=\{1,...,n\}</math>.

==Optimization==

Minimizing (or, equivalently, maximizing) a pseudo-Boolean function is [[NP-hard]]. This can easily be seen by formulating, for example, the [[maximum cut]] problem as maximizing a pseudo-Boolean function.<ref name="boroshammer" Boros2002/>

===Submodularity===

The [[submodular set function]]s can be viewed as a special class of pseudo-Boolean functions, which is equivalent to the condition

:<math> f(\boldsymbol{x}) + f(\boldsymbol{y}) \ge f(\boldsymbol{x} \wedge \boldsymbol{y}) + f(\boldsymbol{x} \vee \boldsymbol{y}), \; \forall \boldsymbol{x}, \boldsymbol{y}\in \mathbf{B}^n\,. </math>

This is an important class of pseudo-boolean functions, because they can be [[Submodular set function#Submodular Minimizationset function minimization|minimized in polynomial time]]. Note that minimization of a submodular function is a polynomially solvable problem independent on the presentation form, for e.g. pesudo-Boolean polynomials, opposite to maximization of a submodular function which is NP-hard, Alexander Schrijver (2000).

===Roof Duality===

If ''f'' is a quadratic polynomial, a concept called ''roof duality'' can be used to obtain a lower bound for its minimum value.<ref name="boroshammer"Boros2002>{{cite journal |last1=Boros and|first1=E. |last2=Hammer, |first2=P. L. |year=2002 |title=Pseudo-Boolean Optimization |journal=[[Discrete Applied Mathematics]] |doi=10.1016/S0166-218X(01)00341-9 |doi-access=free |volume=123 |issue=1–3 |pages=155–225 |hdl=2268/202427 |url=http://orbi.ulg.ac.be/handle/2268/202427|hdl-access=free }}</ref> Roof duality may also provide a partial assignment of the variables, indicating some of the values of a minimizer to the polynomial. Several different methods of obtaining lower bounds were developed only to later be shown to be equivalent to what is now called roof duality.<ref name="boroshammer" Boros2002/>

===Quadratizations===

If the degree of ''f'' is greater than 2, one can always employ ''reductions'' to obtain an equivalent quadratic problem with additional variables. One possible reduction is

:<math> \displaystyle -x_1x_2x_3=\min_{z\in\mathbf{B}}z(2-x_1-x_2-x_3) </math>

There are other possibilities, for example,

:<math> \displaystyle -x_1x_2x_3=\min_{z\in\mathbf{B}}z(-x_1+x_2+x_3)-x_1x_2-x_1x_3+x_1. </math>

Different reductions lead to different results. Take for example the following [[Cubic function|cubic polynomial]]:<ref name="kahlstrandmark"Kahl2011>{{cite conference |last1=Kahl and|first1=F. |last2=Strandmark, |first2=P. |year=2011|title=Generalized Roof Duality for Pseudo-Boolean Optimization |conference=[[International Conference on Computer Vision]] |url=http://www.maths.lth.se/vision/publdb/reports/pdf/kahl-strandmark-iccv-11.pdf}}</ref>

:<math> \displaystyle f(\boldsymbol{x})=-2x_1+x_2-x_3+4x_1x_2+4x_1x_3-2x_2x_3-2x_1x_2x_3. </math>

Using the first reduction followed by roof duality, we obtain a lower bound of -3−3 and no indication on how to assign the three variables. Using the second reduction, we obtain the (tight) lower bound of -2−2 and the optimal assignment of every variable (which is <math> {(0,1,1)}</math>).

===Polynomial Compression Algorithms===

Consider a pseudo-Boolean function <math> f </math> as a mapping from <math>\{-1,1\}^n</math> to <math>\mathbb{R}</math>. Then <math> f(x)= \sum_{I\subseteq [n]}\hat{f}(I)\prod_{i\in I}x_i. </math> Assume that each coefficient <math>\hat{f}(I)</math> is integral. Then for an integer <math>k</math> the problem P of deciding whether <math> f(x) </math> is more or equal to <math>k</math> is NP-complete. It is proved in <ref name=Crowston2011>{{cite journal |last1=Crowston |first1=R. |last2=Fellows |first2=M. |last3=Gutin |first3=G. |last4=Jones |first4=M. |last5=Rosamond |first5=F. |last6=Thomasse |first6=S. |last7=Yeo |first7=A. |year=2011 |title=Simultaneously Satisfying Linear Equations Over GF(2): MaxLin2 and Max-r-Lin2 Parameterized Above Average. |journal=Proc. Of FSTTCS 2011 |arxiv=1104.1135 |bibcode=2011arXiv1104.1135C}}</ref> that in polynomial time we can either solve P or reduce the number of variables to <math> O(k^2\log k).</math> Let <math>r</math> be the degree of the above multi-linear polynomial for <math>f</math>. Then <ref name=Crowston2011/> proved that in polynomial time we can either solve P or reduce the number of variables to <math>r(k-1)</math>.

Then for an integer <math> k </math> the problem P of deciding whether <math> f(x) </math> is more or equal to <math> k </math> is NP-complete. It is proved in <ref name="crow">Crowston et al., 2011</ref> that in polynomial time we can either solve P or reduce the number of variables to <math> O(k^2\log k).</math> Let <math> r </math> be the degree of the above multi-linear polynomial for <math> f </math>. Then <ref name="crow">Crowston et al., 2011</ref> proved that in polynomial time we can either solve P or reduce the number of variables to <math> r(k-1) </math>.

==See also==

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==References==

* {{cite journal |last=Ishikawa | first=H. |year=2011 |title=Transformation of general binary MRF minimization to the first order case |journal= [[IEEE Transactions on Pattern Analysis and Machine Intelligence]] |year=2011|volume=33|number=6|pages=1234–1249| doi=10.1109/tpami.2010.91 |pmid=20421673 |citeseerx=10.1.1.675.2183| |s2cid=17314555 |volume=33 |number=6 |pages=1234–1249}}

* {{cite journal|author1=Boros, E.|author2=Hammer, P. L. |title=Pseudo-Boolean Optimization|journal= [[Discrete Applied Mathematics]] |year=2002|volume=123|issue=1–3 |pages=155–225 |doi=10.1016/S0166-218X(01)00341-9|url=http://orbi.ulg.ac.be/handle/2268/202427 }}

* {{cite journal |last=O'Donnell |first=Ryan | author-link = Ryan O'Donnell (computer scientist)|year=2008 |title=Some topics in analysis of Boolean functions |journal={{ECCC |2008|08|055}}|yearissn=20081433-8092 |url=httphttps://www.eccc.uni-trierweizmann.deac.il/eccc-reportsreport/2008/TR08-055/}}

* {{cite journal|author1=Crowston, R. |author2=Fellows, M. | author3= Gutin, G. |author4=Jones, M. | author5= Rosamond, F. |author6=Thomasse, S. | author7= Yeo, A.|title=Simultaneously Satisfying Linear Equations Over GF(2): MaxLin2 and Max-r-Lin2 Parameterized Above Average.|journal=Proc. Of FSTTCS 2011|year=2011|arxiv=1104.1135|bibcode=2011arXiv1104.1135C}}

* {{cite conference |author1last1=Rother, |first1=C. |author2last2=Kolmogorov, |first2=V. |author3last3=Lempitsky, |first3=V. |author4last4=Szummer, |first4=M. |year=2007 |title=Optimizing Binary MRFs via Extended Roof Duality |conference= [[Conference on Computer Vision and Pattern Recognition]] |year=2007|url=http://research.microsoft.com/pubs/67978/cvpr07-QPBOpi.pdf}}

* {{cite journal|last=Ishikawa | first=H. |title=Transformation of general binary MRF minimization to the first order case|journal= [[IEEE Transactions on Pattern Analysis and Machine Intelligence]] |year=2011|volume=33|number=6|pages=1234–1249| doi=10.1109/tpami.2010.91|pmid=20421673|citeseerx=10.1.1.675.2183| s2cid=17314555 }}

* Alexander{{cite journal |last=Schrijver. |first=Alexander |date=November 2000 |title=A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time. |journal=Journal of Combinatorial Theory, Series B, Volume|doi=10.1006/jctb.2000.1989 |volume=80, Issue |issue=2, November|pages=346–355|doi-access=free 2000, Pages 346-355.}}

* {{cite conference|author1=Kahl, F.|author2=Strandmark, P. |title=Generalized Roof Duality for Pseudo-Boolean Optimization| conference= [[International Conference on Computer Vision]] |year=2011|url=http://www.maths.lth.se/vision/publdb/reports/pdf/kahl-strandmark-iccv-11.pdf}}

* {{cite journal|last=O'Donnell|first=Ryan|title=Some topics in analysis of Boolean functions|journal={{ECCC|2008|08|055}}|year=2008|url=http://www.eccc.uni-trier.de/eccc-reports/2008/TR08-055/}}

* {{cite conference|author1=Rother, C.|author2=Kolmogorov, V.|author3=Lempitsky, V.|author4=Szummer, M.|title=Optimizing Binary MRFs via Extended Roof Duality|conference= [[Conference on Computer Vision and Pattern Recognition]] |year=2007|url=http://research.microsoft.com/pubs/67978/cvpr07-QPBOpi.pdf}}

* Alexander Schrijver. A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time. Journal of Combinatorial Theory, Series B, Volume 80, Issue 2, November 2000, Pages 346-355.

[[Category:Mathematical optimization]]