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| [[Image:Graph fractional coloring.svg|right|thumb| 5:2-coloring of [[Dodecahedron|Dodecahedral graph]]. A 4:2-coloring of
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| '''Fractional coloring''' is a topic in a young branch of [[graph theory]] known as [[fractional graph theory]]. It is a generalization of ordinary [[graph coloring]]. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a ''set'' of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common.
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| Fractional graph coloring can be viewed as the [[linear programming relaxation]] of traditional graph coloring. Indeed, fractional coloring problems are much more amenable to a linear programming approach than traditional coloring problems.
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| ==Definitions==
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| [[File:Fractional coloring of C5.png|thumb|Above:A 3:1-coloring of the cycle on 5 vertices, and the corresponding 6:2-coloring.<br/>Below: A 5:2 coloring of the same graph.]]
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| A '''''b''-fold coloring''' of a graph ''G'' is an assignment of sets of size ''b'' to vertices of a graph such that adjacent vertices receive disjoint sets.
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| An '''''a'':''b''-coloring''' is a ''b''-fold coloring out of ''a'' available colors.
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| The '''''b''-fold chromatic number''' χ<sub>''b''</sub>(''G'') is the least ''a'' such that an ''a'':''b''-coloring exists.
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| The '''fractional chromatic number''' χ<sub>f</sub>(''G'') is defined to be
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| :<math>\chi_{f}(G) = \lim_{b \to \infty}\frac{\chi_{b}(G)}{b} = \inf_{b}\frac{\chi_{b}(G)}{b}</math>
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| Note that the limit exists because χ<sub>''b''</sub>(''G'') is ''[[subadditive function|subadditive]]'', meaning χ<sub>''a''+''b''</sub>(''G'') ≤ χ<sub>''a''</sub>(''G'') + χ<sub>''b''</sub>(''G'').
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| The fractional chromatic number can equivalently be defined in probabilistic terms. χ<sub>f</sub>(''G'') is the smallest ''k'' for which there exists a probability distribution over the [[Independent_set_(graph_theory)|independent sets]] of ''G'' such that for each vertex ''v'', given an independent set ''S'' drawn from the distribution,
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| :<math>\Pr(v\in S) \geq \frac{1}{k}</math>.
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| Some properties of χ<sub>''f''</sub>(''G''):
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| :<math>\chi_f(G)\ge n(G)/\alpha(G)</math>
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| :<math>\omega(G) \le \chi_f(G) \le \chi(G)</math>.
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| Here n(''G'') is the [[glossary of graph theory#Basics|order]] of ''G'', α(''G'') is the [[independence number]], ω(''G'') is the [[clique number]], and χ(''G'') is the [[chromatic number]].
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| ==Linear Programming (LP) Formulation==
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| The fractional chromatic number χ<sub>f</sub>(''G'') of a graph ''G'' can be obtained as a solution to a [[linear programming|linear program]]. Let <math>\mathcal{I}</math>(''G'') be the set of all independent sets of ''G'', and let <math>\mathcal{I}</math>(''G'',''x'') be the set of all those independent sets which include vertex ''x''. For each independent set ''I'', define a nonnegative real variable ''x<sub>I</sub>''. Then χ<sub>f</sub>(''G'') is the minimum value of
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| : <math>\sum_{I\in\mathcal{I}(G)} x_I\,</math>,
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| : subject to <math>\sum_{I\in\mathcal{I}(G,x)} x_I \ge 1</math> for each vertex <math>x</math>.
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| The [[dual problem|dual]] of this linear program computes the "fractional clique number", a relaxation to the rationals of the integer concept of [[clique number]]. That is, a weighting of the vertices of ''G'' such that the total weight assigned to any independent set is at most ''1''. The [[strong duality]] theorem of linear programming guarantees that the optimal solutions to both linear programs have the same value. Note however that each linear program may have size that is exponential in the number of vertices of ''G'', and that computing the fractional chromatic number of a graph is [[NP-hard]].<ref>[[Carsten Lund]] and [[Mihalis Yannakakis]]: "[http://dx.doi.org/10.1145/185675.306789 On the hardness of approximating minimization problems]", J. ACM 41:5(1994), p. 960-981.</ref> This stands in contrast to the problem of fractionally coloring the edges of a graph, which can be solved in polynomial time. This is a straightforward consequence of Edmonds' matching polytope theorem.<ref>Bruce Hajek and Galen Sasaki: "[http://dx.doi.org/10.1109/18.21215 Link scheduling in polynomial time]", IEEE Transactions on Information Theory 34:5(1988), p. 910-917.</ref><ref name=schrijver>{{cite book|last=Schrijver|first=Alexander|title=Combinatorial Optimization: Polyhedra and Efficiency|year=2003|publisher=Springer-Verlag|location=Berlin ; Heidelberg ; New York, N.Y.|isbn=3540443894|pages=474}}</ref> | |
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| ==Applications==
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| Applications of fractional graph coloring include ''activity scheduling''. In this case, the graph ''G'' is a ''conflict graph'': an edge in ''G'' between the nodes ''u'' and ''v'' denotes that ''u'' and ''v'' cannot be active simultaneously. Put otherwise, the set of nodes that are active simultaneously must be an independent set in graph ''G''.
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| An optimal fractional graph coloring in ''G'' then provides a shortest possible schedule, such that each node is active for (at least) 1 time unit in total, and at any point in time the set of active nodes is an independent set. If we have a solution ''x'' to the above linear program, we simply traverse all independent sets ''I'' in an arbitrary order. For each ''I'', we let the nodes in ''I'' be active for <math>x_I</math> time units; meanwhile, each node not in ''I'' is inactive.
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| In more concrete terms, each node of ''G'' might represent a ''radio transmission'' in a wireless communication network; the edges of ''G'' represent ''interference'' between radio transmissions. Each radio transmission needs to be active for 1 time unit in total; an optimal fractional graph coloring provides a minimum-length schedule (or, equivalently, a maximum-bandwidth schedule) that is conflict-free.
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| ===Comparison with traditional graph coloring===
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| If one further required that each node must be active ''continuously'' for 1 time unit (without switching it off and on every now and then), then traditional graph [[vertex coloring]] would provide an optimal schedule: first the nodes of color 1 are active for 1 time unit, then the nodes of color 2 are active for 1 time unit, and so on. Again, at any point in time, the set of active nodes is an independent set.
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| In general, fractional graph coloring provides a shorter schedule than non-fractional graph coloring; there is an [[integrality gap]]. It may be possible to find a shorter schedule, at the cost of switching devices (such as radio transmitters) on and off more than once.
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| ==Notes==
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| <references />
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| ==References==
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| *{{citation
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| | last1 = Scheinerman | first1 = Edward R. | author1-link = Ed Scheinerman
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| | last2 = Ullman | first2 = Daniel H.
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| | isbn = 0-471-17864-0
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| | location = New York
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| | publisher = Wiley-Interscience
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| | title = Fractional graph theory
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| | year = 1997}}.
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| *{{citation
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| | last1 = Godsil | first1 = Chris | author1-link = Chris Godsil
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| | last2 = Royle | first2 = Gordon | author2-link = Gordon Royle
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| | isbn = 0-387-95241-1
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| | location = New York
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| | publisher = Springer-Verlag
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| | title = Algebraic Graph Theory
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| | year = 2001}}.
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| [[Category:Graph coloring]]
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