2369 /www/nrich/html/content/id/2369/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><br></br> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/7372">Warm-up</a></li> <li><a href="http://nrich.maths.org/2368">Try this next</a></li> <li><a href="http://nrich.maths.org/4834">Think higher</a></li> <li><a href="http://en.wikipedia.org/wiki/Mathematical_optimization">Read: mathematics</a></li> <li><a href="http://en.wikipedia.org/wiki/Engineering_optimization">Read: science</a></li> <li><a href="http://en.wikipedia.org/wiki/Discrete_mathematics">Explore further</a></li> </ul> <div> </div> <br></br> <p><mdo:image align="top" alt="max flow graph" bgcolor="" height="292" src="5-2max_flow1.gif" width="314"></mdo:image></p> <div>The graph represents a supply network from $A$ to $B$ and the numbers on the edges of the graph show the maximum capacity for flow in each of the sections.</div> <br></br> <div>Imagine any straight line cutting through edges of the graph (but not through vertices) such that $A$ is on one side of the line and $B$ is on the other. All the flow from $A$ to $B$ has to go along the edges cut by your line so the total flow from $A$ to $B$ is less than or equal to the sum of the flows along those edges. Considering all possible such cuts, why is it that the maximum flow from $A$ to $B$ is less than the minimum sum for all cuts? Find the maximum flow in this example.</div> <div><mdo:image align="top" alt="cube graph with max flows" bgcolor="" height="248" src="5-2max_flow2.gif" width="275"></mdo:image></div> <p>In the second example the network is a cube. Find the maximum flow from $A$ to $G$.</p></mdoxml> <?xml version="1.0" encoding="ISO-8859-1"?> <mdoxml version="1.0"> <p class="editorial">Yanqing from Lipson Community College and Andrei from Tudor Vianu National College, Bucharest, Romania both sent correct solutions.</p> <p>Andrei wrote:</p> <p>&quot; As I am not familiar with graph theory, I first read the hint and the notes of the problem, then I looked on the web for the Maximum flow, minimum cut theorem. I found different examples, algorithms of solving such problems, as well a possible list of utilisation.</p> <p>I started with the definition of terms: a cut in a network is a minimal set of edges whose removal separates the network into two components, one containing the source, and the other the sink.&quot;</p> <p>In the first network a cut through $PT$, $RT$, $RU$, $RB$, and $SB$ separates the network showing that the flow cannot be more than $18$ units (the total flow through these edges).</p> <div><mdo:image height="292" width="314" src="5-2max_flow1.gif" alt="flow network 1"></mdo:image></div> <p>This flow is possible by sending: $3$ units along $ASB$, $5$ units along $ASRB$, $5$ units along $ARB$, $2$ units along $ARUTB$, $1$ unit along $AQRTB$ and $2$ units along $APTB$. Hence the maximum flow is $18$ units.</p> <p>In the second network a cut through $BC$, $DC$, $FG$ and $HG$ separates the network so the flow cannot be more than $16$ units (the total flow along these edges).</p> <div><mdo:image height="248" width="275" src="5-2max_flow2.gif" alt="flow network 2"></mdo:image></div> <p>This flow is possible by sending $3$ units along $ABFG$, $4$ units along $ABCG$, $2$ units along $ADCG$, $3$ units along $ADHG$ and $4$ units along $AEHG$. Hence the maximum flow is $16$ units.</p> </mdoxml> <mdoxml version="1.0"> <p> There is a 'Maximum flow, minimum cut' theorem for which the proof is sketched in this question. Finding the solutions requires all cases to be checked. </p> </mdoxml> <mdoxml version="1.0">If your 'cut' were to block the edges it cuts through there would be no flow from $A$ to $B$ but if those edges take as much flow as they can they only allow through the sum of the flows on each of the 'cut' edges. Whatever cut you make you still have to check that sufficient flow reaches that cut to allow through the flow given by that sum.<br></br></mdoxml> 2 3 0 0 0 0 1 Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network. Decision Mathematics and Combinatorics Optimisation Decision Mathematics and Combinatorics Flows in a network Decision Mathematics and Combinatorics Networks/Graph Theory Using, Applying and Reasoning about Mathematics Mathematical modelling Applications engineering Admin Individual