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## MA428

Combinatorial Optimisation

(a) Consider the following network where capacities $u_{i j}, \operatorname{cost} c_{i j}$ and supplies/demands $b(i)$ are shown on the diagram. Supply is indicated by a positive $b(i)$ value and demand by a negative one.

(i) Find the flow that corresponds to the spanning tree structure $(T, L, U)$ and show that it is feasible, where

$$

\begin{gathered}

T=\{(1,2),(2,3),(2,5),(3,6),(4,5)\} \\

L=\{(2,4)\} \text { and } U=\{(1,4),(5,3),(5,6)\} .

\end{gathered}

$$

[2 marks]

(ii) Starting with the spanning tree structure you found in (i) above, perform two iterations of the network simplex algorithm (calculate node-potentials three times and then stop) and indicate the resulting flow. Use the first eligible arc pivot rule giving priority to lower indices. If after a pivot cycle two arcs become restricted then pick the one with the lowest index to be on the tree.

[12 marks]

(iii) State either whether the flow found in (ii) is optimal or if it is not, then give the violated optimality conditions and the corresponding arcs.

[1 mark]

(iv) Is the resulting spanning tree structure found in (ii) degenerate?

[1 mark]

(b) Answer the following questions on Minimum Spanning Trees (MST):

(i) Give the definitions of a minimax path and a minimax spanning tree.

[1 mark]

(ii) Show that the unique path between two vertices in an MST is a minimax path. [3 marks]

(iii) Show that an MST is a minimax spanning tree but a minimax spanning tree is not necessarily an MST.

[3 marks]

(iv) Show that Kruskal’s algorithm produces a minimum spanning tree.

[2 marks]

(a) A set of families go out to dinner together and, to increase their social interaction, they would like to sit at tables so that no two members of the same family are at the same table. Assume that there are $p$ families and that the $i$ th family has $a(i)$ members. Also, assume that $q$ tables are available and that the $j$ th table has a seating capacity of $b(j)$. Formulate the problem of finding a seating arrangement such that no two members of the same family are at the same table as a maximum flow problem on a network $G$. Define $G$ and all the parameters associated with it and describe the maximum flow problem on $G$.

[6 marks]

(b) Let $G=(N, A)$ be a directed graph with arc set and arc costs as in the table below:

(i) Find a reachability tree from node 4 to all other nodes using a breadth first search. Show the LIST and admissible arcs at each step of the algorithm. Give priority to lower indices.

[3 marks]

(ii) Use an appropriate algorithm to find the shortest paths from node 1 to all other nodes. Explain why you used this algorithm. Give the optimal cost and the shortest path tree. Give priority to lower indices.

[6 marks]

Consider the undirected graph $G^{\prime}=(N, E)$, which is the same as the graph $G$ above except all the directed arcs $A$ are now undirected edges in $E$.

(iii) Use two algorithms that we have learned in this course to find a minimum spanning tree for $G^{\prime}$. In both cases give the resulting tree and tree cost, show all your steps, and give priority to lower indices.

[10 marks]

(a) Let $G=(N, E)$ be an undirected graph with edges $E:\{(1,2),(1,3),(2,4),(2,8),(3,4),(3,5)$, $(4,6),(5,6),(5,7),(6,7),(6,8),(7,8)\}$. Let $M=\{(2,8),(3,4),(5,7)\}$ be a matching on $G$.

(i) Perform a breadth first search algorithm to find an alternating tree routed at node 1 . Show your steps and give priority to lower indices.

[4 marks]

(ii) List any augmenting path from node 1 found in the tree in (i). Increase the cardinality of the matching by using an augmenting path.

[1 mark]

(iii) Parts (i) and (ii) above are steps of the matching algorithm for finding maximum cardinality matchings. Can this matching algorithm be used in arbitrary graphs and initial matchings? Explain your answer.

[1 mark]

(b) David is running the summer projects for a master’s degree where the students are matched to summer projects offered by clients. Suppose there are 5 students and 5 projects (each project corresponds to a distinct client). The students rank all the projects and the clients that offer the projects rank all the students. The preferences for students $S$ and clients $C$, where entry $S_{i j}$ $\left(C_{i j}\right)$ is the ranking of project $j$ (student $j$ ) by student $i$ (client $i$ ), 1 being the best, are shown in the matrices below:

\begin{equation}

\begin{gathered}

S=\left(\begin{array}{lllll}

2 & 1 & 5 & 3 & 4 \\

3 & 4 & 1 & 5 & 2 \\

1 & 4 & 2 & 3 & 5 \\

4 & 2 & 3 & 1 & 5 \\

4 & 1 & 5 & 2 & 3

\end{array}\right) \\

C=\left(\begin{array}{lllll}

1 & 2 & 5 & 3 & 4 \\

1 & 2 & 3 & 4 & 5 \\

5 & 2 & 3 & 4 & 1 \\

3 & 1 & 4 & 5 & 2 \\

2 & 5 & 3 & 4 & 1

\end{array}\right)

\end{gathered}

\end{equation}

(i) Use the Gale-Shapley (propose-and-reject) algorithm where students propose and clients accept or reject to find a stable matching. Show all steps of the algorithm.

[6 marks]

Show that, for any instance of the problem:

(ii) Gale-Shapley produces a stable matching.

[3 marks]

(iii) In the stable matching given by the Gale-Shapley algorithm, where students propose, a client never rejects a stable partner.

[4 marks]

(c) In each of $T$ periods you can buy, sell, or hold for later sale some product subject to the following constraints: in each period $i$ you have to sell at least $\alpha_{i}$ units (due to prior agreements that you made), you can buy at most $\beta_{i}$ units, and you can hold at most $\gamma_{i}$ units of the product for the next time period. Let $x_{i}, y_{i}, z_{i}$ be the selling price, purchase price, and the inventory carrying costs in period $i$. What buy-sell-hold policy should you adopt to maximize total profit in the next $T$ periods? Formulate the problem as a network flow problem for $T=4$.

[6 marks]

## How to understand Minimum cost flows?

Min cost flow problem.

– Send finished good from plants to customers.

– $s_{y}=$ net supply / demand at vertex $v . \quad$ (sum of supply = sum of demand)

– $c_{\mathrm{ww}}=$ unit shipping cost from $v$ to $w$.

(positive or negative)

– $u_{v w}=$ capacity of edge $v-w$.

(infinity ok)

– Goal: satisfy demand at minimize cost.

Mail carrier problem.

– Post office located at node $5 .$

– Find minimum length route that starts and ends at $s$ and visits each road at least once.

– Need to traverse roads more than once unless graph is Eulerian.

## How to deal with Weighted matching problem?

Hall’s Theorem

– Given a bipartite graph G=(LUR,E), where $|\mathrm{L}|=\mid \mathrm{R}$

– It contains a perfect matching if and only if:

– For every subset $S \subseteq L,|\Gamma(S)| \geq|S|$

– $(\Gamma(S)$ is the set of vertices adjacent to $S)$

Proof on Induction

– Let $\mathrm{n}=|\mathrm{L}|=|\mathrm{R}|$

– When $n=1$, trivial

– Suppose it holds for all $n \leq k$, for $n=k+1$, two cases:

Case I: For every subset $S \subseteq L,|\Gamma(S)| \geq|S|+1$

– Then we arbitrarily put an edge $(\mathrm{u}, \mathrm{v})$ in the matching

– In G-\{u,v\}, it still satisfies the condition $|\Gamma(\mathrm{S})| \geq|\mathrm{S}|$, so the result holds by the induction condition

– Case II: there exists a T\subseteqL which has $|\Gamma(T)|=|T|$, then the subgraphs of $\mathrm{G}$ on:

– $\mathrm{T} \cup \Gamma(\mathrm{T})$

– (L\backslashT) U(R\backslash\Gamma(T))

both satisfies the Hall’s condition

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