Item number | Weight (in Kgs) |
Value (in Rupees) |
---|---|---|

1 | 10 | 60 |

2 | 7 | 28 |

3 | 4 | 20 |

4 | 2 | 24 |

The task is to pick a subset of these items such that their total weight is no more than $$11$$ $$Kgs$$ and their total value is maximized. Moreover, no item may be split. The total value of items picked by an optimal algorithm is denoted by $$V$$_{opt}. A greedy algorithm sorts the items by their value-to-weight ratios in descending order and packs them greedily, starting from the first item in the ordered list. The total value of items picked by the greedy algorithm is denoted by $$V$$_{greedy}.

The value of $$V$$_{opt} $$−$$ $$V$$_{greedy} is ____________.

Choose a value for $$x$$ that will maximize the number of minimum weight spanning trees $$(MWSTs)$$ of $$G.$$ The number of $$MWSTs$$ of $$G$$ for this value of $$x$$ is ______.

**TRUE**?

$$\,\,\,\,\,\,\,\,\,\,{\rm I}.\,\,\,\,\,\,\,\,\,\,$$ If $$e$$ is the lightest edge of __some__ cycle in $$G,$$ then every $$MST$$ of $$G$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$__includes__ $$e$$

$$\,\,\,\,\,\,\,\,{\rm I}{\rm I}.\,\,\,\,\,\,\,\,\,\,$$ If $$e$$ is the heaviest edge of __some__ cycle in $$G,$$ then every $$MST$$ of $$G$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$__excludes__ $$e$$

The largest possible integer value of $$x,$$ for which at least one shortest path between some pair of vertices will contain the edge with weight $$x$$ is _________________.