This project corresponds to the final project of CS 170 at UC Berkeley. The problem asked to be solved relates to Graph Theory, and we are asked to design an algorithm to solve it as quickly and accurate as we can. First, let us address the problem.

## Problem

You have some number of bots lost in the city. Your goal is to find these bots and move them home in the shortest time. You have two operations, both that take some time: remote and scout. A remote moves all bots from a vertex to a neighbour of that vertex. A scout sends a student to a vertex and has the student report if a bot is there.

Formally, your delivery system is for a specific city, i.e. a connected weighted undirected graph $$G=(V,E)$$ such that all edge weights $$w_e>0$$, with a designated home vertex $$h \in V$$. The Guavabots live at a set of locations $$L⊆V$$ with exactly one Guavabot in each starting location; but you do not know this set of starting locations. In addition, you have a set of $$k$$ students that you can send to vertices on the graph to inform you if Guavabots are present there; however, the students are unreliable and are incorrect on a fixed, unknown subset of vertices.

You are given and know the graph $$G$$, edge weights $$w_e$$, home vertex $$h$$, number of Guavabots $$\mid L \mid$$ , and the number of students $$k$$. All graphs are complete, have $$\mid L\mid =5$$ bots, $$n=100$$ vertices, and $$k \in \{10,20,40\}$$ students.

You do not know the starting locations of the Guavabots $$L$$ or the opinions of the students (until you ask for them by scouting).

## Action

You get the bots home by performing a series of actions of your choice. Your available actions are:

• scout on $$(i,v)$$, where $$i$$ is a student and $$v ∈ V$$. In this case, student $$i$$ visits the vertex and reports to you whether they see Guavabots at the vertex (they give this as a yes/no answer). However, their report may be incorrect. You have to wait for the student to move to the vertex and report you the answer, which takes 1 time.

• remote on the directed edge $$(u,v)$$, provided that $$\{u,v\} ∈ E$$. This tells you how many Guavabots there were at $$u$$ that received the command, then all the Guavabots at $$u$$ move to $$v$$. Regardless of the presence of Guavabots at $$u$$, it takes $$w_{uv}$$ time to wait and see if your command has moved any Guavabots.

As the graph is undirected, you can remote along either direction on any edge.

You can only perform one action at a time, and you cannot “undo” an action.

We call an instance of this problem and the sequence of actions you take a “rescue”.

## Constraints

• After you remote along an edg $$\{u,v\}$$ in the direction $$u→v$$, any future scouts on $$u$$ or $$v$$ will fail (as you already know about the existence of Guavabots at either vertex). The only exception is if no bots were remoted along the edge, in which case you can scout on $$v$$ in the future.

• You cannot perform a scout at the home vertex $$h$$, ever.

## Students

Each student $$i$$ can give you a report on any vertex so long you haven’t used it in a remote already. If a student is incorrect on a vertex, they will always report the opposite of the truth; students are incorrect a fixed, but unknown number of times. This means that for each student $$i$$, there is a fixed, unknown subset $$S_i ⊆ V$$ of vertices that they will be wrong at. These sets $$S1,…,Sk$$ are fixed for the input, meaning they are completely determined before your rescue even starts; in other words, the student opinions will not change depending on how your algorithm runs.

There are no guarantees on how correct the students will be beyond the fact that $$\mid S_i \mid$$ $$≤$$ $$\mid V \mid$$ $$/2$$ for all students $$i$$. It is up to you to come up with an algorithm that uses the student reports as effectively as possible.

## Time

Time is measured as follows:

• When you start, the current time is zero.

• There is a fixed time every scout action takes; after every scout your time is incremented by 1.

• After a remote on the edge $$\{u,v\}$$, your time is incremented by $$w_{uv}$$, regardless of the direction you remote on $$\{u,v\}$$

The scout cost 1 is much smaller than most edge weights $$w_e$$.

## Goal

You are given a score once you end the rescue. If the time taken so far is $$t$$ and the number of Guavabots that are home is $$g$$, this score is:

[\frac{100}{\mid L \mid + 1}\left( g + \frac{\alpha}{\alpha + t} \right)]

where $$\alpha ∼ 10^3$$ is a constant among all inputs which is yet to be determined, but is within an order a magnitude of $$10^3$$ and is about the raw time of an average solver that gets all the bots home.

You want to get as high a score as possible. The minimum score for any instance is 0, and the maximum score for any instance is 100 (neither side is necessarily attainable, your score will fall in between).

This scoring function was designed such that returning more bots is always better than spending less time. For example, say that we have 5 bots scattered on a graph:

• If we return 5 bots and take 10000 time, our score is $$≈ 84.8$$

• If we return 2 bots and take 100 time, our score is $$≈ 48.5$$

• If we return 1 bot and take 10 time, our score is $$≈ 33.17$$

Another way to see it: treat your score as a tuple $$(g,t)$$. Scores will be ordered by $$g$$ descending first, then by $$t$$ ascending.

## Solution

I provided two solutions which fit two different situations. I picked the second solution for the best accuracy and efficiency combined. Below I will list both solutions.

### Solution 1. Ignore all students’ opinions

Since there might be correct answer which help our rescue, and there might be wrong answers which would make our algorithm worse, in this solution, I ignore all students’ opinions on whether there is any bot at a specific point.

The process of this algorithm runs as the following:

1. Construct a Minimum Spanning Tree on graph $$G$$.

2. Get the topological sort of all vertices of graph $$G$$.

3. Call rescure on each node by the reversed topological sort order, and skip the home vertex.

This algorithm works better if most students are providing wrong information, but if more than half students are providing correct information, this algorithm deos not take advantage from it.

### Solution 2. Consider students’ opinion

In this solution, I take students’ opinion into consideration. If a vertex is believed by many students that there is at least one bot, I retrieve that vertex first. This algorithm runs as following,

1. Construct a Minimum Spanning Tree on graph $$G$$.

2. Get the topological sort list of all vertices of graph $$G$$.

3. Contruct a dictionary where keys are vertices and values are counts for how many students believe there is at least one bot.

4. Sort the dictionary by values.

5. Loop through all vertices by the reversed sorted dictionary and rescue each vertex by one step so we have a clear knowledge with locations of all bots.

6. Loop through the reversed topological sort and rescue all bots.

This algorithm works better on the situation when students are providing helpful opinions. For further improvement, one could work on that among all vertices, if a student is providing wrong information in the begining, he is likely to provide correct information afterwards.

In the end, I picked the second solution as it gives the best correctness and time efficiency conbined. This algorithm scores 92 out of 100 overall and has a successful rescure rate of 99.87%.