Note that in the sample case, because the first three cows share no cereals in common with the last five cows, we can order those three independently of the last five: in any permutation of the 8 cows, all that matters for the number of cows that go hungry is the order of the first three, and separately, the order of the last five.

More generally, for any input, we can consider the graph where each cereal is a vertex, and each cow is a directed edge from her favorite cereal to her second-favorite cereal. If there is a group of cereals such that every cow has 0 or 2 of her cereals in that group, then we can solve for the optimal order of the cows with 2 cereals in that group independently of solving for the optimal order of the cows with 0 cereals in that group. In graph theoretic terms, we can solve every connected component of the graph (ignoring edge directions for now) independently. Once we have an optimal ordering for each component, we can arbitrarily interlace the orderings (or just concatenate them), and we'll have an optimal ordering for the entire graph.

So now let's focus on a single connected component with $V$ vertices (i.e. cereals) and $E$ edges (i.e. cows). Necessarily, it holds that $E \geq V-1$. Also, if $E > V$, then necessarily, at least $E-V$ cows will go hungry in any ordering. So the question is whether we can always find an ordering in which only $\max(0,E-V)$ cows go hungry.

Let's consider the easiest case: $E = V-1$. Then the connected component is a tree. If we root the tree at an arbitrary vertex and pick the cows in order of increasing depth in the tree, then we can see that every cow is able to pick one of her two cereals. Depth-first-search order also works: all that matters is that an edge is chosen before its "child" edges.

Next, if $E=V$, then the connected component is a tree plus an extra edge. Pick the cow corresponding to that extra edge first. What's left is a tree with a missing vertex (the cereal that the first cow took). Root the tree at that vertex, and again pick the cows in order of increasing depth in the tree. Then every cow is able to pick one of her two cereals.

Finally, consider the case $E>V$. By depth-first search we can find a tree in this component, and there are $E+1-V$ extra edges. Put all but one of these cows at the end of the ordering; we don't care if they go hungry. Now we've reduced to the case with only $V$ edges, and we know that in this case we can find an ordering where no cows go hungry. So overall at most $E-V$ cows go hungry (in fact, exactly $E-V$ cows: the last $E-V$ cows in the ordering).

This shows that in a component with $E$ edges and $V$ vertices, we can always find an ordering where $\max(0, E-V)$ cows go hungry, and we've already seen that this is optimal. For each component, we do a depth-first search to find a spanning tree, pick an extra edge in the component, and then do another depth-first search at the favorite endpoint of that edge, to order the spanning tree edges by depth. Thus, solving each component takes linear time in the component size. Finding the components is also done by repeated depth-first search on unvisited vertices, which takes linear time. Thus, the overall algorithm takes linear time.

Note that all of these depth-first searches ignore the directions of the edges. Danny Mittal's Java code is below, followed by Leon Zhao's C++ code.

import java.io.BufferedReader; import java.io.IOException; import java.io.InputStreamReader; import java.util.*; public class Cereal2 { static List<Cow>[] adj; static int[] seen; static Cow[] edgeHeres; static Cow cycleEdge = null; static List<Integer> answer = new ArrayList<>(); static void dfs1(int a, Cow edgeHere) { if (seen[a] == 1) { if (cycleEdge == null) { cycleEdge = edgeHere; } } else { seen[a] = 1; edgeHeres[a] = edgeHere; for (Cow edge : adj[a]) { if (edgeHere == null || edge.id != edgeHere.id) { dfs1(edge.to, edge); } } } } static void dfs2(int a) { for (Cow edge : adj[a]) { if (seen[edge.to] == 1) { seen[edge.to] = 2; answer.add(edge.id); dfs2(edge.to); } } } public static void main(String[] args) throws IOException { BufferedReader in = new BufferedReader(new InputStreamReader(System.in)); StringTokenizer tokenizer = new StringTokenizer(in.readLine()); int n = Integer.parseInt(tokenizer.nextToken()); int m = Integer.parseInt(tokenizer.nextToken()); adj = new List[m + 1]; seen = new int[m + 1]; edgeHeres = new Cow[m + 1]; for (int a = 1; a <= m; a++) { adj[a] = new ArrayList<>(); } for (int j = 1; j <= n; j++) { tokenizer = new StringTokenizer(in.readLine()); int first = Integer.parseInt(tokenizer.nextToken()); int second = Integer.parseInt(tokenizer.nextToken()); adj[first].add(new Cow(j, first, second, first)); adj[second].add(new Cow(j, second, first, first)); } for (int r = 1; r <= m; r++) { if (seen[r] == 0) { cycleEdge = null; dfs1(r, null); if (cycleEdge == null) { seen[r] = 2; dfs2(r); } else { List<Integer> restOfCycle = new ArrayList<>(); for (int a = cycleEdge.from; a != cycleEdge.to; a = edgeHeres[a].from) { seen[a] = 2; restOfCycle.add(edgeHeres[a].id); } seen[cycleEdge.to] = 2; answer.add(cycleEdge.id); if (cycleEdge.favorite == cycleEdge.to) { Collections.reverse(restOfCycle); } answer.addAll(restOfCycle); for (int a = cycleEdge.from; a != cycleEdge.to; a = edgeHeres[a].from) { dfs2(a); } dfs2(cycleEdge.to); } } } Set<Integer> dontGoHungry = new HashSet<>(answer); for (int j = 1; j <= n; j++) { if (!dontGoHungry.contains(j)) { answer.add(j); } } System.out.println(n - dontGoHungry.size()); StringJoiner joiner = new StringJoiner("\n"); for (int j : answer) { joiner.add("" + j); } System.out.println(joiner); } static class Cow { final int id; final int from; final int to; final int favorite; Cow(int id, int from, int to, int favorite) { this.id = id; this.from = from; this.to = to; this.favorite = favorite; } @Override public String toString() { return "" + id; } } }

#include <bits/stdc++.h> using namespace std; struct edge { int cow; // which cow's choice int to; bool is_first; edge() {}; edge(int cow, int to, bool is_first) : cow(cow), to(to), is_first(is_first) {}; }; int N, M; vector<edge> adj[100001]; bool visited_cycle[100001]; // array for cycle finding bool visited[100001]; // visited array for finding which order of cows we should use bool gets_cereal[100001]; int hungry_cows = 0; queue<int> order; int ignore_edge = -1; int first_cereal = -1; // the cereal we start the search from, if the CC is not a tree then this must be on a cycle void find_cycle(int cur, int prev = -1) { visited_cycle[cur] = true; for (edge next : adj[cur]) { if (visited_cycle[next.to]) { if (first_cereal == -1 && next.to != prev) { if (next.is_first) { first_cereal = next.to; } else { first_cereal = cur; } ignore_edge = next.cow; order.push(next.cow); gets_cereal[next.cow] = true; } } else { find_cycle(next.to, cur); } } } void dfs(int cur) { visited[cur] = true; for (edge next : adj[cur]) { if (!visited[next.to] && next.cow != ignore_edge) { gets_cereal[next.cow] = true; order.push(next.cow); dfs(next.to); } } } int main() { cin >> N >> M; for (int i = 0; i < N; ++i) { int a, b; cin >> a >> b; adj[a].push_back(edge(i+1, b, false)); adj[b].push_back(edge(i+1, a, true)); } for (int i = 1; i <= M; ++i) { first_cereal = -1; ignore_edge = -1; if (!visited[i]) { find_cycle(i); if (first_cereal > 0) { dfs(first_cereal); } else { dfs(i); } } } for (int i = 1; i <= N; ++i) { if (!gets_cereal[i]) { ++hungry_cows; order.push(i); } } cout << hungry_cows << endl; while (!order.empty()) { cout << order.front() << endl; order.pop(); } return 0; }