I have a question about Yandex.Algorithm 2015 round 2.2E: the problem can be reduced to computing the min. bipartite vertex cover that can't fully cover one side and since the dual of min. vertex cover is max. matching, I tried to find the dual of this problem.

I mean, that vertex cover can be described as a linear programming problem

• 
• x_i ≥ 0
• x_uj + x_vj ≥ 1 for each edge e_j = (u_j - v_j), 1 ≤ j ≤ E
• 

Its dual should be

• 
• 
• for 1 ≤ i ≤ N, (sum over edges incident on vertex i in the left part)
• for N + 1 ≤ i ≤ N + M, (same as above for the right part)

y_j are variables assigned to edges (flow through them); are assigned to the left and right part of the graph and if they're fixed, finding the optimal y_j is just bipartite matching/flow. However, this gives the optimal solution as for example for an almost complete bipartite graph K_N, N that's missing just one edge, which doesn't make sense.

What am I missing here?

#include <bits/stdc++.h>
#include "highest_mountain.h"
// iostream is too mainstream
#include <cstdio>
// bitch please
#include <iostream>
#include <algorithm>
#include <cstdlib>
#include <vector>
#include <set>
#include <map>
#include <queue>
#include <stack>
#include <list>
#include <cmath>
#include <iomanip>
#include <time.h>
#include <message.h>
#include <stdio.h>
#define dibs reserve
#define OVER9000 1234567890
#define ALL_THE(CAKE,LIE) for(auto LIE =CAKE.begin(); LIE != CAKE.end(); LIE++)
#define tisic 47
#define soclose 1e-8
#define chocolate win
// so much chocolate
#define patkan 9
#define ff first
#define ss second
#define abs(x) ((x < 0)?-(x):x)
#define uint unsigned int
#define dbl long double
#define pi 3.14159265358979323846
using namespace std;
// mylittledoge

typedef long long cat;

#ifdef DONLINE_JUDGE
	// palindromic tree is better than splay tree!
	#define lld I64d
#endif

#define MASTER_NODE 0

int main() {
	cin.sync_with_stdio(0);
	cin.tie(0);
	cout << fixed << setprecision(10);
	int nodes =NumberOfNodes(), my_id =MyNodeId();
	int N =NumberOfPeaks();

	vector<int> ilen(nodes,N/nodes);
	for(int i =0; i < N%nodes; i++) ilen[i]++;
	int L =0, R =0;
	for(int i =0; i < my_id; i++) L +=ilen[i];
	for(int i =0; i <= my_id; i++) R +=ilen[i];

	vector< pair<cat,int> > H;
	H.dibs(R-L);
	for(int i =L; i < R; i++) {
		cat h =GetHeight(i);
		int s =H.size();
		while(s >= 2 && (H[s-1].ff-h)*(H[s-2].ss-i) > (H[s-2].ff-h)*(H[s-1].ss-i)) {
			H.pop_back();
			s--;}
		H.push_back(make_pair(h,i));}
	int Hs =H.size();

	vector<int> comp1;
	int K =2000;
	for(int i =0; i < Hs; i +=K) comp1.push_back(H[i].ss);
	if(Hs > 0 && comp1.back() != H[Hs-1].ss) comp1.push_back(H[Hs-1].ss);
	PutInt(MASTER_NODE,comp1.size());
	ALL_THE(comp1,it) PutInt(MASTER_NODE,*it);
	Send(MASTER_NODE);

	if(my_id == MASTER_NODE) {
		vector< pair<cat,int> > Hcomp;
		Hcomp.dibs(nodes*2000);
		vector< pair<int,int> > I(nodes,make_pair(N+1,-1));
		vector< pair<int,int> > I2(nodes,make_pair(N+1,-1));
		vector< vector<int> > V(nodes);
		for(int i =0; i < nodes; i++) {
			Receive(i);
			int l =GetInt(i);
			V[i].resize(l);
			for(int j =0; j < l; j++) {
				V[i][j] =GetInt(i);
				I[i].ff =min(I[i].ff,V[i][j]);
				I[i].ss =max(I[i].ss,V[i][j]);}
			}
		for(int i =0; i < nodes; i++) {
			for(int j =0; j < (int)V[i].size(); j++) {
				int a =V[i][j];
				cat h =GetHeight(a);
				int s =Hcomp.size();
				while(s >= 2 && (Hcomp[s-1].ff-h)*(Hcomp[s-2].ss-a) > (Hcomp[s-2].ff-h)*(Hcomp[s-1].ss-a)) {
					Hcomp.pop_back();
					s--;}
				Hcomp.push_back(make_pair(h,a));}
			for(int j =(int)Hcomp.size()-1; j >= 0; j--) {
				if(Hcomp[j].ss < I[i].ff) break;
				I2[i].ff =Hcomp[j].ss;}
			}
		Hcomp.clear();
		for(int i =nodes-1; i >= 0; i--) {
			for(int j =(int)V[i].size()-1; j >= 0; j--) {
				int a =V[i][j];
				cat h =GetHeight(a);
				int s =Hcomp.size();
				while(s >= 2 && (Hcomp[s-1].ff-h)*(Hcomp[s-2].ss-a) < (Hcomp[s-2].ff-h)*(Hcomp[s-1].ss-a)) {
					Hcomp.pop_back();
					s--;}
				Hcomp.push_back(make_pair(h,a));}
			for(int j =(int)Hcomp.size()-1; j >= 0; j--) {
				if(Hcomp[j].ss > I[i].ss) break;
				I2[i].ss =Hcomp[j].ss;}
			}
		for(int i =0; i < nodes; i++) {
			PutInt(i,I2[i].ff);
			PutInt(i,I2[i].ss);
			Send(i);}
		}

	Receive(MASTER_NODE);
	int mi =GetInt(MASTER_NODE);
	int mx =GetInt(MASTER_NODE);
	int l =0, r =Hs-1;
	while(l < Hs && H[l].ss < mi) l++;
	while(r >= 0 && H[r].ss > mx) r--;
	vector<int> comp2l,comp2r;
	for(int i =max(0,l-K); i <= min(Hs-1,r+K); i++) {
		if(i <= l+K) comp2l.push_back(H[i].ss);
		if(i >= r-K) comp2r.push_back(H[i].ss);
		}
	for(int i =0; i < my_id; i++) {
		PutInt(i,comp2l.size());
		ALL_THE(comp2l,it) PutInt(i,*it);
		Send(i);}
	for(int i =my_id+1; i < nodes; i++) {
		PutInt(i,comp2r.size());
		ALL_THE(comp2r,it) PutInt(i,*it);
		Send(i);}

	vector<int> Pi;
	Pi.dibs(Hs+nodes*5000);
	vector< pair<cat,int> > Hi;
	Hi.dibs(Hs+nodes*5000);
	for(int i =0; i < nodes; i++) {
		if(i == my_id) ALL_THE(H,it) Pi.push_back(it->ss);
		else {
			Receive(i);
			l =GetInt(i);
			for(int j =0; j < l; j++) Pi.push_back(GetInt(i));
			}
		}
	ALL_THE(Pi,it) {
		int a =*it;
		cat h =GetHeight(a);
		int s =Hi.size();
		while(s >= 2 && (Hi[s-1].ff-h)*(Hi[s-2].ss-a) > (Hi[s-2].ff-h)*(Hi[s-1].ss-a)) {
			Hi.pop_back();
			s--;}
		Hi.push_back(make_pair(h,a));}
	int inH =0;
	ALL_THE(Hi,it) if(it->ss >= L && it->ss < R) inH++;
	PutInt(MASTER_NODE,inH);
	Send(MASTER_NODE);

	if(my_id == MASTER_NODE) {
		int ans =0;
		for(int i =0; i < nodes; i++) {
			Receive(i);
			ans +=GetInt(i);}
		cout << ans << "\n";}
	}

// look at my code
// my code is amazing