Problem Link : https://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=24&page=show_problem&problem=1760

Solution idea:

This is a classical knapsack problem. Set a correct base case is the most challenging part of this problem. If we read and think carefully about this problem we can see that if the expense exceeds 2000$ then 200$ will be refunded so it will be strictly greater than 2000. Now we can consider 3 things as base case. Those are here-

Let W is expense at current time and C = capacity or Total money.

•  When W > Capacity and W-Capacity > 200 then we can’t buy anything else and the current product so we will return a large negative value to cancel this solution.
• When Capacity < 1800 and Capacity < W then it's clear we can't buy anything and current product so we will cancel this.
•  And Last thing is when we are taking the last product then if W > Capacity and W <= 2000 then we can take these so we will return a Zero. Otherwise we will cancel this one by returning a large negative value.

/*
         +-+ +-+ +-+
         |R| |.| |S|
         +-+ +-+ +-+
 */

#include <bits/stdc++.h>

#define pii             pair <int,int>
#define sc              scanf
#define pf              printf
#define Pi              2*acos(0.0)
#define ms(a,b)         memset(a, b, sizeof(a))
#define pb(a)           push_back(a)
#define MP              make_pair
#define db              double
#define ll              long long
#define EPS             10E-10
#define ff              first
#define ss              second
#define sqr(x)          (x)*(x)
#define D(x)            cout<<#x " = "<<(x)<<endl
#define VI              vector <int>
#define DBG             pf("Hi\n")
#define MOD             100007
#define MAX             10000
#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
#define SZ(a)           (int)a.size()
#define sf(a)           scanf("%d",&a)
#define sfl(a)          scanf("%lld",&a)
#define sff(a,b)        scanf("%d %d",&a,&b)
#define sffl(a,b)       scanf("%lld %lld",&a,&b)
#define sfff(a,b,c)     scanf("%d %d %d",&a,&b,&c)
#define sfffl(a,b,c)    scanf("%lld %lld %lld",&a,&b,&c)
#define loop(i,n)       for(int i=0;i<n;i++)
#define REP(i,a,b)      for(int i=a;i<b;i++)
#define TEST_CASE(t)    for(int z=1;z<=t;z++)
#define PRINT_CASE      printf("Case %d: ",z)
#define all(a)          a.begin(),a.end()
#define intlim          2147483648
#define inf             1000000
#define ull             unsigned long long

using namespace std;


/*----------------------Graph Moves----------------*/
//const int fx[]={+1,-1,+0,+0};
//const int fy[]={+0,+0,+1,-1};
//const int fx[]={+0,+0,+1,-1,-1,+1,-1,+1};   // Kings Move
//const int fy[]={-1,+1,+0,+0,+1,+1,-1,-1};  // Kings Move
//const int fx[]={-2, -2, -1, -1,  1,  1,  2,  2};  // Knights Move
//const int fy[]={-1,  1, -2,  2, -2,  2, -1,  1}; // Knights Move
/*------------------------------------------------*/

/*-----------------------Bitmask------------------*/
//int Set(int N,int pos){return N=N | (1<<pos);}
//int reset(int N,int pos){return N= N & ~(1<<pos);}
//bool check(int N,int pos){return (bool)(N & (1<<pos));}
/*------------------------------------------------*/

int capacity, n;

int z;

int dp[110][10300];
int weight[110],profit[110];

int knapsack(int idx, int w)
{
    if(w>capacity && w-capacity>200) return -100000;
    if(capacity<1800 && w>capacity) return -100000;

    if(idx==n)
    {
        if(w>capacity && w<=2000) return -100000;
        return 0;
    }

    if(dp[idx][w]!=-1) return dp[idx][w];

    return dp[idx][w]=max(profit[idx]+knapsack(idx+1,w+weight[idx]),knapsack(idx+1,w));
}

int main()
{
    ///freopen("in.txt","r",stdin);
    ///freopen("out.txt","w",stdout);
    CIN;
    while(cin>>capacity>>n)
    {
        z++;
        loop(i,n) cin>>weight[i]>>profit[i];

        ms(dp,-1);
        cout<<knapsack(0,0)<<endl;


    }

    return 0;
}

Here is another solution 🙂

/*
         +-+ +-+ +-+
         |R| |.| |S|
         +-+ +-+ +-+
 */

#include <bits/stdc++.h>

#define pii             pair <int,int>
#define sc              scanf
#define pf              printf
#define Pi              2*acos(0.0)
#define ms(a,b)         memset(a, b, sizeof(a))
#define pb(a)           push_back(a)
#define MP              make_pair
#define db              double
#define ll              long long
#define EPS             10E-10
#define ff              first
#define ss              second
#define sqr(x)          (x)*(x)
#define D(x)            cout<<#x " = "<<(x)<<endl
#define VI              vector <int>
#define DBG             pf("Hi\n")
#define MOD             100007
#define MAX             10000
#define CIN             ios_base::sync_with_stdio(0); cin.tie(0)
#define SZ(a)           (int)a.size()
#define sf(a)           scanf("%d",&a)
#define sfl(a)          scanf("%lld",&a)
#define sff(a,b)        scanf("%d %d",&a,&b)
#define sffl(a,b)       scanf("%lld %lld",&a,&b)
#define sfff(a,b,c)     scanf("%d %d %d",&a,&b,&c)
#define sfffl(a,b,c)    scanf("%lld %lld %lld",&a,&b,&c)
#define loop(i,n)       for(int i=0;i<n;i++)
#define REP(i,a,b)      for(int i=a;i<b;i++)
#define TEST_CASE(t)    for(int z=1;z<=t;z++)
#define PRINT_CASE      printf("Case %d: ",z)
#define all(a)          a.begin(),a.end()
#define intlim          2147483648
#define inf             1000000
#define ull             unsigned long long

using namespace std;


/*----------------------Graph Moves----------------*/
//const int fx[]={+1,-1,+0,+0};
//const int fy[]={+0,+0,+1,-1};
//const int fx[]={+0,+0,+1,-1,-1,+1,-1,+1};   // Kings Move
//const int fy[]={-1,+1,+0,+0,+1,+1,-1,-1};  // Kings Move
//const int fx[]={-2, -2, -1, -1,  1,  1,  2,  2};  // Knights Move
//const int fy[]={-1,  1, -2,  2, -2,  2, -1,  1}; // Knights Move
/*------------------------------------------------*/

/*-----------------------Bitmask------------------*/
//int Set(int N,int pos){return N=N | (1<<pos);}
//int reset(int N,int pos){return N= N & ~(1<<pos);}
//bool check(int N,int pos){return (bool)(N & (1<<pos));}
/*------------------------------------------------*/

int capacity, n;

pii input[110];
int z;
int dp[110][10210];
int cases[110][10210];
bool test=0;

int knapsack(int idx, int w)
{
    if(w==capacity)
    {
        return 0;
    }
    if(idx>=n) return 0;
    int &ret=dp[idx][w];
    int &cas=cases[idx][w];

    if(cas==z) return ret;

    cas=z;

    int p=0,q=0;

    if(w+input[idx].ff<=capacity)
        p=input[idx].ss+knapsack(idx+1,w+input[idx].ff);
    else if(w+input[idx].ff>capacity)
    {
        if((w+input[idx].ff>2000) && (w+input[idx].ff<=capacity+200) && !test)
        {
            test=1;
            capacity+=200;
            p=input[idx].ss+knapsack(idx+1,w+input[idx].ff);
            test=0;
            capacity-=200;
        }
    }
    q=knapsack(idx+1,w);

    return ret=max(p,q);
}

int main()
{
    ///freopen("in.txt","r",stdin);
    ///freopen("out.txt","w",stdout);

    while(cin>>capacity>>n)
    {
        z++;

        loop(i,n) cin>>input[i].ff>>input[i].ss;
        test=0;
        sort(input,input+n);
        cout<<knapsack(0,0)<<endl;

    }
    return 0;
}