Problem:
Huge Knapsack Problem

You have N items that you want to put them into a knapsack. Item i has value vi and weight wi.

You want to find a subset of items to put such that:

The total value of the items is as large as possible.

The items have combined weight at most W, that is capacity of the knapsack.

Find the maximum total value of items in the knapsack.

Input

N W
v1 w1
v2 w2
:
vN wN

The first line consists of the integers N and W. In the following N lines, the value and weight of the i-th item are given.

Output

Print the maximum total values of the items in a line.

Constraints

1 ≤ N ≤ 40

1 ≤ vi ≤ 1015

1 ≤ wi ≤ 1015

1 ≤ W ≤ 1015

Sample Input 1

4 5
4 2
5 2
2 1
8 3

Sample Output 1

13

Sample Input 2

2 20
5 9
4 10

Sample Output 2

9

Solution in JAVA (judge verdict: Accepted)

/**
 * Created at 19:38 on 2019-08-16
 */

import java.io.*;
import java.util.*;
import java.util.function.Predicate;

public class Main {

  static FastScanner sc = new FastScanner();
  static Output out = new Output(System.out);

  static final int[] dx = {0, 1, 0, -1};
  static final int[] dy = {-1, 0, 1, 0};

  static final long MOD = (long) (1e9 + 7);
  static final long INF = Long.MAX_VALUE / 2;

  public static class Solver {
    public Solver() {

      int N = sc.nextInt();

      long maxW = sc.nextLong();

      long[] V = new long[N];
      long[] W = new long[N];
      for (int i=0; i<N; i++) {
        V[i] = sc.nextLong();
        W[i] = sc.nextLong();
      }

      Baggage half[] = new Baggage[1<<N/2];

      for (Bit bit = new Bit(); !bit.get(N/2); bit.next()) {
        long w = 0, v = 0;
        for (int i = bit.nextSetBit(0); i>=0; i=bit.nextSetBit(i+1)) {
          w += W[i];
          v += V[i];
        }
        half[(int)bit.toLong()] = new Baggage(w, v);
      }

      Arrays.sort(half, (o1, o2) -> Compare.L(o1.weight, o2.weight));

      int validCount = 1;
      for (int i=1; i<(1<<N/2); i++) {
        if (half[i].value > half[validCount-1].value) {
          half[validCount++] = half[i];
        }
      }

      long ans = 0;

      for (Bit bit = new Bit(); !bit.get(N-N/2); bit.next()) {
        long w = 0, v = 0;

        for (int i = bit.nextSetBit(0); i>=0; i = bit.nextSetBit(i+1)) {
          w += W[N/2+i];
          v += V[N/2+i];
        }

        if (w > maxW) continue;

        final long wLimit = maxW - w;
        int index = BinarySearch.binarySearch((i) -> half[i].weight <= wLimit, 0, validCount, false);
        ans = Math.max(v + half[index].value, ans);
      }

      out.println(ans);

    }

    public static class Compare {

      public static int L(long l1, long l2) {
        if (l1 == l2) return 0;
        else if (l1 < l2) return -1;
        else return 1;
      }

    }

    public class Baggage {
      long weight, value;
      Baggage(long w, long v) {
        weight = w;
        value = v;
      }
    }


    public static class BinarySearch {

      /*
       * [l, r)の区間でchecker.test()を実行する
       * rightOK ... 右側OK左側NGならtrue, 右側NG左側OKならfalse
       */
      public static long binarySearchLong(Predicate<Long> checker, long l, long r, boolean rightOK) {
        long ng, ok;

        if (rightOK) {
          ok = r-1;
          ng = l;
          if (!checker.test(ok)) {
            return r;
          }
          if (checker.test(ng)) {
            return l;
          }
        } else {
          ok = l;
          ng = r-1;
          if (!checker.test(ok)) {
            return l-1;
          }
          if (checker.test(ng)) {
            return r-1;
          }
        }

        /* ok と ng のどちらが大きいかわからないことを考慮 */
        while (Math.abs(ok - ng) > 1) {
          long mid = (ok + ng) / 2;

          if (checker.test(mid)) ok = mid;
          else ng = mid;
        }

        return ok;
      }

      /*
       * [l, r)の区間でchecker.test()を実行する
       */
      public static int binarySearch(Predicate<Integer> checker, int l, int r, boolean rightOK) {

        int ng, ok;

        if (rightOK) {
          ok = r-1;
          ng = l;
          if (!checker.test(ok)) {
            return r;
          }
          if (checker.test(ng)) {
            return l;
          }
        } else {
          ok = l;
          ng = r-1;
          if (!checker.test(ok)) {
            return l-1;
          }
          if (checker.test(ng)) {
            return r-1;
          }
        }

        while (Math.abs(ok - ng) > 1) {
          int mid = (ok + ng) / 2;

          if (checker.test(mid)) ok = mid;
          else ng = mid;
        }

        return ok;
      }

      public static int lowerBound(long[] a, long v) {

        if (a[0] >= v) return 0;

        int l = 0, r = a.length;
        int mid = (r + l) / 2;

        while (r - l > 1) {

          if (a[mid] >= v) r = mid;
          else l = mid;

          mid = (r + l) / 2;
        }

        return r;

      }

      public static int upperBound(long[] a, long v) {

        if (a[0] > v) return 0;

        int l = 0, r = a.length;
        int mid = (r + l) / 2;

        while (r - l > 1) {

          if (a[mid] > v) r = mid;
          else l = mid;

          mid = (r + l) / 2;
        }

        return r;

      }

      public static int count(long[] a, long v) {
        int lower = lowerBound(a, v);
        int upper = upperBound(a, v);
        return upper - lower;
      }

    }


    public class Bit extends BitSet {

      public void set(long l) {
        clear();
        for (int i=0; i<64; i++) {
          if (((l>>i)&1) == 1) set(i, true);
        }
      }

      public long toLong() {
        long res = 0;
        for (int i = nextSetBit(0); i >= 0; i = nextSetBit(i+1)) {
          res += 1<<i;
        }
        return res;
      }

      public void next() {
        for (int i=0; ; i++) {
          if (get(i)) {
            set(i, false);
          } else {
            set(i, true);
            break;
          }
        }
      }

    }

  }

  public static void main(String[] args) {
    new Solver();
    out.flush();
  }

  static class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    private boolean hasNextByte() {
      if (ptr < buflen) {
        return true;
      } else {
        ptr = 0;
        try {
          buflen = in.read(buffer);
        } catch (IOException e) {
          e.printStackTrace();
        }
        if (buflen <= 0) {
          return false;
        }
      }
      return true;
    }

    private int readByte() {
      if (hasNextByte()) return buffer[ptr++];
      else return -1;
    }

    private static boolean isPrintableChar(int c) {
      return 33 <= c && c <= 126;
    }

    private void skipUnprintable() {
      while (hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;
    }

    public boolean hasNext() {
      skipUnprintable();
      return hasNextByte();
    }

    public String next() {
      if (!hasNext()) throw new NoSuchElementException();
      StringBuilder sb = new StringBuilder();
      int b = readByte();
      while (isPrintableChar(b)) {
        sb.appendCodePoint(b);
        b = readByte();
      }
      return sb.toString();
    }

    public long nextLong() {
      if (!hasNext()) throw new NoSuchElementException();
      long n = 0;
      boolean minus = false;
      int b = readByte();
      if (b == '-') {
        minus = true;
        b = readByte();
      }
      if (b < '0' || '9' < b) {
        throw new NumberFormatException();
      }
      while (true) {
        if ('0' <= b && b <= '9') {
          n *= 10;
          n += b - '0';
        } else if (b == -1 || !isPrintableChar(b)) {
          return minus ? -n : n;
        } else {
          throw new NumberFormatException();
        }
        b = readByte();
      }
    }

    public int nextInt() {
      return (int) nextLong();
    }

    public int[] nextIntArray(int N, boolean oneBased) {
      if (oneBased) {
        int[] array = new int[N + 1];
        for (int i = 1; i <= N; i++) {
          array[i] = sc.nextInt();
        }
        return array;
      } else {
        int[] array = new int[N];
        for (int i = 0; i < N; i++) {
          array[i] = sc.nextInt();
        }
        return array;
      }
    }

    public long[] nextLongArray(int N, boolean oneBased) {
      if (oneBased) {
        long[] array = new long[N + 1];
        for (int i = 1; i <= N; i++) {
          array[i] = sc.nextLong();
        }
        return array;
      } else {
        long[] array = new long[N];
        for (int i = 0; i < N; i++) {
          array[i] = sc.nextLong();
        }
        return array;
      }
    }
  }

  static class Output extends PrintWriter {

    public Output(PrintStream ps) {
      super(ps);
    }

    public void print(int[] a, String separator) {
      for (int i = 0; i < a.length; i++) {
        if (i == 0) print(a[i]);
        else print(separator + a[i]);
      }
      println();
    }

    public void print(long[] a, String separator) {
      for (int i = 0; i < a.length; i++) {
        if (i == 0) print(a[i]);
        else print(separator + a[i]);
      }
      println();
    }

    public void print(String[] a, String separator) {
      for (int i = 0; i < a.length; i++) {
        if (i == 0) print(a[i]);
        else print(separator + a[i]);
      }
      println();
    }

    public void print(ArrayList a, String separator) {
      for (int i = 0; i < a.size(); i++) {
        if (i == 0) print(a.get(i));
        else print(separator + a.get(i));
      }
      println();
    }
  }

}