Smallest Sufficient Team

Smallest Sufficient Team

What is this problem about?

"Smallest Sufficient Team" is a challenging "Hard" problem that combines elements of set theory, bit manipulation, and optimization. Imagine you have a list of required skills and a set of people, each possessing a subset of those skills. Your goal is to form a team that covers every single required skill, using the fewest number of people possible.

This is a variation of the classic "Set Cover Problem," which is known to be NP-hard. However, in most interview settings, the number of skills is small (e.g., less than 16), which allows for an efficient solution using dynamic programming and bitmasks. The problem asks for the actual team members, not just the size of the team, adding an extra layer of complexity to the tracking of the optimal solution.

Why is this asked in interviews?

Companies like Google and TCS ask this question to see how candidates handle high-complexity optimization problems. It tests your ability to use Bit Manipulation to represent sets compactly, which is a crucial skill for memory-efficient programming. Furthermore, it evaluates your proficiency with Dynamic Programming (DP), specifically how to build a solution incrementally and how to reconstruct the path (the team members) from the DP table.

Algorithmic pattern used

The primary algorithmic pattern is Dynamic Programming with Bitmasking. We represent the "set of skills covered" as a binary integer (the mask), where the $i$-th bit is 1 if the $i$-th skill is covered. We maintain a DP table dp[mask], which stores the smallest list of people needed to achieve that specific skill set. For each person, we iterate through the current dp table and see if adding that person to an existing team would result in a better (smaller) team for a new, larger mask.

Example explanation

Suppose the required skills are ["Java", "Python", "SQL"] (3 skills, so mask ranges from 0 to 7).

• Person A has ["Java"]. Mask: 001.
• Person B has ["Python", "SQL"]. Mask: 110.
• Person C has ["Java", "Python"]. Mask: 011. We start with dp[0] = [].
• Process A: dp[001] becomes [A].
• Process B: dp[110] becomes [B], and dp[111] (A+B) becomes [A, B].
• Process C: dp[011] becomes [C]. Also check dp[011 | 110] which is dp[111]. A+B had 2 people, B+C also has 2 people. We keep one of them. The smallest team covering 111 is [A, B] or [B, C].

Common mistakes candidates make

A common mistake is trying to use a greedy approach (e.g., always picking the person with the most missing skills). While intuitive, greedy fails for set cover problems. Another issue is inefficiently updating the DP table; you should iterate through the existing masks in a way that doesn't use the same person multiple times for the same team in a single step. Finally, many candidates forget to handle the "reconstruction" part—they calculate the minimum team size but fail to return the actual indices of the people.

Interview preparation tip

To prepare for the "Smallest Sufficient Team interview question," study the "Bitmask DP" pattern. This pattern is very common for problems involving small sets (usually $N < 20$). Practice problems like "Traveling Salesperson" or "Counting Tilings" to get comfortable with bitwise operations (|, &, <<). For this specific problem, focus on how to store and update the "best team" for each mask efficiently, perhaps by storing indices instead of full lists to save space.