The Open Question: Do All Np-Complete Problems Have #P-Complete Counting Versions?
Defining #P-completeness
The complexity class #P consists of counting problems associated with decision problems in NP. Specifically, #P contains the counting versions of NP problems where the goal is to count the number of solutions or witnesses that satisfy the decision problem. For example, while the NP-complete problem SAT asks “Is there a satisfying assignment to this Boolean formula?”, the #P-complete problem #SAT asks “How many satisfying assignments are there for this Boolean formula?”
A counting problem #X is #P-complete if:
- #X is in #P
- Every problem in #P can be reduced to #X in polynomial time
This means that #P-complete problems are the hardest problems in #P in the sense that if any #P-complete problem can be solved in polynomial time, then every problem in #P can. An example of a #P-complete problem is #SAT, the counting version of the NP-complete decision problem SAT.
The Relationship Between NP and #P
While NP problems focus on the existence of solutions or witnesses that satisfy certain conditions, #P problems require counting the actual number of solutions. Despite this difference, there is thought to be an intimate connection between the two classes. In particular, it is widely believed that all NP-complete problems have #P-complete counting versions.
Intuitively, counting the solutions seems at least as hard as determining if one solution exists. For typical NP-complete problems like SAT, counting all solutions requires finding all solutions, not just determining if one exists. There are also connections through Turing reductions – an oracle for the counting problem could solve the decision problem in one query by checking if the count is at least one.
More formally, Leslie Valiant showed that #P forms an upper bound on NP by giving a polynomial-time Turing reduction from any NP problem to some #P problem. This provides theoretical evidence that all NP problems have harder counting counterparts in #P.
Open Questions Remaining
Despite substantial evidence and intuition, it remains an open question whether all NP-complete problems have #P-complete counting versions. This has not been formally proven, and resolving this question either way would have significant implications in complexity theory.
If true, it would further demonstrate the intricately connected relationship between decision and counting for computationally hard problems. All NP-complete problems would have an associated #P-complete counting problem at least as hard. However, proving this may require new techniques beyond current knowledge.
On the other hand, even a single counter-example of an NP-complete problem without a #P-hard counting version would be very surprising. This would demonstrate gaps in our understanding and require reassessing links between decision and counting. Significant revisions to complexity theory hierarchies might be needed.
Current Approaches and Limitations
The main evidence that all NP-complete problems have #P equivalents relies on Leslie Valiant’s #P counting class. Valiant defined #P and showed that NP is contained within #P via polynomial-time Turing reductions.
Specifically, Valiant gave a reduction showing that for any NP problem, there is an associated #P problem such that the counting problem can solve the decision problem. An oracle that counts solutions could simply check if the count is nonzero. This implies NP ⊆ #P.
However, the opposite containment has not been shown. Proving #P ⊆ NP would require reducing counting problems to decision problems. But counting problems seem inherently harder than their decision counterparts, so reducing them may require techniques and insights not currently available.
Another barrier is that the counting problems maintain more information about the solutions, which cannot be exploited by an NP oracle. So while NP machines can use #P oracles, allowing #P machines to access NP oracles seems far weaker.
Future Research Directions
Going forward, complexity theorists may pursue different avenues to relate the power of NP and #P:
- Find novel reductions between NP and #P that avoid limitations of previous approaches
- Explore fine-grained complexity to relate specific NP-complete and #P-complete problems rather than the classes as a whole
- Relate the polynomial-time hierarchy to the #P hierarchy to better understand structural similarities
- Study the space of problems in NP and #P rather than just complete problems to identify gaps in containment
Through these efforts, researchers may make progress on this central open question on the connections between counting and decision. Settling the question either way would advance our understanding of the boundaries between easy and hard problems in computer science.
