Relating counting complexity to non-uniform probability measures

A standard method for designing randomized algorithms to approximately count the number of solutions of a problem in $#$P, is by constructing a rapidly mixing Markov chain converging to the uniform distribution over this set of solutions. This construction is not always an easy task, and it is conjectured that it is not always possible. We want to investigate other possibilities for using Markov Chains in relation to counting, and whether we can relate algorithmic counting to other, non-uniform, probability distributions over the set we want to count. In this paper we present a family of probability distributions over the set of solutions of a problem in TotP, and show how they relate to counting; counting is equivalent to computing their normalizing factors. We analyse the complexity of sampling, of computing the normalizing factor, and of computing the size support of these distributions. The latter is also equivalent to counting. We also show how the above tasks relate to each other, and to other problems in complexity theory as well. In short, we prove that sampling and approximating the normalizing factor is easy. We do this by constructing a family of rapidly mixing Markov chains for which these distributions are stationary. At the same time we show that exactly computing the normalizing factor is TotP-hard. However the reduction proving the latter is not approximation preserving, which conforms with the fact that TotP-hard problems are inapproximable if NP $\neq$ RP. The problem we consider is the Size-of-Subtree, a TotP-complete problem under parsimonious reductions. Therefore the results presented here extend to any problem in TotP. TotP is the Karp-closure of self-reducible problems in $#$P, having decision version in P.

💡 Research Summary

The paper investigates the relationship between counting complexity and non‑uniform probability measures, focusing on the class TotP – the Karp‑closure of self‑reducible #P problems whose decision versions lie in P. While the standard approach to approximate counting in #P is to design a rapidly mixing Markov chain whose stationary distribution is uniform over the solution space, this is often difficult or impossible for certain structures (e.g., perfect binary trees). The authors therefore ask whether counting can be related to other, non‑uniform distributions and whether such distributions admit efficient sampling and approximation algorithms.

The concrete problem studied is Size‑of‑Subtree, a TotP‑complete problem under parsimonious reductions. An instance consists of a succinct description of a subtree S of a perfect binary tree T of height n; the goal is to compute |V(S)|, the number of nodes in S. The authors define a family of probability distributions {π_S} over the nodes of any binary tree S: for a node u at depth i, π_S(u) = α·2^{‑i}, where α is a normalizing constant chosen so that the probabilities sum to one. This distribution heavily weights nodes near the root and decays exponentially with depth, thereby breaking the symmetry that makes uniform random walks on T mix slowly.

A Markov chain P_S is constructed with transition probabilities p_S(i,j) = 1/2 if j is the parent of i, 1/4 if j is a child, and 0 otherwise (self‑loop probability is the remainder). By adding a lazy step (probability 1/2 of staying put) the chain becomes lazy, time‑reversible, and its stationary distribution is exactly π_S. Using conductance arguments, the authors prove that the conductance Φ(H) of the underlying weighted graph satisfies Φ(H) ≥ 1/(4(n+1)), which implies a mixing time τ_root(ε) = O(n²·log n·log(1/ε)). Hence sampling from π_S can be performed in randomized polynomial time.

The central technical contribution is the analysis of the computational complexity of three related tasks: (1) sampling from π_S, (2) computing the normalizing factor α_S, and (3) computing the support size |V(S)|. The paper shows:

• Sampling (1) is easy: the rapidly mixing chain yields a polynomial‑time randomized algorithm.
• Approximating α_S within any relative error (1 ± ζ) with confidence 1 − δ is also easy: an FPRAS is obtained by estimating expectations via the Markov chain samples.
• Exactly computing α_S is TotP‑hard under Turing reductions. The reduction is parsimonious: knowing α_S for all prefixes of S (subtrees S_i containing nodes up to depth i) allows one to recover the exact node counts at each depth, and thus |V(S)|. Consequently, exact computation of α_S is as hard as the original counting problem.
• Computing the support size |V(S)| is likewise TotP‑hard (parsimonious reduction). Moreover, |V(S)| can be expressed as a telescoping sum of the inverses of the normalizing factors of the prefix subtrees: |V(S)| = α_{S_n}^{‑1} − ∑{k=0}^{n‑1} α{S_k}^{‑1}. Hence exact support size computation reduces to exact α‑computation.
• Additive (absolute) approximation of |V(S)| is achievable in randomized polynomial time via the same sampling‑based FPRAS for α_S. However, multiplicative (relative) approximation of |V(S)| is impossible unless NP = P (or NP = RP for randomized algorithms), mirroring the known inapproximability of TotP‑hard problems.

The authors also discuss how these results extend to any problem in TotP. For any TotP problem #A, there exists a binary tree S whose nodes correspond bijectively to the solutions of #A; thus the same family of non‑uniform distributions and the associated Markov chain can be employed. Consequently, the paper provides a general framework showing that while exact counting remains TotP‑hard, sampling and relative‑error approximation of appropriately chosen non‑uniform measures are tractable.

In summary, the work demonstrates that for TotP‑complete problems, one can construct rapidly mixing Markov chains whose stationary distributions are non‑uniform yet encode the counting information in their normalizing constants. Sampling from these distributions is efficient, and the normalizing constants admit an FPRAS, but exact computation remains TotP‑hard. This bridges the gap between counting and sampling beyond the uniform case and suggests new avenues for studying counting complexity via non‑uniform probabilistic measures.