Randomized Communication Complexity

• Randomized Communication Complexity is the study of minimal communication bits needed by parties using shared or private randomness to achieve distributed computation with bounded error.
• Key lower bound techniques include the rectangle (corruption) bound, generalized discrepancy, and partition bound, which unifies and strengthens previous methods.
• Its applications extend to streaming lower bounds, extended formulations, and direct product theorems, deeply impacting distributed computing and circuit complexity.

Randomized communication complexity investigates the minimal amount of communication, measured in bits, required for two or more parties to compute a function (or relation) of distributed inputs when they are allowed to use random bits as part of the protocol. It contrasts with deterministic communication complexity by enabling protocols with probabilistic transitions and a bounded error probability, often yielding exponentially lower resource requirements for solving the same task. This field provides foundational lower bound techniques for distributed computing, streaming, circuit complexity, optimization, and related domains.

1. Fundamental Principles

Randomized communication protocols are functions of the parties' private inputs and shared random bits (either public-coin or private-coin). Such protocols compute an output with correctness probability at least $1-\epsilon$ for every valid input. The public-coin model assumes the parties share randomness, simplifying analysis via Yao's minimax principle, while the private-coin model can simulate the public-coin with a logarithmic overhead. The central measure is $R^{\text{pub}}_\epsilon(f)$, the minimum number of communicated bits in a protocol that computes $f$ erring with probability at most $\epsilon$.

Protocols may operate in various architectures: two-way, one-way, simultaneous message passing (SMP), or multiparty settings. The complexity can be distributional (under a fixed input distribution) or worst-case.

2. Classical Lower Bound Techniques

Several techniques and bounds have historically underpinned lower bounds in randomized communication complexity:

• Rectangle (Corruption) Bound: Based on the impossibility of covering too much mass of one output’s inputs with large monochromatic rectangles. It is formulated as a linear program, optimizing weights on rectangles subject to accuracy and normalization constraints for only one output value at a time.
• Generalized Discrepancy / $\gamma_2$ Bound: Relies on matrix factorization norms (e.g., Linial–Shraibman) and shows that functions with small discrepancy require large communication.
• Partition Bound: This is a master method unifying and subsuming other LP-based bounds (0910.4266). For a (possibly partial) function $f\colon X\times Y\to Z$, the $\epsilon$-partition bound $\text{prt}_\epsilon(f)$ is defined as the value of the following linear program:

$\min \sum_{z,R} w_{z,R}$

subject to

$$\forall (x,y)\in\text{dom}(f): \sum_{R:(x,y)\in R} w_{f(x,y),R} \geq 1-\epsilon$$

$\forall (x,y): \sum_{z,R:(x,y)\in R} w_{z,R} = 1$

$w_{z,R}\geq 0$

The crucial result is $$R^{\text{pub}}_\epsilon(f) \geq \log \text{prt}_\epsilon(f)$$. The partition bound is always at least as strong as the rectangle (corruption) bound and the generalized discrepancy bound.

• Quadratic Separations: There are explicit functions (e.g., Tribes) where the partition bound yields $\Omega(n)$, while previous bounds such as approximate degree or adversary only give $O(n^{1/3})$ (0910.4266). This demonstrates that the partition bound can yield provably larger lower bounds for specific functions.

3. Relations to Other Methods and Advanced Comparisons

Various lower bound techniques are tightly characterized in terms of their strength and limitations:

Lower Bound Type	Formulation	Strength Compared to Partition Bound
Rectangle/Corruption	LP (one-sided)	$$\text{prt}_\epsilon(f)\geq \text{srec}_\epsilon(f)\geq \text{rec}_\epsilon(f)$$
Generalized Discrepancy	Norm-based (factorization, $\gamma_2$)	Partition bound strictly subsumes
Approximate Degree (Query)	Min deg. poly. $\epsilon$-approximating $f$	Partition bound can be quadratically larger

The partition bound is a "master" method: any lower bound that can be expressed as decomposing randomized protocols into a convex combination of deterministic protocols (rectangles/partitions) is at most as strong as the partition bound, because the LP captures the optimal convex decomposition subject to error constraints.

4. Randomized Complexity Model Variations

Randomized communication complexity extends to multiple models and scenarios:

• Multiparty and Symmetrization: For $k$-player problems (e.g., $k$-XOR, $k$-AND, $k$-MAJ), the symmetrization technique reduces lower bounds for $k$-player protocols to two-party hardness, proving, for example, $\Omega(nk)$ lower bounds in both blackboard and message-passing models (Phillips et al., 2011).
• Distributional Complexity and Correlation: One can analyze complexity $$R^{I\leq k}_\epsilon(f)=\max_{\mu: I(X:Y)\leq k} D^\mu_\epsilon(f)$$ where correlation (mutual information) in the input distribution is limited. E.g., for Disjointness:

$$R^{I\leq k}_\epsilon(\text{DISJ}) = \Theta\big(\sqrt{n(k+1)}\big)$$

interpolates between product-distribution ($k=0$) and full correlation ($k=n$) regimes (Bottesch et al., 2015).

5. Applications and Implications

Randomized communication complexity has driven profound results across several domains:

• Streaming Lower Bounds: The lower bounds for Gap-Hamming-Distance ($\Omega(n)$ communication for $\Theta(\sqrt{n})$ gap) imply nearly optimal space lower bounds for streaming problems, such as frequency moments and distinct elements, across $k$-pass models (Chakrabarti et al., 2010).
• Extended Formulations and Polyhedral Complexity: The randomized communication complexity of computing the slack matrix in expectation equals the base-2 logarithm of its nonnegative rank. This directly characterizes the size of the smallest extended formulation for a polytope (via Yannakakis' theorem). Allowing randomized instead of deterministic protocols can significantly reduce extended formulation sizes, as shown for the spanning tree and perfect matching polytopes (Faenza et al., 2011).

Polytope	Deterministic Protocol	Randomized Protocol	Extension Complexity
Spanning Tree	Exponential	$O(n^3)$	$O(n^3)$
Perfect Matching	No small deterministic	$O(1.42^n)$	$O(1.42^n)$

• Direct Product Theorems: Strong direct product results hold in the public-coin one-way (Jain, 2010) and bounded-round (Jain et al., 2012) models. If communication is less than $k$ times the complexity for a single instance, the success probability for all $k$ copies is exponentially small. Tight characterizations (e.g., robust min-entropy) underlie these theorems.

6. Structural Separations, Limitations, and Recent Developments

Recent work has demonstrated striking separations among randomized, quantum, and classical lower-bound techniques:

• Super-Quadratic Separations: There exist total functions $F$ with randomized complexity $\Omega(Q(F)^{2.5})$ [$Q(F)$ quantum complexity], showing quantum advantage is provable but bounded for total functions (Anshu et al., 2016).
• Partition Bound Gaps: The gap between partition bound and partition number can be nearly quadratic, i.e., $\Omega(\text{UN}(F)^{2-o(1)})$ separation ($\text{UN}(F)$ = unambiguous nondeterministic complexity).
• Implicit Graph Representations: For matrices of sign-rank $3$ and certain geometric graphs (e.g., unit disk graphs), constant randomized communication complexity—and thus efficient implicit representations—only arise when arbitrarily large instances of the Greater-Than problem (half-graphs) are excluded. Otherwise, complexity is $\Theta(\log n)$ (Harms et al., 2023).
• Counterexamples to Stability Conjectures: There exist matrix families for which every $O(\sqrt{n})\times O(\sqrt{n})$ submatrix has constant randomized complexity, but the global matrix requires $\Theta(\log n)$ bits (Hambardzumyan et al., 2021).

7. Open Problems and Future Directions

Key open questions highlighted across the literature include:

• Whether classes admitting constant-cost randomized communication protocols for adjacency (e.g., via reductions to Equality) always permit efficient implicit representations, particularly for bounded sign-rank beyond $3$ (Harms et al., 2023).
• Whether the partition bound or robust entropy measures can be efficiently computed for broader function classes and fully characterize randomized query or communication complexity beyond symmetric or composed functions.
• Identifying the optimal communication/space tradeoffs for distributed and streaming variants of classic problems (e.g., collision, matrix rank, triangle detection), especially in constrained models where error, round, and sharing regimes interact in subtle ways (Göös et al., 2022, Sherstov et al., 26 Oct 2024, Chang et al., 17 Dec 2024).
• The precise separation between randomized and quantum models for various total functions and delineating the ultimate power of information complexity and direct product techniques (Anshu et al., 2016).

Randomized communication complexity remains a central tool in complexity theory, offering not only tight lower bounds for distributed protocols but also deep structural insights into the interplay between randomness, information, and computation across distributed models.