Multiparty Communication Complexity Research Articles

Lower Bounds on OBDD Proofs with Several Orders

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

Public vs. private randomness in simultaneous multi-party communication complexity

In simultaneous number-in-hand multi-party communication protocols, we have k players, who each receive a private input, and wish to compute a joint function of their inputs. The players simultaneously each send a single message to a referee, who then outputs the value of the function. The cost of the protocol is the total number of bits sent to the referee.For two players, it is known that giving the players a public (shared) random string is much more useful than private randomness: public-coin protocols can be unboundedly better than deterministic protocols, while private-coin protocols can only give a quadratic improvement on deterministic protocols.We extend the two-player gap to multiple players, and show that the private-coin communication complexity of a k-player function f is at least Ω(D(f)) for any k≥2. Perhaps surprisingly, this bound is tight: although one might expect the gap between private-coin and deterministic protocols to grow with the number of players, we show that the All-Equality function, where each player receives n bits of input and the players must determine if their inputs are all the same, can be solved by a private-coin protocol with O˜(nk+k) bits. Since All-Equality has deterministic complexity Θ(nk), this shows that sometimes the gap scales only as the square root of the number of players, and consequently the number of bits each player needs to send actually decreases as the number of players increases. We also consider the Exists-Equality function, where we ask whether there is a pair of players that received the same input, and prove a nearly-tight bound of Θ˜(kn) for it.

A note on multiparty communication complexity and the Hales–Jewett theorem

For integers n and k, the density Hales–Jewett numbercn,k is defined as the maximal size of a subset of [k]n that contains no combinatorial line. We show that for k≥3 the density Hales–Jewett number cn,k is equal to the maximal size of a cylinder intersection in the problem Partn,k of testing whether k subsets of [n] form a partition. It follows that the communication complexity, in the Number On the Forehead (NOF) model, of Partn,k, is equal to the minimal size of a partition of [k]n into subsets that do not contain a combinatorial line. Thus, the bound in [7] on Partn,k using the Hales–Jewett theorem is in fact tight, and the density Hales–Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity.As a simple application we prove a lower bound on cn,k, similar to the lower bound in [19] which is roughly cn,k/kn≥exp⁡(−O(log⁡n)1/⌈log2⁡k⌉). This lower bound follows from a protocol for Partn,k. It is interesting to better understand the communication complexity of Partn,k as this will also lead to the better understanding of the Hales–Jewett number. The main purpose of this note is to motivate this study.

Separation of Unbounded-Error Models in Multi-Party Communication Complexity

We construct a simple function that has small unbounded-error communication complexity in the $k$-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with subexponential advantage over random guessing has cost essentially $\Omega(\sqrt{n}\,/\,4^k)$ bits. This separates these classes up to $k \leq \delta\log n$ players for any constant $\delta < 1/4$, and gives the largest known separation by an explicit function in this regime of $k$. Our analysis is elementary and self-contained, inspired by the methods of Goldmann, Håstad, and Razborov (Computational Complexity, 1992). After initial publication of our work as a preprint (ECCC, 2016), Sherstov pointed out to us that an alternative proof of an $\Omega((n\,/\,4^k)^{1/7})$ separation is implicit in his prior work (SICOMP, 2016). Furthermore, based on his prior work (SICOMP, 2013 and SICOMP, 2016), Sherstov gave an alternative proof of our constructive $\Omega(\sqrt{n}\,/\,4^k)$ separation and also produced a stronger non-constructive $\Omega(n\,/\,4^k)$ separation. These results are explained in Sherstov's preprint (ECCC, 2016) and in article 14/22 in Theory of Computing. Our result has the following consequence for boolean threshold circuits. Let $\text{THR}$ and $\text{MAJ}$ denote the classes of linear threshold functions that have unbounded weights and polynomially bounded weights, respectively. Further, let $\text{PAR}_k$ ($\text{ANY}_k$) denote the class of functions that are parities of $k$ bits (any $k$-junta, respectively). Denote by $\text{THR} \circ \text{PAR}_k$ the class of depth-2 circuits where the top gate computes a linear threshold function, and the bottom gates compute functions in $\text{PAR}_k$. For every $2 \le k \le \delta \log n$, we show that there exists a function computable by a linear-size $\text{THR} \circ \text{PAR}_k$ circuit, but requires $\exp\left({n^{\Omega(1)}}\right)$-size circuits in the class $\text{MAJ} \circ \text{SYM} \circ \text{ANY}_{k-1}$, where the gates in the middle layer compute symmetric functions. Applying a result of Goldmann et al. (loc. cit.) to the above, similar lower bounds on the size of circuits of the form $\text{MAJ} \circ \text{THR} \circ \text{ANY}_{k-1}$ for computing the function follow. The main technical ingredient of our proof is to exhibit a composed function of the form $f \circ \text{XOR}$ which has exponentially small discrepancy while $f$ has sign-degree just 1. An interesting aspect of our composed function is that the block size of the inner XOR function is fixed to 1, independent of $k$, the number of players. A preliminary version of this paper appeared as ECCC technical report TR 16-095.

Lower Bounds for Number-in-Hand Multiparty Communication Complexity, Made Easy

In this paper we prove lower bounds on randomized multiparty communication complexity, mainly in the message-passing model, where messages are sent player-to-player. Some of our results apply to the blackboard model, where each message is written on a blackboard for all players to see. We introduce a new technique for proving such bounds, called symmetrization, which is natural, intuitive, and often easy to use. For example, for the problem where each of $k$ players gets a bit-vector of length $n$, and the goal is to compute the coordinatewise XOR of these vectors, we prove a tight lower bound of $\Omega(nk)$ in the blackboard model. For the same problem with AND instead of XOR, we prove a lower bound of roughly $\Omega(nk)$ in the message-passing model (assuming $k \le n/3200$) and $\Omega(n \log k)$ in the blackboard model. We also prove lower bounds for bitwise majority, for a graph-connectivity problem, and for other problems; the technique seems applicable to a wide range of other problems as well. All of our lower bounds allow randomized communication protocols with two-sided error. We also use the symmetrization technique to prove several direct-sum-like results for multiparty communication.

The Multiparty Communication Complexity of Set Disjointness

In the $k$-party set disjointness problem, the goal is to determine whether given subsets $S_1,S_2,\dots,S_k\subseteq\{1,2,\dots,n\}$ have empty intersection. We study this problem in the number-on-the-forehead model of communication, where the $i$th party knows all the sets except for $S_i,$ and prove a lower bound of $\Omega(n/4^k)^{1/4}$ on the randomized and nondeterministic communication complexity. This lower bound is close to tight. Previous lower bounds for set disjointness with $k\geq3$ parties were weaker than $\Omega(n/2^{k^3})^{1/(k+1)}.$ We also prove that solving $\ell$ instances of set disjointness requires $\ell\cdot\Omega(n/4^k)^{1/4}$ bits of communication, even to achieve correctness probability exponentially close to $1/2.$ This gives the first direct product result for multiparty set disjointness, solving an open problem due to Beame et al. (2005). Finally, we construct a read-once $\{\wedge,\vee\}$-circuit of depth $3$ with exponentially small discrepancy for up to $k\approx\frac12\l...

Special Section on the Forty-Fourth Annual ACM Symposium on Theory of Computing (STOC 2012)

This issue of SICOMP contains seven specially selected papers from the Forty-Fourth Annual ACM Symposium on Theory of Computing, otherwise known as STOC 2012, held May 19 to 22 in New York, New York. The papers here were chosen to represent both the excellence and the broad range of the STOC program. The papers have been revised and extended by the authors and subjected to the standard thorough reviewing process of SICOMP. The program committee consisted of Richard Cleve, Parikshit Gopalan, Jason Hartline, Tom Hayes, Anna Karlin, Sanjeev Khanna, Andrew McGregor, Rina Panigrahy, Toniann Pitassi, Ran Raz, Charles Rackoff, Satish Rao, Oded Regev, Dana Ron, Guy Rothblum, Amin Saberi, Rahul Santhanam, Shubhangi Saraf, Daniel Spielman, Madhur Tulsiani, Suresh Venkatasubramanian, Avi Wigderson, and David Williamson. They selected 90 papers out of 303 submissions. We briefly describe the papers that appear here. In “The Multiparty Communication Complexity of Set Disjointness,” Alexander Sherstov presents an $\Ome...

The Cost of Fault Tolerance in Multi-Party Communication Complexity

Multi-party communication complexity involves distributed computation of a function over inputs held by multiple distributed players. A key focus of distributed computing research, since the very beginning, has been to tolerate failures. It is thus natural to ask “If we want to compute a certain function in a fault-tolerant way, what will the communication complexity be?” For this question, this article will focus specifically on (i) tolerating node crash failures, and (ii) computing the function over general topologies (instead of, e.g., just cliques). One way to approach this question is to first develop results in a simpler failure-free setting, and then “amend” the results to take into account failures' impact. Whether this approach is effective largely depends on how big a difference failures can make. This article proves that the impact of failures is significant, at least for the Sum aggregate function in general topologies: As our central contribution, we prove that there exists (at least) an exponential gap between the non-fault-tolerant and fault-tolerant communication complexity of S um . This gap attests that fault-tolerant communication complexity needs to be studied separately from non-fault-tolerant communication complexity, instead of being considered as an “amended” version of the latter. Such exponential gap is not obvious: For some other functions such as the M ax aggregate function, the gap is only logarithmic. Part of our results are obtained via a novel reduction from a new two-party problem U nion S ize CP that we introduce. U nion S ize CP comes with a novel cycle promise , which is the key enabler of our reduction. We further prove that this cycle promise and U nion S ize CP likely play a fundamental role in reasoning about fault-tolerant communication complexity.

The NOF Multiparty Communication Complexity of Composed Functions

We study the k-party "number on the forehead" communication complexity of composed functions $${f \circ \vec{g}}$$f?g?, where $${f:\{0,1\}^n \to \{\pm 1\}}$$f:{0,1}n?{±1}, $${\vec{g} = (g_1,\ldots,g_n)}$$g?=(g1,?,gn), $${g_i : \{0,1\}^k \to \{0,1\}}$$gi:{0,1}k?{0,1} and for $${(x_1,\ldots,x_k) \in (\{0,1\}^n)^k}$$(x1,?,xk)?({0,1}n)k, $${f \circ \vec{g}(x_1,\ldots,x_k) = f(\ldots,g_i(x_{1,i},\ldots,x_{k,i}), \ldots)}$$f?g?(x1,?,xk)=f(?,gi(x1,i,?,xk,i),?). When $${\vec{g} = (g, g,\ldots, g)}$$g?=(g,g,?,g), we denote $${f \circ \vec{g}}$$f?g? by $${f \circ g}$$f?g. We show that there is an $${O({\rm log}^3 n)}$$O(log3n) cost simultaneous protocol for SYM $${\circ g}$$?g when k > 1 + log n, SYM is any symmetric function and g is any function. When k > 1 + 2 log n, our simultaneous protocol applies to SYM $${\circ \, \vec{g}}$$?g? with $${\vec{g}}$$g? being a vector of n arbitrary functions. We also get a non-simultaneous protocol for SYM $${\circ \, \vec{g}}$$?g? of cost $${O(n/2^k \cdot {\rm log}\, n+ k {\rm log}\, n)}$$O(n/2k·logn+klogn) for any k ? 2. In the setting of k ≤ 1 + log n, we study more closely functions of the form MAJORITY $${\circ g}$$?g, MOD m$${\circ g}$$?g and NOR $${\circ g}$$?g, where the latter two are generalizations of the well-known and studied functions generalized inner product and disjointness, respectively. We characterize the communication complexity of these functions with respect to the choice of g. In doing so, we answer a question posed by Babai et al. (SIAM J Comput 33:137---166, 2003) and determine the communication complexity of MAJORITY?QCSB k, where QCSB k is the "quadratic character of the sum of the bits" function. In the second part of our paper, we utilize the connection between the `number on the forehead' model and Ramsey theory to construct a large set without a k-dimensional corner (k-dimensional generalization of a k-term arithmetic progression) in $${(\mathbb{F}_{2}^{n})^k}$$(F2n)k, thereby obtaining the first non-trivial bound on the corresponding Ramsey number. Furthermore, we give an explicit coloring of [N] × [N] without a monochromatic two-dimensional corner and use this to obtain an explicit three-party protocol of cost $${O(\sqrt{n})}$$O(n) for the EXACTN function. For x1,x2,x3n-bit integers, EXACTN(x1,x2,x3) = ?1 iff x1 + x2 + x3 = N.

One-round multi-party communication complexity of distinguishing sums

We consider an instance of the following problem: Parties P1,…,Pk each receive an input xi, and a coordinator (distinct from each of these parties) wishes to compute f(x1,…,xk) for some predicate f. We are interested in one-round protocols where each party sends a single message to the coordinator; there is no communication between the parties themselves. What is the minimum communication complexity needed to compute f, possibly with bounded error?We prove tight bounds on the one-round communication complexity when f corresponds to the promise problem of distinguishing sums (namely, determining which of two possible values the {xi} sum to) or the problem of determining whether the {xi} sum to a particular value. Similar problems were studied previously by Nisan and in concurrent work by Viola. Our proofs rely on basic theorems from additive combinatorics, but are otherwise elementary.

Tensor Rank and Strong Quantum Nondeterminism in Multiparty Communication

In this paper we study quantum nondeterminism in multiparty communication. There are three (possibly) different types of nondeterminism in quantum computation: i) strong, ii) weak with classical proofs, and iii) weak with quantum proofs. Here we focus on the first one. A strong quantum nondeterministic protocol accepts a correct input with positive probability, and rejects an incorrect input with probability 1. In this work we relate strong quantum nondeterministic multiparty communication complexity to the rank of the communication tensor in the Number-On-Forehead and Number-In-Hand models. In particular, by extending the definition proposed by de Wolf to {\it nondeterministic tensor-rank} ($nrank$), we show that for any boolean function $f$ when there is no prior shared entanglement between the players, 1) in the Number-On-Forehead model, the cost is upper-bounded by the logarithm of $nrank(f)$; 2) in the Number-In-Hand model, the cost is lower-bounded by the logarithm of $nrank(f)$. Furthermore, we show that when the number of players is $o(\log\log n)$ we have that $NQP\nsubseteq BQP$ for Number-On-Forehead communication.

Finding, Minimizing, and Counting Weighted Subgraphs

For a pattern graph $H$ on $k$ nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of $H$ in a given large graph $G$ on $n$ nodes, as well as finding minimum weight copies in both node-weighted and edge-weighted graphs. Our results include the following: 1. The number of copies of an $H$ with an independent set of size $s$ can be computed exactly in $O^*(2^s n^{k-s+3})$ time and $poly(n,k)+O(2^s n^2)$ space, or in $O^*(4^s n^{k-s+3})$ time and $poly(n,k)$ space. (The $O^*$ notation omits $poly(k)$ factors.) To obtain these algorithms we provide fast algorithms for computing the permanent of a $k \times n$ matrix over rings and semirings. 2. The number of copies of any $H$ having minimum (or maximum) node-weight (with arbitrary real weights on nodes) can be found in $O(n^{\omega k/3} + n^{2k/3+o(1)})$ time, where $\omega < 2.38$ is the matrix multiplication exponent and $k$ is divisible by $3$. Similar results hold for other values of $k$. Also, the number of copies having exactly a prescribed weight can be found within this time. These algorithms extend the technique of Czumaj and Lingas [in Proceedings of SODA 2007, ACM, New York, SIAM, Philadelphia, 2007, pp. 986--994] and give a new (algorithmic) application of multiparty communication complexity. 3. Finding an edge-weighted triangle of weight exactly $0$ in general graphs requires $\Omega(n^{3-\varepsilon})$ time for all $\varepsilon > 0$, unless the $3$SUM problem on $N$ numbers can be solved in $O(N^{2 - \varepsilon})$ time. This suggests that the edge-weighted problem is much harder than its node-weighted version.

Hadamard tensors and lower bounds on multiparty communication complexity

We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the Number on the Forehead model of Chandra, Furst, and Lipton. We define an analog of the Hadamard property of matrices for tensors in multiple dimensions and show that any k-party communication problem represented by a Hadamard tensor must have ?(n/2 k ) multiparty communication complexity. We also exhibit constructions of Hadamard tensors. This allows us to obtain ?(n/2 k ) lower bounds on multiparty communication complexity for a new class of explicitly defined Boolean functions.

Multiparty Communication Complexity and Threshold Circuit Size of $\ensuremath{\sfAC}^0$

We prove an $n^{\Omega(1)}/4^k$ lower bound on the randomized k-party communication complexity of depth 4 $\ensuremath {\sf AC}^0$ functions in the number-on-forehead (NOF) model for up to $\Theta(\log n)$ players. These are the first nontrivial lower bounds for general NOF multiparty communication complexity for any $\ensuremath {\sf AC}^0$ function for $\omega(\log\log n)$ players. For nonconstant k the bounds are larger than all previous lower bounds for any $\ensuremath {\sf AC}^0$ function even for simultaneous communication complexity. Our lower bounds imply the first superpolynomial lower bounds for the simulation of $\ensuremath {\sf AC}^0$ by $\ensuremath {\sf MAJ\circ SYM\circ AND}$ circuits, showing that the well-known quasi-polynomial simulations of $\ensuremath {\sf AC}^0$ by such circuits due to Allender (1989) and Yao (1990) are qualitatively optimal,-1pt even for formulas of small constant depth. We also exhibit a depth 5 formula in ${\ensuremath {\sf NP}^{cc}_k}-{\ensuremath {\sf BPP}^{cc}_k}$ for k up to $\Theta(\log n)$ and derive $\Omega(2^{\sqrt{\log n}/\sqrt{k}})$ lower bound on the randomized k-party NOF communication complexity of set disjointness for up to $\Theta(\log^{1/3} n)$ players, which is significantly larger than the $O(\log\log n)$ players allowed in the best previous lower bounds for multiparty set disjointness. We prove other strong results for depth 3 and 4 $\ensuremath {\sf AC}^0$ functions.

The Pattern Matrix Method

We develop a novel technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function $f: \{0,1\}^n \to\{0,1\}$, and let $A_f$ be the matrix whose columns are each an application of $f$ to some subset of the variables $x_1,x_2,\ldots,x_{4n}.$ We prove that $A_f$ has bounded-error communication complexity $\Omega(d),$ where $d$ is the approximate degree of $f.$ This result remains valid in the quantum model, regardless of prior entanglement. In particular, it gives a new and simple proof of Razborov's breakthrough quantum lower bounds for disjointness and other symmetric predicates. We further characterize the discrepancy, approximate rank, and approximate trace norm of $A_f$ in terms of well-studied analytic properties of $f,$ broadly generalizing several recent results on small-bias communication and agnostic learning. The method of this paper has also enabled important progress in multiparty communication complexity.

Partition arguments in multiparty communication complexity

Consider the “Number in Hand” multiparty communication complexity model, where k players holding inputs x 1 , … , x k ∈ { 0 , 1 } n communicate to compute the value f ( x 1 , … , x k ) of a function f known to all of them. The main lower bound technique for the communication complexity of such problems is that of partition arguments: partition the k players into two disjoint sets of players and find a lower bound for the induced two-party communication complexity problem. In this paper, we study the power of partition arguments. Our two main results are very different in nature: (i) For randomized communication complexity, we show that partition arguments may yield bounds that are exponentially far from the true communication complexity. Specifically, we prove that there exists a 3 -argument function f whose communication complexity is Ω ( n ) , while partition arguments can only yield an Ω ( log n ) lower bound. The same holds for nondeterministiccommunication complexity. (ii) For deterministic communication complexity, we prove that finding significant gaps between the true communication complexity and the best lower bound that can be obtained via partition arguments, would imply progress on a generalized version of the “log-rank conjecture” in communication complexity. We also observe that, in the case of computing relations (search problems), very large gaps do exist. We conclude with two results on the multiparty “fooling set technique”, another method for obtaining communication complexity lower bounds.

A Separation of NP and coNP in Multiparty Communication Complexity

We prove that coNP * MA in the number-on-forehead model of multiparty communication complexity for up to k = (1 e) log n players, where e > 0 is any con- stant. Specifically, we construct an explicit function F : (f0; 1g n ) k ! f0; 1g with co- nondeterministic complexity O(log n) and Merlin-Arthur complexity n W(1) . The problem was open for k > 3. As a corollary, we obtain an explicit separation of NP and coNP for up to k = (1 e) log n players, complementing an independent result by Beame et al. (2010) who separate these classes nonconstructively for up to k = 2 (1 e)n players.

Separating Deterministic from Randomized Multiparty Communication Complexity

We solve some fundamental problems in the number-on-forehead (NOF) $k$-player communication model. We show that there exists a function which has at most logarithmic communication complexity for randomized protocols with one-sided false-positives error probability of 1/3, but which has linear communication complexity for deterministic protocols, and in fact, even for the more powerful nondeterministic protocols. The result holds for every $\epsilon > 0$ and every $k \le 2^{(1-\epsilon)n}$ players, where $n$ is the number of bits on each player's forehead. As a consequence, we obtain the NOF communication class separation $\mathsf{coRP} \not\subset \mathsf{NP}$. This in particular implies that $\mathsf{P} \neq \mathsf{RP}$ and $\mathsf{NP} \neq \mathsf{coNP}$. We also show that for every $\epsilon > 0$ and every $k \le n^{1-\epsilon}$, there exists a function which has constant randomized complexity for public coin protocols but at least logarithmic complexity for private coin protocols. No larger gap between private and public coin protocols is possible. Our lower bounds are existential; no explicit function is known to satisfy nontrivial lower bounds for $k \ge \log n$ players. However, for every $\epsilon > 0$ and every $k \le (1-\epsilon) \cdot \log n$ players, the $\mathsf{NP} \ne \mathsf{coNP}$ separation (and even the $\mathsf{coNP} \not\subset \mathsf{MA}$ separation) was obtained independently by Gavinsky and Sherstov (2010) using an explicit construction. In this work, for $k \le (1/9) \cdot \log n$ players, we exhibit an explicit function which has communication complexity $O(1)$ for public coin protocols and $\Omega(\log n)$ for deterministic protocols. This improves the best previously known deterministic lower bound for a function with efficient randomized protocols, which was $\Omega(\log \log n)$, given by Beigel, Gasarch, and Glenn (2006). It follows from our existential result that any function that is complete for the class of functions with polylogarithmic nondeterministic $k$-player communication complexity does not have polylogarithmic deterministic complexity. We show that the set intersection function, which is complete in the number-in-hand model, is not complete in the NOF model under cylindrical reductions.

SIGEST

Suppose three people walk into a room from three different entrances. Before they enter, a hat is placed on each person. Each hat is either red or blue, but the wearer of the hat doesn't know its color. They then see each other and their hats for the first time, and need to guess simultaneously whether their own hat is red or blue. (No mirrors allowed!) Each person can either guess “red,” “blue,” or “pass.” They win a prize collectively if they all are correct, including at least one correct guess and the remainder passes. What is the optimal percentage of times they can win? There is an obvious strategy for 50%: two pass and the third says red (or blue). But there is a strategy that achieves 75% success (without any ordering of the three people). If you search “hat guessing games” online, you will find it along with some simple explanations of how achieving 75% success is feasible. Hat guessing games have attracted increased attention in the discrete mathematics community in the past decade. They are far more than fun parlor games—they have applications in a variety of areas. The most established is to coding theory, which deals with the design of efficient and error correcting methods for data transmission. Another application is to the design of deterministic auction mechanisms. Hat guessing games also have linkages to various important problems in theoretical computer science, including multiparty communication complexity. This issue's SIGEST selection, from the SIAM Journal on Discrete Mathematics, provides a brief review of the state of the art in analyzing hat guessing games as well as important new research results. Appropriately titled “Hat Guessing Games,” it is co-authored by Steve Butler, Mohammad Hajiaghayi, Robert Kleinberg, and Tom Leighton. The paper begins with the basic problem, where there are n players, k hat colors, and a deterministic guessing strategy known to both the players and the “adversary” placing the hats on each person. The problem is to find the deterministic guessing strategy that maximizes the percentage of persons guessing their own hat color correctly. (Note that this version of the problem is different from the collective success in the initial example described above, and hence there is no “pass” guess.) The paper summarizes the state of the art for this problem and then considers several variants, including the version where people see only a subset of the other participants (which has applications to index coding problems). It also shows an interesting and perhaps intuitive result: under certain conditions, optimal strategies are unbiased, meaning that each player guesses each color had with equal probability. It concludes with a nice discussion of open research questions. “Hat Guessing Games” provides a very nice and understandable glance into an important area of discrete mathematics, in a way that should be accessible to most SIAM Review readers. The mathematics, which comes from graph theory and combinatorics, is widely accessible, and the exposition is particularly nice. And besides, it may provide you with a new idea for your next party!

Quantum multiparty communication complexity and circuit lower bounds

We define a quantum model for multiparty communication complexity and prove a simulation theorem between the classical and quantum models. As a result, we show that if the quantum k-party communication complexity of a function f is Ω(n/2k), its classical k-party communication is Ω(n/2k/2). Finding such an f would allow us to prove strong classical lower bounds for k ≥ log n players and make progress towards solving a major open question about symmetric circuits.