Razborov [J. ACM, 2016] exhibited the following surprisingly strong trade-off phenomenon in propositional proof complexity: for a parameter $k = k(n)$, there exists $k$-CNF formulas over $n$ variables, having resolution refutations of $O(k)$ width, but every tree-like refutation of width $n^{1-\epsilon}/k$ needs size $\text{exp}\big(n^{\Omega(k)}\big)$. We extend this result to tree-like ... more >>>

Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities, is an outstanding problem that has received a lot of attention after its introduction by Raz and Tzamaret (2008). Very recently, Efremenko, Garlik and Itsykson (2023) proved the first exponential lower bounds ... more >>>

We relate various complexity measures like sensitivity, block sensitivity, certificate complexity for multi-output functions to the query complexities of such functions. Using these relations, we improve upon the known relationship between pseudo-deterministic query complexity and deterministic query complexity for total search problems: We show that pseudo-deterministic query complexity is at ... more >>>

A classic result of Nisan [SICOMP '91] states that the deterministic decision tree depth complexity of every total Boolean function is at most the cube of its randomized decision tree depth complexity. The question whether randomness helps in significantly reducing the size of decision trees appears not to have been ... more >>>

We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f \circ g$ as long as the gadget $g$ satisfies a property that we call stifling. We observe that several simple ... more >>>

We establish an $\epsilon$-sensitive hierarchy separation for monotone arithmetic computations. The notion of $\epsilon$-sensitive monotone lower bounds was recently introduced by Hrubes [Computational Complexity'20]. We show the following:

(1) There exists a monotone polynomial over $n$ variables in VNP that cannot be computed by $2^{o(n)}$ size monotone ... more >>>

We show that there is a family of monotone multilinear polynomials over $n$ variables in VP, such that any monotone arithmetic circuit for it would be of size $2^{\Omega(n)}$. Before our result, strongly exponential lower bounds on the size of monotone circuits were known only for computing explicit polynomials in ... more >>>

Valiant (1980) showed that general arithmetic circuits with negation can be exponentially more powerful than monotone ones. We give the first qualitative improvement to this classical result: we construct a family of polynomials $P_n$ in $n$ variables, each of its monomials has positive coefficient, such that $P_n$ can be computed ... more >>>

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $\Theta(n)$. This improves upon the ... more >>>

Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function $f : \{-1, 1\}^n \to \{-1, 1\}$ and $\bullet : \{-1, 1\}^2 \to \{-1, 1\}$ the two-party bounded-error quantum communication complexity of $(f \circ \bullet)$ is $O(Q(f) \log n)$, where $Q(f)$ is the bounded-error quantum query complexity of $f$. ... more >>>

Lifting theorems are theorems that relate the query complexity of a function $f:\left\{ 0,1 \right\}^n\to \left\{ 0,1 \right\}$ to the communication complexity of the composed function $f\circ g^n$, for some “gadget” $g:\left\{ 0,1 \right\}^b\times \left\{ 0,1 \right\}^b\to \left\{ 0,1 \right\}$. Such theorems allow transferring lower bounds from query complexity to ... more >>>

We demonstrate a lower bound technique for linear decision lists, which are decision lists where the queries are arbitrary linear threshold functions.
We use this technique to prove an explicit lower bound by showing that any linear decision list computing the function $MAJ \circ XOR$ requires size $2^{0.18 n}$. This ... more >>>

The canonical problem that gives an exponential separation between deterministic and randomized communication complexity in the classical two-party communication model is `Equality'. In this work, we show that even allowing access to an `Equality' oracle, deterministic protocols remain exponentially weaker than randomized ones. More precisely, we exhibit a total function ... more >>>

We construct a simple and total XOR function $F$ on $2n$ variables that has only $O(\sqrt{n})$ spectral norm, $O(n^2)$ approximate rank and $n^{O(\log n)}$ approximate nonnegative rank. We show it has polynomially large randomized bounded-error communication complexity of $\Omega(\sqrt{n})$. This yields the first exponential gap between the logarithm of the ... more >>>

We develop a technique for proving lower bounds in the setting of asymmetric communication, a model that was introduced in the famous works of Miltersen (STOC'94) and Miltersen, Nisan, Safra and Wigderson (STOC'95). At the core of our technique is a novel simulation theorem: Alice gets a $p \times n$ ... more >>>

Proving super-polynomial lower bounds against depth-2 threshold circuits of the form THR of THR is a well-known open problem that represents a frontier of our understanding in boolean circuit complexity. By contrast, exponential lower bounds on the size of THR of MAJ circuits were shown by Razborov and Sherstov (SIAM ... more >>>

We show a new duality between the polynomial margin complexity of $f$ and the discrepancy of the function $f \circ$ XOR, called an XOR function. Using this duality,
we develop polynomial based techniques for understanding the bounded error (BPP) and the weakly-unbounded error (PP) communication complexities of XOR functions. ... more >>>

We prove a randomized communication-complexity lower bound for a composed OrderedSearch $\circ$ IP — by lifting the randomized query-complexity lower-bound of OrderedSearch to the communication-complexity setting. We do this by extending ideas from a paper of Raz and Wigderson. We think that the techniques we develop will be useful in ... more >>>

We consider the problem of elimination in communication complexity, that was first raised by Ambainis et al. and later studied by Beimel et al. for its connection to the famous direct sum question. In this problem, let $f:\{0,1\}^n \to \{0,1\}$ be any boolean function. Alice and Bob get $k$ inputs ... more >>>

We prove tight network topology dependent bounds on the round complexity of computing well studied $k$-party functions such as set disjointness and element distinctness. Unlike the usual case in the CONGEST model in distributed computing, we fix the function and then vary the underlying network topology. This complements the recent ... more >>>

We show that a simple function has small unbounded error communication complexity in the $k$-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with sub-exponential advantage over random guessing has cost essentially $\Omega\left(\frac{\sqrt{n}}{4^k}\right)$ bits. Such a separation was first shown for $k=2$ independently by Buhrman et al. ['07] ... more >>>

We provide the first communication lower bounds that are sensitive to the network topology for computing natural and simple functions by point to point message passing protocols for the `Number in Hand' model. All previous lower bounds were either for the broadcast model or assumed full connectivity of the network. ... more >>>

In the `Number-on-Forehead' (NOF) model of multiparty communication, the input is a $k \times m$ boolean matrix $A$ (where $k$ is the number of players) and Player $i$ sees all bits except those in the $i$-th row, and the players communicate by broadcast in order to evaluate a specified ... more >>>

We study the $k$-party `number on the forehead' communication complexity of composed functions $f \circ \vec{g}$, where $f:\{0,1\}^n \to \{\pm 1\}$, $\vec{g} = (g_1,\ldots,g_n)$, $g_i : \{0,1\}^k \to \{0,1\}$ and for $(x_1,\ldots,x_k) \in (\{0,1\}^n)^k$, $f \circ \vec{g}(x_1,\ldots,x_k) = f(\ldots,g_i(x_{1,i},\ldots,x_{k,i}), \ldots)$. When $\vec{g} = (g,g,\ldots,g)$ we denote $f \circ \vec{g}$ by ... more >>>

We give a new upper bound for the Group and Quasigroup
Isomorphism problems when the input structures
are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with $O(\log\log n)$ depth and $O(\log^2 n)$ nondeterministic bits, ... more >>>

We study solution sets to systems of generalized linear equations of the following form:
$\ell_i (x_1, x_2, \cdots , x_n)\, \in \,A_i \,\, (\text{mod } m)$,
where $\ell_1, \ldots ,\ell_t$ are linear forms in $n$ Boolean variables, each $A_i$ is an arbitrary subset of $\mathbb{Z}_m$, and $m$ is a composite ... more >>>

We extend the 'Generalized Discrepancy' technique suggested by Sherstov to the `Number on the Forehead' model of multiparty communication. This allows us to prove strong lower bounds of n^{\Omega(1)} on the communication needed by k players to compute the Disjointness function, provided $k$ is a constant. In general, our method ... more >>>

We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'Number on the Forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the Degree-Discrepancy Lemma in the recent work of Sherstov (STOC'07). ... more >>>

We study languages with bounded communication complexity in the multiparty "input on the forehead" model with worst-case partition. In the two party case, it is known that such languages are exactly those that are recognized by programs over commutative monoids. This can be used to show that these languages can ... more >>>

Let m,q > 1 be two integers that are co-prime and A be any subset of Z_m. Let P be any multi-linear polynomial of degree d in n variables over Z_m. We show that the MOD_q boolean function on n variables has correlation at most exp(-\Omega(n/(m2^{m-1})^d)) with the boolean function ... more >>>