Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated `lifted' function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. Several important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, from its current near-linear size down to polylogarithmic in the number of inputs N of the original function or, ideally, constant. The near-linear size bound was shown by Lovett, Meka, Mertz, Pitassi and Zhang using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with the Index function of near-linear size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following; 1) The conjecture of Lovett et al. is false when the size of the Index gadget is log N-ω(1). 2) Also, the Inner-Product function, which satisfies the disperser property at size O(log N), does not have this property when its size is log N-ω(1). 3) Nonetheless, using Index gadgets of size at least 4, we prove a lifting theorem for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs. 4) Using the ideas from this lifting theorem, we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(⊕) refutation size, which yields many new exponential lower bounds on such proofs.

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be the maximum $k$-party NOF randomized communication complexity of $\mathcal{G}$. We show: (1) The Generalized Inner Product function $GIP^k_n$ cannot be computed in $FORMULA[s]\circ \mathcal{G}$ on more than $1/2+\varepsilon$ fraction of inputs for $$ s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right).$$ As a corollary, we get an average-case lower bound for $GIP^k_n$ against $FORMULA[n^{1.99}]\circ PTF^{k-1}$. (2) There is a PRG of seed length $n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right)$ that $\varepsilon$-fools $FORMULA[s] \circ \mathcal{G}$. For $FORMULA[s] \circ LTF$, we get the better seed length $O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right)$. This gives the first non-trivial PRG (with seed length $o(n)$) for intersections of $n$ half-spaces in the regime where $\varepsilon \leq 1/n$. (3) There is a randomized $2^{n-t}$-time $\#$SAT algorithm for $FORMULA[s] \circ \mathcal{G}$, where $$t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}.$$ In particular, this implies a nontrivial #SAT algorithm for $FORMULA[n^{1.99}]\circ LTF$. (4) The Minimum Circuit Size Problem is not in $FORMULA[n^{1.99}]\circ XOR$. On the algorithmic side, we show that $FORMULA[n^{1.99}] \circ XOR$ can be PAC-learned in time $2^{O(n/\log n)}$.

We prove a new query-to-communication lifting for randomized protocols, with inner product as gadget. This allows us to use a much smaller gadget, leading to a more efficient lifting. Prior to this work, such a theorem was known only for deterministic protocols, due to Chattopadhyay et al. and Wu et al. The only query-to-communication lifting result for randomized protocols, due to G\o\os, Pitassi and Watson, used the much larger indexing gadget. Our proof also provides a unified treatment of randomized and deterministic lifting. Most existing proofs of deterministic lifting theorems use a measure of information known as thickness. In contrast, G\o\os, Pitassi and Watson used blockwise min-entropy as a measure of information. Our proof uses the blockwise min-entropy framework to prove lifting theorems in both settings in a unified way.

Non-signaling strategies are a generalization of quantum strategies that have been studied in physics over the past three decades. Recently, they have found applications in theoretical computer science, including to proving inapproximability results for linear programming and to constructing protocols for delegating computation. A central tool for these applications is probabilistically checkable proof (PCPs) systems that are sound against non-signaling strategies. In this paper we show, assuming a certain geometrical hypothesis about noise robustness of non-signaling proofs (or, equivalently, about robustness to noise of solutions to the Sherali-Adams linear program), that a slight variant of the parallel repetition of the exponential-length constant-query PCP construction due to Arora et al. (JACM 1998) is sound against non-signaling strategies with constant locality. Our proof relies on the analysis of the linearity test and agreement test (also known as the direct product test) in the non-signaling setting.

We show that Cutting Planes (CP) proofs are hard to find: Given an unsatisfiable formula $F$, 1) It is NP-hard to find a CP refutation of $F$ in time polynomial in the length of the shortest such refutation; and 2)unless Gap-Hitting-Set admits a nontrivial algorithm, one cannot find a tree-like CP refutation of $F$ in time polynomial in the length of the shortest such refutation. The first result extends the recent breakthrough of Atserias and M\"uller (FOCS 2019) that established an analogous result for Resolution. Our proofs rely on two new lifting theorems: (1) Dag-like lifting for gadgets with many output bits. (2) Tree-like lifting that simulates an $r$-round protocol with gadgets of query complexity $O(\log r)$ independent of input length.

We study projective dimension, a graph parameter, denoted by pd( G ) for a bipartite graph G , introduced by Pudlák and Rödl (1992). For a Boolean function f (on n bits), Pudlák and Rödl associated a bipartite graph G f and showed that size of the optimal branching program computing f , denoted by bpsize( f ), is at least pd( G f ) (also denoted by pd( f )). Hence, proving lower bounds for pd( f ) implies lower bounds for bpsize( f ). Despite several attempts (Pudlák and Rödl (1992), Rónyai et al. (2000)), proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We observe that there exist a Boolean function f for which the gap between the pd( f ) and bpsize( f )) is 2 Ω( n ) . Motivated by the argument in Pudlák and Rödl (1992), we define two variants of projective dimension: projective dimension with intersection dimension 1 , denoted by upd( f ), and bitwise decomposable projective dimension , denoted by bitpdim( f ). We show the following results: (a) We observe that there exist a Boolean function f for which the gap between upd( f ) and bpsize( f ) is 2 Ω( n ) . In contrast, we also show that the bitwise decomposable projective dimension characterizes size of the branching program up to a polynomial factor. That is, there exists a constant c > 0 and for any function f , bitpdim( f )/6 ≤ bpsize( f ) ≤ (bitpdim( f )) c . (b) We introduce a new candidate family of functions f for showing super-polynomial lower bounds for bitpdim( f ). As our main result, for this family of functions, we demonstrate gaps between pd( f ) and the above two new measures for f : pd( f ) = O (√ n ) upd( f ) = Ω ( n ) bitpdim( f ) = Ω ( n 1.5 / log n ). We adapt Nechiporuk’s techniques for our linear algebraic setting to prove the best-known bpsize lower bounds for bitpdim. Motivated by this linear algebraic setting of our main result, we derive exponential lower bounds for two restricted variants of pd( f ) and upd( f ) by observing that they are exactly equal to well-studied graph parameters—bipartite clique cover number and bipartite partition number, respectively.

One of the central problems in complexity theory is to prove super-logarithmic depth bounds for circuits computing a problem in P , i.e., to prove that P is not contained in NC1. As an approach for this question, Karchmer, Raz and Wigderson [5] proposed a conjecture called the KRW conjecture, which if true, would imply that P is not cotained in NC1. Since proving this conjecture is currently considered an extremely difficult problem, previous works by Edmonds, Impagliazzo, Rudich and Sgall [1], Håstad and Wigderson [3] and Gavinsky, Meir, Weinstein and Wigderson [2] considered weaker variants of the conjecture. In this work we significantly improve the parameters in these variants, achieving almost tight lower bounds. 2012 ACM Subject Classification Theory of computation → Communication complexity, Theory of computation → Circuit complexity, Theory of computation → Complexity classes

Baxter permutations are a class of permutations which are in bijection with a class of floorplans that arise in chip design called mosaic floorplans. We study a subclass of mosaic floorplans called Hierarchical Floorplans of Order k defined from mosaic floorplans by placing certain geometric restrictions. This naturally leads to studying a subclass of Baxter permutations. This subclass of Baxter permutations are characterized by pattern avoidance. We establish a bijection, between the subclass of floorplans we study and a subclass of Baxter permutations, based on the analogy between decomposition of a floorplan into smaller blocks and block decomposition of permutations. Apart from the characterization, we also answer combinatorial questions on these classes. We give an algebraic generating function (but without a closed form solution) for the number of permutations, an exponential lower bound on growth rate, and a linear time algorithm for deciding membership in each subclass. Based on the recurrence relation describing the class, we also give a polynomial time algorithm for enumeration. We finally prove that Baxter permutations are closed under inverse based on an argument inspired from the geometry of the corresponding mosaic floorplans. This proof also establishes that the subclass of Baxter permutations we study are also closed under inverse. Characterizing permutations instead of the corresponding floorplans can be helpful in reasoning about the solution space and in designing efficient algorithms for floorplanning.

We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation of a function f is the characteristic vector of the minimum sized set of negated variables needed in any DeMorgan circuit computing f. We prove trade-off results between the depth and the weight/structure of the orientation vectors in any circuit C computing the CLIQUE function on an n vertex graph. We prove that if C is of depth d and each gate computes a Boolean function with orientation of weight at most w (in terms of the inputs to C), then d ×w must be Ω(n). In particular, if the weights are \(o(\frac{n}{\log^k n})\), then C must be of depth ω(log k n). We prove a barrier for our general technique. However, using specific properties of the CLIQUE function (used in [4]) and the Karchmer-Wigderson framework [11], we go beyond the limitations and obtain lower bounds when the weight restrictions are less stringent.

A floorplan is a rectangular dissection which describes the relative placement of electronic modules on the chip. It is called a mosaic floorplan if there are no empty rooms or cross junctions in the rectangular dissection. We study a subclass of mosaic floorplans called hierarchical floorplans of order $k$ (abbreviated HFO-${k}$). A floorplan is HFO-$k$ if it can be obtained by starting with a single rectangle and recursively embedding mosaic floorplans of at most $k$ rooms inside the rooms of intermediate floorplans. When $k=2$ this is exactly the class of slicing floorplans as the only distinct floorplans with two rooms are a room with a vertical slice and a room with a horizontal slice respectdeively. And embedding such a room is equivalent to slicing the parent room vertically/horizontally. In this paper we characterize permutations corresponding to the Abe-labeling of HFO-$k$ floorplans and also give an algorithm for identification of such permutations in linear time for any particular $k$. We give a recurrence relation for exact number of HFO-5 floorplans with $n$ rooms which can be easily extended to any $k$ also. Based on this recurrence we provide a polynomial time algorithm to generate the number of HFO-$k$ floorplans with $n$ rooms. Considering its application in VLSI design we also give moves on HFO-$k$ family of permutations for combinatorial optimization using simulated annealing etc. We also explore some interesting properties of Baxter permutations which have a bijective correspondence with mosaic floorplans.

Given a rectangle R, a Rectangular Dissection (RD) is a subdivision of R into smaller rectangles by non-intersecting vertical or horizontal segments. A basic rectangle in a rectangular dissection is a rectangle which does not contain any other rectangles inside. In this paper we study the number of rectangular partitions which can be decomposed hierarchically. A rectangular partition is said to be a Hierarchical Rectangular Dissection (HRD) of order k if the rectangular dissection can be obtained by starting from a single rectangle by embedding rectangular dissections of at most k basic rectangles hierarchically. When k = 2 this is exactly the class of guillotine rectangular dissections. Rectangular dissections are studied in VLSI circuit design under the name of floorplans. Ackerman et al. proved that point-free rectangular dissections are in bijective correspondence with Baxter permutations. We characterize hierarchical rectangular dissections of order k, a sub-class of point-free rectangular dissections, based on the Baxter permutations corresponding to them. We provide a recurrence relation for the distinct number of hierarchical rectangular partitions of order k with n rooms by proving that they are in bijective correspondence with a family of rooted trees called generating trees of order k. Based on this recurrence relation we give a polynomial time algorithm for generating the number of such distinct rectangular dissections. This gives rise to family of integer sequences I = {Ik} where Ik,i correspond to the number different HRDk with i rooms. Based on the characterization of permutations corresponding to HRD’s of order k we prove that there are at least 3 n k which are in-decomposable, that is it is not HRDk for

Private P2P networks are peer-to-peer (P2P) networks that only allow some mutually trusted computers to share resources. In this paper, we consider a special type of Private P2P system - the Secure-group P2P (SGP) system. SGP systems form an overlay network consisting of private groups which allow only members belonging to the same group to see and share resources. We introduce Galaxia: a semi-decentralized system that creates and manages SGP networks. We discuss the rationale behind adopting a semi-decentralized approach, discuss various issues that need to be resolved when dealing with SGP networks and provide the details of the Galaxia architecture