Algebraic and Combinatorial Methods in Computational Complexity

Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. The algebraic theme continues in some of the most exciting recent progress in computational complexity. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Another surprising connection is that the algebraic techniques invented to show lower bounds now prove useful to develop efficient algorithms. For example, Williams showed how to use the polynomial method to obtain faster all-pair-shortest-path algorithms. This emphases once again the central role of algebra in computer science. The seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and this seminar can play an important role in educating a diverse community about the latest new techniques, spurring further progress

Quantum state testing beyond the polarizing regime and quantum triangular discrimination

The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$) captures computational difficulties of the time-bounded quantum state testing problem with respect to the trace distance, known as the Quantum State Distinguishability Problem (QSDP) introduced by Watrous (FOCS 2002). However, QSDP is in $\mathsf{QSZK}$ merely within the constant polarizing regime, similar to its classical counterpart shown by Sahai and Vadhan (JACM 2003) due to the polarization lemma (error reduction for SDP). Recently, Berman, Degwekar, Rothblum, and Vasudevan (TCC 2019) extended the $\mathsf{SZK}$ containment for SDP beyond the polarizing regime via the time-bounded distribution testing problems with respect to the triangular discrimination and the Jensen-Shannon divergence. Our work introduces proper quantum analogs for these problems by defining quantum counterparts for triangular discrimination. We investigate whether the quantum analogs behave similarly to their classical counterparts and examine the limitations of existing approaches to polarization regarding quantum distances. These new $\mathsf{QSZK}$-complete problems improve $\mathsf{QSZK}$ containments for QSDP beyond the polarizing regime and establish a simple $\mathsf{QSZK}$-hardness for the quantum entropy difference problem (QEDP) defined by Ben-Aroya, Schwartz, and Ta-Shma (ToC 2010). Furthermore, we prove that QSDP with some exponentially small errors is in $\mathsf{PP}$, while the same problem without error is in $\mathsf{NQP}$.Comment: 31 pages. v3: added a simple QSZK-hardness proof for QEDP, updated a correct version of Theorem 5.1(2), and improved presentation. v2: minor change

Information Value of Two-Prover Games

We introduce a generalization of the standard framework for studying the difficulty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) - in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol. In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game). We show that if one can win the game with at least (1 - epsilon)-probability by communicating at most epsilon bits of information, then one can win n copies with probability at least 2^{-O(epsilon n)}. This gives an intuitive explanation why Raz\u27s counter-example to strong parallel repetition [Raz2008] (the odd cycle game) is a counter-example to strong parallel repetition - one can win the odd-cycle game on a cycle of length $m$ by communicating O(m^{-2})-bits where m is the number of vertices. Conversely, for projection games, we show that if one can win n copies with probability larger than (1-epsilon)^n, then one can win one copy with at least (1 - O(epsilon))-probability by communicating O(epsilon) bits of information. By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games. The main technical tool is the "Chi-Squared Lemma" which bounds the information cost of the protocol in terms of Chi-Squared distance, instead of usual divergence. This avoids the square loss from using Pinsker\u27s Inequality

Information Value of Two-Prover Games

We introduce a generalization of the standard framework for studying the difficulty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) - in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol. In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game). We show that if one can win the game with at least (1 - epsilon)-probability by communicating at most epsilon bits of information, then one can win n copies with probability at least 2^{-O(epsilon n)}. This gives an intuitive explanation why Raz\u27s counter-example to strong parallel repetition [Raz2008] (the odd cycle game) is a counter-example to strong parallel repetition - one can win the odd-cycle game on a cycle of length $m$ by communicating O(m^{-2})-bits where m is the number of vertices. Conversely, for projection games, we show that if one can win n copies with probability larger than (1-epsilon)^n, then one can win one copy with at least (1 - O(epsilon))-probability by communicating O(epsilon) bits of information. By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games. The main technical tool is the "Chi-Squared Lemma" which bounds the information cost of the protocol in terms of Chi-Squared distance, instead of usual divergence. This avoids the square loss from using Pinsker\u27s Inequality

Near-Quadratic Lower Bounds for Two-Pass Graph Streaming Algorithms

We prove that any two-pass graph streaming algorithm for the $s$-$t$ reachability problem in $n$-vertex directed graphs requires near-quadratic space of $n^{2-o(1)}$ bits. As a corollary, we also obtain near-quadratic space lower bounds for several other fundamental problems including maximum bipartite matching and (approximate) shortest path in undirected graphs. Our results collectively imply that a wide range of graph problems admit essentially no non-trivial streaming algorithm even when two passes over the input is allowed. Prior to our work, such impossibility results were only known for single-pass streaming algorithms, and the best two-pass lower bounds only ruled out $o(n^{7/6})$ space algorithms, leaving open a large gap between (trivial) upper bounds and lower bounds

A new distance measure and corresponding TOPSIS method for interval-valued intuitionistic fuzzy sets in multi-attribute decision-making

Strengthening the evaluation of teaching satisfaction plays a crucial role in guiding teachers to improve their teaching quality and competence, as well as in aiding educational institutions in the formulation of effective teaching reforms and plans. The evaluation process for teaching satisfaction is usually regarded as a typical multi-attribute decision-making (MADM) process, which inherently possesses uncertainty and fuzziness due to the subjective nature of human cognition. In order to improve the subtle discrimination of evaluation information data and enhance the accuracy of the evaluation results, we have developed an integrated MADM method by combining a new distance measure and an improved TOPSIS method for interval-valued intuitionistic fuzzy sets (IvIFSs). First, a novel distance measure for IvIFSs based on triangular divergence is proposed to capture the differences between two IvIFSs, and some properties of this distance measure are investigated. Then, the superiority of this new distance measure is compared with some existing distance measures. Afterward, an improved TOPSIS method is also established based on the proposed triangular distance under the interval-valued intuitionistic fuzzy setting. Besides, to illustrate the practicality of the new method, a numerical example is presentedto evaluate mathematics teaching satisfaction. Moreover, a comparative analysis that includes existing TOPSIS methods, is presented to demonstrate the superiority of the given method. The comparison outcomes show that the proposed technique can effectively discern uncertainties or subtle differences in IvIFSs, resulting in more accurate and comprehensive evaluation results for teaching satisfaction. Overall, the findings of this study emphasize the importance of incorporating the new distance measure in MADM. The proposed approach serves as a valuable tool for decision-makers to compare and evaluate alternatives effectively