Graph Polynomials and Group Coloring of Graphs

Let $\Gamma$ be an Abelian group and let $G$ be a simple graph. We say that $G$ is $\Gamma$-colorable if for some fixed orientation of $G$ and every edge labeling $\ell:E(G)\rightarrow \Gamma$, there exists a vertex coloring $c$ by the elements of $\Gamma$ such that $c(y)-c(x)\neq \ell(e)$, for every edge $e=xy$ (oriented from $x$ to $y$). Langhede and Thomassen proved recently that every planar graph on $n$ vertices has at least $2^{n/9}$ different $\mathbb{Z}_5$-colorings. By using a different approach based on graph polynomials, we extend this result to $K_5$-minor-free graphs in the more general setting of field coloring. More specifically, we prove that every such graph on $n$ vertices is $\mathbb{F}$-$5$-choosable, whenever $\mathbb{F}$ is an arbitrary field with at least $5$ elements. Moreover, the number of colorings (for every list assignment) is at least $5^{n/4}$.Comment: 14 page

Covering grids with multiplicity

Given a finite grid in $\mathbb{R}^2$, how many lines are needed to cover all but one point at least $k$ times? Problems of this nature have been studied for decades, with a general lower bound having been established by Ball and Serra. We solve this problem for various types of grids, in particular showing the tightness of the Ball--Serra bound when one side is much larger than the other. In other cases, we prove new lower bounds that improve upon Ball--Serra and provide an asymptotic answer for almost all grids. For the standard grid $\{0,\ldots,n-1\} \times \{0,\ldots,n-1\}$, we prove nontrivial upper and lower bounds on the number of lines needed. To prove our results, we combine linear programming duality with some combinatorial arguments

Subspace coverings with multiplicities

We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k-1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$. The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and F\"uredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k \ge 2^{n-d-1}$ we have $f(n,k,d)=2^d k - \left \lfloor \frac{k}{2^{n-d}} \right \rfloor$, while for $n > 2^{2^d k-k-d+1}$ we have $f(n,k,d)= n + 2^dk-d-2$, and also study the transition between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.Comment: 15 page

근사 연산에 대한 계산 검증 연구

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 천정희.Verifiable Computing (VC) is a complexity-theoretic method to secure the integrity of computations. The need is increasing as more computations are outsourced to untrusted parties, e.g., cloud platforms. Existing techniques, however, have mainly focused on exact computations, but not approximate arithmetic, e.g., floating-point or fixed-point arithmetic. This makes it hard to apply them to certain types of computations (e.g., machine learning, data analysis, and scientific computation) that inherently require approximate arithmetic. In this thesis, we present an efficient interactive proof system for arithmetic circuits with rounding gates that can represent approximate arithmetic. The main idea is to represent the rounding gate into a small sub-circuit, and reuse the machinery of the Goldwasser, Kalai, and Rothblum's protocol (also known as the GKR protocol) and its recent refinements. Specifically, we shift the algebraic structure from a field to a ring to better deal with the notion of ``digits'', and generalize the original GKR protocol over a ring. Then, we represent the rounding operation by a low-degree polynomial over a ring, and develop a novel, optimal circuit construction of an arbitrary polynomial to transform the rounding polynomial to an optimal circuit representation. Moreover, we further optimize the proof generation cost for rounding by employing a Galois ring. We provide experimental results that show the efficiency of our system for approximate arithmetic. For example, our implementation performed two orders of magnitude better than the existing system for a nested 128 x 128 matrix multiplication of depth 12 on the 16-bit fixed-point arithmetic.계산검증 기술은 계산의 무결성을 확보하기 위한 계산 복잡도 이론적 방법이다. 최근 많은 계산이 클라우드 플랫폼과 같은 제3자에게 외주됨에 따라 그 필요성이 증가하고 있다. 그러나 기존의 계산검증 기술은 비근사 연산만을 고려했을 뿐, 근사 연산 (부동 소수점 또는 고정 소수점 연산)은 고려하지 않았다. 따라서 본질적으로 근사 연산이 필요한 특정 유형의 계산 (기계 학습, 데이터 분석 및 과학 계산 등)에 적용하기 어렵다는 문제가 있었다. 이 논문은 반올림 게이트를 수반하는 산술 회로를 위한 효율적인 대화형 증명 시스템을 제시한다. 이러한 산술 회로는 근사 연산을 효율적으로 표현할 수 있으므로, 근사 연산에 대한 효율적인 계산 검증이 가능하다. 주요 아이디어는 반올림 게이트를 작은 회로로 변환한 후, 여기에 Goldwasser, Kalai, 및 Rothblum의 프로토콜 (GKR 프로토콜)과 최근의 개선을 적용하는 것이다. 구체적으로, 대수적 객체를 유한체가 아닌 ``숫자''를 보다 잘 처리할 수 있는 환으로 치환한 후, 환 위에서 적용 가능하도록 기존의 GKR 프로토콜을 일반화하였다. 이후, 반올림 연산을 환에서 차수가 낮은 다항식으로 표현하고, 다항식 연산을 최적의 회로 표현으로 나타내는 새롭고 최적화된 회로 구성을 개발하였다. 또한, 갈루아 환을 사용하여 반올림을 위한 증명 생성 비용을 더욱 최적화하였다. 마지막으로, 실험을 통해 우리의 근사 연산 검증 시스템의 효율성을 확인하였다. 예를 들어, 우리의 시스템은 구현 시, 16 비트 고정 소수점 연산을 통한 깊이 12의 반복된 128 x 128 행렬 곱셈의 검증에 있어 기존 시스템보다 약 100배 더 나은 성능을 보인다.1 Introduction 1 1.1 Verifiable Computing 2 1.2 Verifiable Approximate Arithmetic 3 1.2.1 Problem: Verification of Rounding Arithmetic 3 1.2.2 Motivation: Verifiable Machine Learning (AI) 4 1.3 List of Papers 5 2 Preliminaries 6 2.1 Interactive Proof and Argument 6 2.2 Sum-Check Protocol 7 2.3 The GKR Protocol 10 2.4 Notation and Cost Model 14 3 Related Work 15 3.1 Interactive Proofs 15 3.2 (Non-)Interactive Arguments 17 4 Interactive Proof for Rounding Arithmetic 20 4.1 Overview of Our Approach and Result 20 4.2 Interactive Proof over a Ring 26 4.2.1 Sum-Check Protocol over a Ring 27 4.2.2 The GKR Protocol over a Ring 29 4.3 Verifiable Rounding Operation 31 4.3.1 Lowest-Digit-Removal Polynomial over Z_{p^e} 32 4.3.2 Verification of Division-by-p Layer 33 4.4 Delegation of Polynomial Evaluation in Optimal Cost 34 4.4.1 Overview of Our Circuit Construction 35 4.4.2 Our Circuit for Polynomial Evaluation 37 4.4.3 Cost Analysis 40 4.5 Cost Optimization 45 4.5.1 Galois Ring over Z_{p^e} and a Sampling Set 45 4.5.2 Optimization of Prover's Cost for Rounding Layers 47 5 Experimental Results 50 5.1 Experimental Setup 50 5.2 Verifiable Rounding Operation 51 5.2.1 Effectiveness of Optimization via Galois Ring 51 5.2.2 Efficiency of Verifiable Rounding Operation 53 5.3 Comparison to Thaler's Refinement of GKR Protocol 54 5.4 Discussion 57 6 Conclusions 60 6.1 Towards Verifiable AI 61 6.2 Verifiable Cryptographic Computation 62 Abstract (in Korean) 74Docto