arXiv subjects

Siu On Chan

Publications and source records attributed to Siu On Chan.

8 recordsLinked to original sources

arXiv2026

Strongly Refuting Random CSP without Literals

Under what condition is a random constraint satisfaction problem hard to refute by the sum-of-squares (SoS) algorithm? A sufficient condition is t-wise uniformity, that is, each constraint has a t-wise uniform distribution of satisfying assignments, as shown by the lower bounds of Kothari, Mori, O'Donnell, and Witmer (STOC 2017). This condition is also necessary for random CSPs given by a predicate and uniformly random literals, due to the constant-degree SoS refutation of Allen, O'Donnell, and Witmer (FOCS 2015). For higher degree, Raghavendra, Rao, and Schramm (STOC 2017) gave a refutation for Boolean random CSPs with uniformly random literals, matching the lower bounds optimally in terms of the three-way tradeoff between constraint density, SoS degree, and strength of refutation. Two long-standing open problems are to find a more general sufficient condition for SoS lower bounds, and to refute similar random CSPs not involving literals. We show that for a general random k-CSP, the necessary and sufficient hardness condition is not t-wise uniformity, but t-wise independence. We generalize the optimal three-way tradeoff to any random k-CSP, without assuming a Boolean domain or uniformly random literals. Our analysis involves new Kikuchi matrices for odd order and for asymmetric tensors. It also uses the global correlation rounding technique of Barak, Raghavendra, and Steurer (FOCS 2011). To avoid the running-time penalty of this technique, we also give a spectral refutation algorithm.

cs.CC↗

arXiv2022

Rethinking Graph Neural Networks for the Graph Coloring Problem

Graph coloring, a classical and critical NP-hard problem, is the problem of assigning connected nodes as different colors as possible. However, we observe that state-of-the-art GNNs are less successful in the graph coloring problem. We analyze the reasons from two perspectives. First, most GNNs fail to generalize the task under homophily to heterophily, i.e., graphs where connected nodes are assigned different colors. Second, GNNs are bounded by the network depth, making them possible to be a local method, which has been demonstrated to be non-optimal in Maximum Independent Set (MIS) problem. In this paper, we focus on the aggregation-combine GNNs (AC-GNNs), a popular class of GNNs. We first define the power of AC-GNNs in the coloring problem as the capability to assign nodes different colors. The definition is different with previous one that is based on the assumption of homophily. We identify node pairs that AC-GNNs fail to discriminate. Furthermore, we show that any AC-GNN is a local coloring method, and any local coloring method is non-optimal by exploring the limits of local methods over sparse random graphs, thereby demonstrating the non-optimality of AC-GNNs due to its local property. We then prove the positive correlation between model depth and its coloring power. Moreover, we discuss the color equivariance of graphs to tackle some practical constraints such as the pre-fixing constraints. Following the discussions above, we summarize a series of rules a series of rules that make a GNN color equivariant and powerful in the coloring problem. Then, we propose a simple AC-GNN variation satisfying these rules. We empirically validate our theoretical findings and demonstrate that our simple model substantially outperforms state-of-the-art heuristic algorithms in both quality and runtime.

cs.LG↗

arXiv2020

The Gambler's Problem and Beyond

We analyze the Gambler's problem, a simple reinforcement learning problem where the gambler has the chance to double or lose the bets until the target is reached. This is an early example introduced in the reinforcement learning textbook by Sutton and Barto (2018), where they mention an interesting pattern of the optimal value function with high-frequency components and repeating non-smooth points. It is however without further investigation. We provide the exact formula for the optimal value function for both the discrete and the continuous cases. Though simple as it might seem, the value function is pathological: fractal, self-similar, derivative taking either zero or infinity, and not written as elementary functions. It is in fact one of the generalized Cantor functions, where it holds a complexity that has been uncharted thus far. Our analyses could provide insights into improving value function approximation, gradient-based algorithms, and Q-learning, in real applications and implementations.

stat.ML↗

arXiv2016

Approximate Constraint Satisfaction Requires Large LP Relaxations

We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are exactly as powerful as programs arising from a constant number of rounds of the Sherali-Adams hierarchy. In particular, any polynomial-sized linear program for Max Cut has an integrality gap of 1/2 and any such linear program for Max 3-Sat has an integrality gap of 7/8.

cs.CC↗

arXiv2015

On the Worst-Case Approximability of Sparse PCA

It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1) a simple and efficient algorithm that achieves an $n^{-1/3}$-approximation; 2) NP-hardness of approximation to within $(1-\varepsilon)$, for some small constant $\varepsilon > 0$; 3) SSE-hardness of approximation to within any constant factor; and 4) an $\exp\exp\left(Ω\left(\sqrt{\log \log n}\right)\right)$ ("quasi-quasi-polynomial") gap for the standard semidefinite program.

stat.ML↗

arXiv2015

Random Walks and Evolving Sets: Faster Convergences and Limitations

Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more combinatorial graph structures, and show some implications in approximating small-set expansion. On the other hand, we provide examples showing the limitations of using random walks and evolving sets in disproving the small-set expansion hypothesis. - We define a combinatorial analog of the spectral gap, and use it to prove the convergence of non-lazy random walks. A corollary is a tight lower bound on the small-set expansion of graph powers for any graph. - We prove that random walks converge faster when the robust vertex expansion of the graph is larger. This provides an improved analysis of the local graph partitioning algorithm using the evolving set process. - We give an example showing that the evolving set process fails to disprove the small-set expansion hypothesis. This refutes a conjecture of Oveis Gharan and shows the limitations of local graph partitioning algorithms in approximating small-set expansion.

cs.DS↗

arXiv2015

Sum of Squares Lower Bounds from Pairwise Independence

We prove that for every $ε>0$ and predicate $P:\{0,1\}^k\rightarrow \{0,1\}$ that supports a pairwise independent distribution, there exists an instance $\mathcal{I}$ of the $\mathsf{Max}P$ constraint satisfaction problem on $n$ variables such that no assignment can satisfy more than a $\tfrac{|P^{-1}(1)|}{2^k}+ε$ fraction of $\mathcal{I}$'s constraints but the degree $Ω(n)$ Sum of Squares semidefinite programming hierarchy cannot certify that $\mathcal{I}$ is unsatisfiable. Similar results were previously only known for weaker hierarchies.

cs.CC↗

arXiv2012

On extracting common random bits from correlated sources on large alphabets

Suppose Alice and Bob receive strings $X=(X_1,...,X_n)$ and $Y=(Y_1,...,Y_n)$ each uniformly random in $[s]^n$ but so that $X$ and $Y$ are correlated . For each symbol $i$, we have that $Y_i = X_i$ with probability $1-\eps$ and otherwise $Y_i$ is chosen independently and uniformly from $[s]$. Alice and Bob wish to use their respective strings to extract a uniformly chosen common sequence from $[s]^k$ but without communicating. How well can they do? The trivial strategy of outputting the first $k$ symbols yields an agreement probability of $(1 - \eps + \eps/s)^k$. In a recent work by Bogdanov and Mossel it was shown that in the binary case where $s=2$ and $k = k(\eps)$ is large enough then it is possible to extract $k$ bits with a better agreement probability rate. In particular, it is possible to achieve agreement probability $(k\eps)^{-1/2} \cdot 2^{-k\eps/(2(1 - \eps/2))}$ using a random construction based on Hamming balls, and this is optimal up to lower order terms. In the current paper we consider the same problem over larger alphabet sizes $s$ and we show that the agreement probability rate changes dramatically as the alphabet grows. In particular we show no strategy can achieve agreement probability better than $(1-\eps)^k (1+δ(s))^k$ where $δ(s) \to 0$ as $s \to \infty$. We also show that Hamming ball based constructions have {\em much lower} agreement probability rate than the trivial algorithm as $s \to \infty$. Our proofs and results are intimately related to subtle properties of hypercontractive inequalities.

cs.IT↗