Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2304.08615

Revisiting Block-Diagonal SDP Relaxations for the Clique Number of the Paley Graphs

Vladimir A. Kobzar·Krishnan Mody
View original source ↗Read source document ↗Export metadata ↓

Abstract

This work addresses the block-diagonal semidefinite program (SDP) relaxations for the clique number of the Paley graphs. The size of the maximal clique (clique number) of a graph is a classic NP-complete problem; a Paley graph is a deterministic graph where two vertices are connected if their difference is a quadratic residue (square) in a finite field with the number of elements given by certain primes and prime powers. Improving the upper bound for the Paley graph clique number for prime powers that are non-squares is an open problem in combinatorics. Moreover, since quadratic residues exhibit pseudorandom properties, Paley graphs are related to the construction of deterministic restricted isometries, an open problem in compressed sensing. Recent work provides numerical evidence that the current upper bounds can be improved by the sum-of-squares (SOS) relaxations. In particular, the bounds given by the SOS relaxations of degree 4 (SOS-4) have been empirically observed to be growing at an order smaller than square root of the prime. However, computations of SOS-4 appear to be intractable with respect to large graphs. Gvozdenovic et al. introduced a more computationally efficient block-diagonal hierarchy of SDPs and computed the values of these SDPs of degrees 2 (L2) for the Paley graph clique numbers associated with primes p less or equal to 809, which bound from above the corresponding SOS-4 relaxations. We compute the values of the L2 relaxations for p's between 821 and 997. Our results provide some numerical evidence that these relaxations, and therefore also the SOS-4 relaxations, may be scaling at an order smaller than the square root of p. However, due to the size of the SDPs, we have not been able to compute L2 relaxations for p's greater than 997. Therefore, our scaling estimate is not conclusive and presents an interesting open problem for further study.

Explore related subjects

cs.DScs.CCmath.COmath.NTmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vladimir A. Kobzar, Krishnan Mody. 2023-09-16. Revisiting Block-Diagonal SDP Relaxations for the Clique Number of the Paley Graphs. https://arxiv.org/abs/2304.08615

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Marked Edge Walk: A Novel MCMC Algorithm for Sampling of Graph Partitions

Novel Markov Chain Monte Carlo (MCMC) methods have enabled the generation of large ensembles of redistricting plans modeled as a graph partitioning problem. However, existing algorithms such as Reversible Recombination (RevReCom) and Metropolized Forest Recombination (MFR) have strong preferences for distributions related to the spanning tree measure. In this paper we introduce the Marked Edge Walk (MEW), a novel Markov chain proposal for sampling from the space of graph partitions. The walk operates on the space of spanning trees with marked edges, allowing for calculable transition probabilities for use in the Metropolis-Hastings algorithm. Empirical results on real-world dual graphs show convergence under a broad class of target distributions less constrained by spanning tree counts, including policy-based distributions, such as competitiveness on New Hampshire that are independent of spanning trees, and compactness and partisan symmetry distributions on New Hampshire and Texas that, while related to spanning trees, can now be properly targeted with a smaller degree of spanning tree bias, which represents an advancement in flexible ensemble generation.

cs.DS↗
arXiv2026

The Price of Privacy For Approximating Max-CSP

We study approximation algorithms for Maximum Constraint Satisfaction Problems (Max-CSPs) under differential privacy (DP) where the constraints are considered sensitive data. Information-theoretically, we aim to classify the best approximation ratios possible for a given privacy budget $\varepsilon$. In the high-privacy regime ($\varepsilon \ll 1$), we show that any $\varepsilon$-DP algorithm cannot beat a random assignment by more than $O(\varepsilon)$ in the approximation ratio. We devise a polynomial-time algorithm which matches this barrier under the assumptions that the instances are bounded-degree and triangle-free. Finally, we show that one or both of these assumptions can be removed for specific CSPs--such as Max-Cut or Max $k$-XOR--albeit at the cost of computational efficiency.

cs.DS↗
arXiv2026

Distributed Santa Claus via Global Rounding

In this paper, we initiate the study of a new class of problems in the CONGEST model: Mixed packing and covering linear programs (LP). Previously, the design of optimization algorithms in the distributed setting was heavily focused on solving linear programs with either packing or covering constraints. We are the first to explore the class of linear programs with both packing and covering constraints by providing a general-purpose CONGEST solver for such LPs and by studying the sequentially well-studied Santa Claus problem as a central representative. This NP-hard problem can be modeled as a bipartite graph of children and gifts where an edge indicates that a child desires a gift. The goal is to assign the gifts to the children such that the least happy child is as happy as possible. Even though this is a well-studied problem in the sequential setting, we provide the first results in the distributed setting. In particular, we show that the complexity of computing an $\mathcal{O}(\log n/\log \log n)$-approximation is $\widehat Θ(\sqrt n+D)$ rounds, where our $\widetildeΩ(\sqrt n+D)$-round lower bound even holds for any approximation.

cs.DS↗