arXiv subjects

Charles Carlson

Publications and source records attributed to Charles Carlson.

5 recordsLinked to original sources

arXiv2024

Efficient algorithms for the Potts model on small-set expanders

An emerging trend in approximate counting is to show that certain `low-temperature' problems are easy on typical instances, despite worst-case hardness results. For the class of regular graphs one usually shows that expansion can be exploited algorithmically, and since random regular graphs are good expanders with high probability the problem is typically tractable. Inspired by approaches used in subexponential-time algorithms for Unique Games, we develop an approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition. In such graphs it may not suffice to explore the state space of the model close to ground states, and a novel feature of our method is to efficiently find a larger set of `pseudo-ground states' such that it is enough to explore the model around each pseudo-ground state.

cs.DS↗

arXiv2020

Lower bounds for Max-Cut in $H$-free graphs via semidefinite programming

For a graph $G$, let $f(G)$ denote the size of the maximum cut in $G$. The problem of estimating $f(G)$ as a function of the number of vertices and edges of $G$ has a long history and was extensively studied in the last fifty years. In this paper we propose an approach, based on semidefinite programming (SDP), to prove lower bounds on $f(G)$. We use this approach to find large cuts in graphs with few triangles and in $K_r$-free graphs.

cs.DS↗

arXiv2018

Improving the smoothed complexity of FLIP for max cut problems

Finding locally optimal solutions for max-cut and max-$k$-cut are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei showed that the smoothed complexity of FLIP for max-cut in complete graphs is $O(ϕ^5n^{15.1})$, where $ϕ$ is an upper bound on the random edge-weight density and $n$ is the number of vertices in the input graph. While Angel et al.'s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress towards improving the run-time bound. We prove that the smoothed complexity of FLIP in complete graphs is $O(ϕn^{7.83})$. Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for max-$3$-cut in complete graphs is polynomial and for max-$k$-cut in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest towards addressing the smoothed complexity of FLIP for max-$k$-cut in complete graphs for larger constants $k$.

cs.DS↗

arXiv2017

Optimal Lower Bounds for Sketching Graph Cuts

We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+ε$ error must use $Ω(n\log n/ε^2)$ bits of space in the worst case, improving the $Ω(n/ε^2)$ bound of Andoni et al. and matching the best known upper bound achieved by spectral sparsifiers. Our proof is based on a rigidity phenomenon for cut (and spectral) approximation which may be of independent interest: any two $d-$regular graphs which approximate each other's cuts significantly better than a random graph approximates the complete graph must overlap in a constant fraction of their edges.

cs.DS↗

arXiv2017

Invertibility and Largest Eigenvalue of Symmetric Matrix Signings

The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of identifying symmetric signings of matrices with natural spectral properties. Our results are twofold: 1. We show NP-completeness for the following three problems: verifying whether a given matrix has a symmetric signing that is positive semi-definite/singular/has bounded eigenvalues. However, we also illustrate that the complexity could substantially differ for input matrices that are adjacency matrices of graphs. 2. We exhibit a stark contrast between invertibility and the above-mentioned spectral properties: we show a combinatorial characterization of matrices with invertible symmetric signings and design an efficient algorithm using this characterization to verify whether a given matrix has an invertible symmetric signing. Next, we give an efficient algorithm to solve the search problem of finding an invertible symmetric signing for matrices whose support graph is bipartite. We also provide a lower bound on the number of invertible symmetric signed adjacency matrices. Finally, we give an efficient algorithm to find a minimum increase in support of a given symmetric matrix so that it has an invertible symmetric signing. We use combinatorial and spectral techniques in addition to classic results from matching theory. Our combinatorial characterization of matrices with invertible symmetric signings might be of independent interest.

cs.DM↗