arXiv subjects

Charlie Carlson

Publications and source records attributed to Charlie Carlson.

10 recordsLinked to original sources

arXiv2025

Hardness of Approximation for Shortest Path with Vector Costs

We obtain hardness of approximation results for the $\ell_p$-Shortest Path problem, a variant of the classic Shortest Path problem with vector costs. For every integer $p \in [2,\infty)$, we show a hardness of $\Omega(p(\log n / \log^2\log n)^{1-1/p})$ for both polynomial- and quasi-polynomial-time approximation algorithms. This nearly matches the approximation factor of $O(p(\log n / \log\log n)^{1-1/p})$ achieved by a quasi-polynomial-time algorithm of Makarychev, Ovsiankin, and Tani (ICALP 2025). No hardness of approximation results were previously known for any $p < \infty$. We also present results for the case where $p$ is a function of $n$. For $p = \infty$, we establish a hardness of $\tilde\Omega(\log^2 n)$, improving upon the previous $\tilde\Omega(\log n)$ hardness result. Our result nearly matches the $O(\log^2 n)$ approximation guarantee of the quasi-polynomial-time algorithm by Li, Xu, and Zhang (ICALP 2025). Finally, we present asymptotic bounds on higher-order Bell numbers, which might be of independent interest.

cs.DS↗

arXiv2024

Optimal Mixing for Randomly Sampling Edge Colorings on Trees Down to the Max Degree

We address the convergence rate of Markov chains for randomly generating an edge coloring of a given tree. Our focus is on the Glauber dynamics which updates the color at a randomly chosen edge in each step. For a tree $T$ with $n$ vertices and maximum degree $\Delta$, when the number of colors $q$ satisfies $q\geq\Delta+2$ then we prove that the Glauber dynamics has an optimal relaxation time of $O(n)$, where the relaxation time is the inverse of the spectral gap. This is optimal in the range of $q$ in terms of $\Delta$ as Dyer, Goldberg, and Jerrum (2006) showed that the relaxation time is $\Omega(n^3)$ when $q=\Delta+1$. For the case $q=\Delta+1$, we show that an alternative Markov chain which updates a pair of neighboring edges has relaxation time $O(n)$. Moreover, for the $\Delta$-regular complete tree we prove $O(n\log^2{n})$ mixing time bounds for the respective Markov chain. Our proofs establish approximate tensorization of variance via a novel inductive approach, where the base case is a tree of height $\ell=O(\Delta^2\log^2{\Delta})$, which we analyze using a canonical paths argument.

cs.DM↗

arXiv2024

Flip Dynamics for Sampling Colorings: Improving $(11/6-\epsilon)$ Using a Simple Metric

We present improved bounds for randomly sampling $k$-colorings of graphs with maximum degree $\Delta$; our results hold without any further assumptions on the graph. The Glauber dynamics is a simple single-site update Markov chain. Jerrum (1995) proved an optimal $O(n\log{n})$ mixing time bound for Glauber dynamics whenever $k>2\Delta$ where $\Delta$ is the maximum degree of the input graph. This bound was improved by Vigoda (1999) to $k > (11/6)\Delta$ using a "flip" dynamics which recolors (small) maximal 2-colored components in each step. Vigoda's result was the best known for general graphs for 20 years until Chen et al. (2019) established optimal mixing of the flip dynamics for $k > (11/6 - \epsilon ) \Delta$ where $\epsilon \approx 10^{-5}$. We present the first substantial improvement over these results. We prove an optimal mixing time bound of $O(n\log{n})$ for the flip dynamics when $k \geq 1.809 \Delta$. This yields, through recent spectral independence results, an optimal $O(n\log{n})$ mixing time for the Glauber dynamics for the same range of $k/\Delta$ when $\Delta=O(1)$. Our proof utilizes path coupling with a simple weighted Hamming distance for "unblocked" neighbors.

cs.DM↗

arXiv2023

Approximation Algorithms for Quantum Max-$d$-Cut

We initiate the algorithmic study of the Quantum Max-$d$-Cut problem, a quantum generalization of the well-known Max-$d$-Cut problem. The Quantum Max-$d$-Cut problem involves finding a quantum state that maximizes the expected energy associated with the projector onto the antisymmetric subspace of two, $d$-dimensional qudits over all local interactions. Equivalently, this problem is physically motivated by the $SU(d)$-Heisenberg model, a spin glass model that generalized the well-known Heisenberg model over qudits. We develop a polynomial-time randomized approximation algorithm that finds product-state solutions of mixed states with bounded purity that achieve non-trivial performance guarantees. Moreover, we prove the tightness of our analysis by presenting an algorithmic gap instance for Quantum Max-d-Cut problem with $d \geq 3$.

quant-ph↗

arXiv2023

Improved Distributed Algorithms for Random Colorings

We study distributed versions of Markov Chain Monte Carlo (MCMC) algorithms for generating random $k$-colorings of an input graph with maximum degree $\Delta$. In the sequential setting, the Glauber dynamics is the simple MCMC algorithm which updates the color at a randomly chosen vertex in each step. Fischer and Ghaffari (2018), and independently Feng, Hayes, and Yin (2018), presented a parallel and distributed version of the Glauber dynamics which converges in $O(\log{n})$ rounds for $k>(2+\varepsilon)\Delta$ for any $\varepsilon>0$. We present the distributed flip dynamics and prove $O(n\log{n})$ mixing for $k>(11/6-\delta)\Delta$ for a fixed $\delta>0$. Our new Markov chain is a generalization of the distributed Glauber dynamics previously analyzed, and is a parallel and distributed version of the more general flip dynamics considered in the sequential setting which recolors local maximal two-colored components in each step. While the distributed Glauber dynamics and the sequential flip dynamics are symmetric Markov chains, and hence their stationary distribution is uniformly distributed over colorings, our distributed flip dynamics is not symmetric and hence the stationary distribution is unclear.

cs.DC↗

arXiv2023

Approximation Algorithms for Norm Multiway Cut

We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph $G$ into $k$ parts so as to separate $k$ given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced $\ell_p$-norm Multiway, a generalization of the problem, in which the goal is to minimize the $\ell_p$ norm of the edge boundaries of $k$ parts. We provide an $O(\log^{1/2} n\log^{1/2+1/p} k)$ approximation algorithm for this problem, improving upon the approximation guarantee of $O(\log^{3/2} n \log^{1/2} k)$ due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an $O(\log^{1/2} n \log^{7/2} k)$ approximation algorithm with a weaker oracle and an $O(\log^{1/2} n \log^{5/2} k)$ approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no $n^{1/4-\varepsilon}$ approximation algorithm for every $\varepsilon > 0$ assuming the Hypergraph Dense-vs-Random Conjecture.

cs.DS↗

arXiv2023

Approximately counting independent sets in dense bipartite graphs via subspace enumeration

We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph -- in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to ``high-temperature'' problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments. The proof exploits the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e. bounded threshold rank), which via subspace enumeration lets us enumerate small cuts efficiently.

cs.DS↗

arXiv2023

A quantum advantage over classical for local max cut

We compare the performance of a quantum local algorithm to a similar classical counterpart on a well-established combinatorial optimization problem LocalMaxCut. We show that a popular quantum algorithm first discovered by Farhi, Goldstone, and Gutmannn [1] called the quantum optimization approximation algorithm (QAOA) has a computational advantage over comparable local classical techniques on degree-3 graphs. These results hint that even small-scale quantum computation, which is relevant to the current state-of the art quantum hardware, could have significant advantages over comparably simple classical computation.

quant-ph↗

arXiv2022

Algorithms for the ferromagnetic Potts model on expanders

We give algorithms for approximating the partition function of the ferromagnetic $q$-color Potts model on graphs of maximum degree $d$. Our primary contribution is a fully polynomial-time approximation scheme for $d$-regular graphs with an expansion condition at low temperatures (that is, bounded away from the order-disorder threshold). The expansion condition is much weaker than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models; we use extremal graph theory and applications of Karger's algorithm to count cuts that may be of independent interest. It is \#BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of \#BIS are rare. We also obtain efficient algorithms in the Gibbs uniqueness region for bounded-degree graphs. While our high temperature proof follows more standard polymer model analysis, our result holds in the largest known range of parameters $d$ and $q$.

cs.DS↗

arXiv2021

Computational thresholds for the fixed-magnetization Ising model

The ferromagnetic Ising model is a model of a magnetic material and a central topic in statistical physics. It also plays a starring role in the algorithmic study of approximate counting: approximating the partition function of the ferromagnetic Ising model with uniform external field is tractable at all temperatures and on all graphs, due to the randomized algorithm of Jerrum and Sinclair. Here we show that hidden inside the model are hard computational problems. For the class of bounded-degree graphs we find computational thresholds for the approximate counting and sampling problems for the ferromagnetic Ising model at fixed magnetization (that is, fixing the number of $+1$ and $-1$ spins). In particular, letting $\beta_c(\Delta)$ denote the critical inverse temperature of the zero-field Ising model on the infinite $\Delta$-regular tree, and $\eta_{\Delta,\beta,1}^+$ denote the mean magnetization of the zero-field $+$ measure on the infinite $\Delta$-regular tree at inverse temperature $\beta$, we prove, for the class of graphs of maximum degree $\Delta$: 1. For $\beta < \beta_c(\Delta)$ there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for all magnetizations $\eta$. 2. For $\beta > \beta_c(\Delta)$, there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for magnetizations $\eta$ such that $|\eta| >\eta_{\Delta,\beta,1}^+ $. 3. For $\beta > \beta_c(\Delta)$, there is no FPRAS for the fixed-magnetization Ising model for magnetizations $\eta$ such that $|\eta| <\eta_{\Delta,\beta,1}^+ $ unless NP=RP\@.

cs.DS↗