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 \(\Omega\left( p(\log n / \log^2 \log n)^{1 - 1/p} \right)\) for both polynomial- and quasi-polynomial-time approximation algorithms. This nearly matches the approximation factor of \(O\left( p(\log n / \log \log n)^{1 - 1/p} \right)\) achieved by a quasi-polynomial-time algorithm of Makarychev, Ovsiankin, and Tani (ICALP 2025). No hardness of approximation results were previously known for any \(p \lt \infty\). We also present results for the case where \(p\) is a function of \(n\).

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 Δ, when the number of colors q satisfies q ≥ Δ + 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 Δ as Dyer, Goldberg, and Jerrum (2006) showed that the relaxation time is Ω(n3) when q = Δ + 1. For the case q = Δ + 1, we show that an alternative Markov chain which updates a pair of neighboring edges has relaxation time O (n ). Moreover, for the Δ-regular complete tree we prove O (n log2 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 ℓ = O (Δ2 log2 Δ), which we analyze using a canonical paths argument.

We present improved bounds for randomly sampling k-colorings of graphs with maximum degree Δ; 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 > 2Δ where Δ is the maximum degree of the input graph. This bound was improved by Vigoda (1999) to k > (11/6)Δ 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 — ε )Δ where ε ≈ 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 > 1.809Δ. Our proof utilizes path coupling with a simple weighted Hamming distance for “unblocked” neighbors.

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. Our contributions include a novel extension of the method of graph containers that differs considerably from other recent low-temperature algorithms. The additional key insights come from spectral graph theory and have previously been successful in approximation algorithms. As a result, we can overcome some limitations that seem inherent to the aforementioned class of algorithms. In particular, we exploit 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.

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.

Markov Chain Monte Carlo (MCMC) algorithms are a widely-used algorithmic tool for sampling from high-dimensional distributions, a notable example is the equilibirum distribution of graphical models. The Glauber dynamics, also known as the Gibbs sampler, is the simplest example of an MCMC algorithm; the transitions of the chain update the configuration at a randomly chosen coordinate at each step. Several works have studied distributed versions of the Glauber dynamics and we extend these efforts to a more general family of Markov chains. An important combinatorial problem in the study of MCMC algorithms is random colorings. Given a graph G of maximum degree Δ and an integer k ≥ Δ+1, the goal is to generate a random proper vertex k-coloring of G. Jerrum (1995) proved that the Glauber dynamics has O(nlog{n}) mixing time when k > 2Δ. 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+ε)Δ for any ε > 0. We improve this result to k > (11/6-δ)Δ for a fixed δ > 0. This matches the state of the art for randomly sampling colorings of general graphs in the sequential setting. Whereas previous works focused on distributed variants of the Glauber dynamics, our work presents a parallel and distributed version of the more general flip dynamics presented by Vigoda (2000) (and refined by Chen, Delcourt, Moitra, Perarnau, and Postle (2019)), which recolors local maximal two-colored components in each step.

We give algorithms for approximating the partition function of the ferromagnetic Potts model on d-regular expanding graphs. We require much weaker expansion 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, using extremal graph theory and applications of Karger’s algorithm to counting 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 believe that these methods can shed more light on other important problems such as sub-exponential algorithms for approximate counting problems.

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.

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, 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.

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 (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for max-cut in complete graphs is ( O Φ 5 n 15.1 ), where Φ is an upper bound on the random edge-weight density and Φ is the number of vertices in the input graph. While Angel, Bubeck, Peres, and Wei’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 toward improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O (Φ n 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 toward showing smoothed polynomial complexity of FLIP for MAX - k - CUT in complete graphs for larger constants k .

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 finding symmetric signings of matrices with natural spectral properties. Our results are the following: 1) We characterize matrices that have an invertible signing: a symmetric matrix has an invertible symmetric signing if and only if the support graph of the matrix contains a perfect 2-matching. Further, we present an efficient algorithm to search for an invertible symmetric signing. 2) We use the above-mentioned characterization to give an algorithm to find a minimum increase in the support of a given symmetric matrix so that it has an invertible symmetric signing. 3) We show NP-completeness of the following problems: verifying whether a given matrix has a symmetric signing that is singular or has bounded eigenvalues. However, we also illustrate that the complexity could differ substantially for input matrices that are adjacency matrices of graphs. We use combinatorial techniques in addition to classic results from matching theory.

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 [ER17] showed that the smoothed complexity of FLIP for MAX-CUT in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres and Wei [ABPW17] showed that the smoothed complexity of FLIP for MAX-CUT in complete graphs is O(ϕ5n15.1), 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(ϕn7.83). 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 showing smoothed polynomial complexity of FLIP for MAX-k-CUT in complete graphs for larger constants k.

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 + ∊ error must use Ω(n log n/∊2) bits of space in the worst case, improving the Ω(n/∊2) bound of [ACK+16] and matching the best known upper bound achieved by spectral sparsifiers [BSS12]. 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.