One natural question in the area of sublinear time algorithms asks whether we can distinguish between graphs with max-cut value at least 1-ε from graphs with max-cut value at most 1/2+ε in the adjacency list model where we can make degree queries and neighbor queries. Chiplunkar, Kapralov, Khanna, Mousavifar, and Peres (FOCS' 18) showed that in graphs of bounded degree, one cannot hope for a factor 1/2+ε approximation to the max-cut value in time n^{1/2+o(ε)}. Recently, Peng and Yoshida (SODA '23) obtained o(n) time algorithms which can distinguish expanders with max-cut value at least 1-ε from expanders with small max-cut value (their running time is n^{1/2+O(ε)}). In this paper, going beyond the results of Peng-Yoshida, we develop sublinear time algorithms for this problem on clusterable graphs (which is a graph class with a good community structure). Our algorithms run in ≈ n^{0.5001+ O(ε)} time. A natural extension of Peng-Yoshida approach does not seem to work for clusterable graphs. Indeed, their random walk based technique tracks the 𝓁₂ length of random walk vectors and they exploit the difference in the length of these vectors to tell apart expanders with large cut value from expanders with small cut-value. Such approaches fail to be reliable when graph has loosely connected clusters. Taking inspiration from [Ashish Chiplunkar et al., 2018], we exploit the more refined geometry of spectra of clusterable graphs which leads to our sublinear time implementation. We prove a novel spectral lemma which shows that in a spectral expander 2 - λ_{n-1} ≥ Ω(λ₂). This lemma is leveraged to show that there is a suitable difference between spectra of clusterable graphs with large cut value and spectra of clusterable graphs with small cut value. We use this gap to obtain our sublinear time implementation. To do this, we obtain a nuanced understanding of the eigenvector structure of clusterable graphs and in particular, we show that the eigenvectors of the normalized Laplacian of a clusterable graph, corresponding to eigenvalues which are close to 2 have a small infinity norm.

In a graph bisection problem, we are given a graph $G$ with two equally-sized unlabeled communities, and the goal is to recover the vertices in these communities. A popular heuristic, known as spectral clustering, is to output an estimated community assignment based on the eigenvector corresponding to the second-smallest eigenvalue of the Laplacian of $G$. Spectral algorithms can be shown to provably recover the cluster structure for graphs generated from probabilistic models, such as the Stochastic Block Model (SBM). However, spectral clustering is known to be non-robust to model mis-specification. Techniques based on semidefinite programming have been shown to be more robust, but they incur significant computational overheads. In this work, we study the robustness of spectral algorithms against semirandom adversaries. Informally, a semirandom adversary is allowed to ``helpfully'' change the specification of the model in a way that is consistent with the ground-truth solution. Our semirandom adversaries in particular are allowed to add edges inside clusters or increase the probability that an edge appears inside a cluster. Semirandom adversaries are a useful tool to determine the extent to which an algorithm has overfit to statistical assumptions on the input. On the positive side, we identify a wide range of semirandom adversaries under which spectral bisection using the _unnormalized_ Laplacian is strongly consistent, i.e., it exactly recovers the planted partitioning. On the negative side, we show that in many of these settings, _normalized_ spectral bisection outputs a partitioning that makes a classification mistake on a constant fraction of the vertices. Finally, we demonstrate numerical experiments that complement our theoretical findings.

Abstract Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) repaving road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c), (d) are restrictions to different graph classes. We show that (a) does not admit polynomial-time algorithms (assuming $${{\,\mathrm{\texttt {P}}\,}}\ne {{\,\mathrm{\texttt {NP}}\,}}$$ P ≠ NP ), even for relaxed variants of the problem (assuming $${{\,\mathrm{\texttt {P}}\,}}\ne {{\,\mathrm{\texttt {PSPACE}}\,}}$$ P ≠ PSPACE ). For (b), (c), (d), we present polynomial-time algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer $$k\ge 2$$ k ≥ 2 ) contiguous vertices on a shortest path can be changed at a time.

Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time, so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) revamping road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c) and (d) are restrictions to different graph classes. We show that (a) is intractable, even for relaxed variants of the problem. For (b), (c) and (d), we present efficient algorithms to solve the respective problems. We also generalise the problem to when at most k (for some k >= 2) contiguous vertices on a shortest path can be changed at a time.

We study the problem of approximating a center under the Ulam metric. The Ulam metric, defined over a set of permutations over [n], is the minimum number of move operations (deletion plus insertion) to transform one permutation into another. The Ulam metric is a simpler variant of the general edit distance metric. It provides a measure of dissimilarity over a set of rankings/permutations. In the center problem, given a set of permutations, we are asked to find a permutation (not necessarily from the input set) that minimizes the maximum distance to the input permutations. This problem is also referred to as maximum rank aggregation under Ulam. So far, we only know of a folklore 2-approximation algorithm for this NP-hard problem. Even for constantly many permutations, we do not know anything better than an exhaustive search over all n! permutations. In this paper, we achieve a (3/2 - 1/(3m))-approximation of the Ulam center in time n^O(m² ln m), for m input permutations over [n]. We therefore get a polynomial time bound while achieving better than a 3/2-approximation for constantly many permutations. This problem is of special interest even for constantly many permutations because under certain dissimilarity measures over rankings, even for four permutations, the problem is NP-hard. In proving our result, we establish a surprising connection between the approximate Ulam center problem and the closest string with wildcards problem (the center problem over the Hamming metric, allowing wildcards). We further study the closest string with wildcards problem and show that there cannot exist any (2-ε)-approximation algorithm (for any ε > 0) for it unless 𝖯 = NP. This inapproximability result is in sharp contrast with the same problem without wildcards, where we know of a PTAS.