Jul 2026· Embedded Systems and Applications· pp. 105:1-105:19· 0 citations· 41 references
Computer Science
TL;DR
A near-linear-time algorithm is given and there is no combinatorial algorithm improving over the state-of-the-art $\mathcal{O}(m|E| + N)$ bound for any $b\ge 4$.
Abstract
We consider directed graphs in which the nodes are labeled with strings. A walk in such a graph naturally corresponds to the concatenation of the visited nodes'labels. These graphs are widely used in bioinformatics to compactly describe large collections of highly similar genomes. Given such a graph $G=(V,E)$ and a pattern of length $m$, we seek a walk whose corresponding string has an occurrence of the pattern. We call this the SMLG problem. Amir et al. [J. Algorithms, 2000] showed that SMLG can be solved in $\mathcal{O}(m|E| + N)$ time, where $N$ is the total length of all node labels. Equi et al. [ACM Trans. Algorithms, 2023] showed that this is essentially optimal (under SETH). The existing lower bound assumes that the sought walk is of length $\Theta(|V|)$. Thus, we might be able to bypass this lower bound by restricting the walk length to $b-1$, which naturally reduces to having as input a directed graph whose set of nodes is partitioned into $b$ blocks. Then, we seek a walk in this graph that starts in the first block and ends in the last block. We call this the $b$-SMBG problem. We provide a more fine-grained classification that essentially settles the complexity of $b$-SMBG parameterized by $b$: (1) We give a near-linear-time algorithm for $b=3$. (2) We show that there is no combinatorial algorithm improving over the state-of-the-art $\mathcal{O}(m|E| + N)$ bound for any $b\ge 4$. (3) We also present a fast matrix multiplication-based algorithm yielding an improvement for $b \in \mathcal{O}(1)$, which is conditionally optimal. (4) Finally, we show that under SETH, for any $b \in \omega(\log |V|)$, no algorithm can improve over the state of the art.
The planted subgraph detection problem asks whether a random graph contains a hidden structured subgraph. In the classical formulation, the entire adjacency matrix is observed and one distinguishes between an Erd\H{o}s--R\'enyi random graph and one obtained by planting a copy of a prescribed graph inside an Erd\H{o}s--R\'enyi random graph. The statistical and computational limits of this problem under full observation are now well understood, even for arbitrary planted subgraphs. In this paper, we investigate an information-limited version of the problem in which the planted structure is an arbitrary sequence of graphs $\Gamma=(\Gamma_n)_{n\geq1}$, where $\Gamma_n$ is embedded in an ambient graph on $n$ vertices, but the observer does not have access to the full adjacency matrix. Instead, information is acquired through a limited number of non-adaptive edge queries. We study the minimum query complexity required for reliable detection. We derive general information-theoretic lower bounds and complementary algorithmic upper bounds on the query complexity as functions of the query budget and structural properties of the planted graph. The proposed algorithms exploit three distinct structural mechanisms: dense local motifs, high-degree vertices, and global edge density. We establish matching bounds, up to polylogarithmic factors, for several broad families of planted graphs, including clique-like, bounded-cover, and hub-dominated graph classes. Our framework substantially generalizes existing query-complexity results for planted clique and planted dense subgraph models, providing a unified treatment of arbitrary planted subgraphs under restricted graph access.
The Graph Edit Distance (GED) is a widely used graph similarity measure asking for the minimum cost of a sequence of edits transforming one (labeled) graph into another. The considered edit operations are deletion, insertion, and relabeling of nodes and edges. Special cases include the Graph Isomorphism problem, as well as many other graph problems that ask for the existence or minimum cost of a certain substructure, like the Traveling Salesman or Maximum Clique problem. We present a novel exponential time algorithm to compute the exact GED and a corresponding edit sequence in $O^*(4 + \varepsilon)^n$ time and polynomial space, provided one of the two graphs admits strictly sublinear balanced separators. In particular, the claimed runtime holds if one of the graphs is $K_h$-minor free (e.g., planar), or has bounded treewidth, which is the case for many real-world applications (e.g., all instances in GEDLIB). This substantially improves the best known worst-case running time bounds of $O^*(n!)$ for these graph classes.
Laura Bülte, Philip Mayer, Lars Müller et al.· 0 citations
We prove an elementary yet powerful combinatorial lemma: in any rooted tree with $L$ leaves, the number of nodes whose depth is smaller than the number of their leaf descendants is at most $L$. For any string $T$ of length $n$, a direct application of this lemma to the suffix trie of $T$ yields that the number of substrings of $T$ whose length is smaller than their number of occurrences in $T$ is at most $n$. This combinatorial insight leads to space-efficient data structures with optimal query times for string counting problems via the following algorithmic framework: store the counts for the at most $n$ ``frequent''substrings of $T$ in a preprocessing step, and use a reporting query to count for the ``infrequent''substrings. Our framework acts as a convenient black box, lifting indexes with reporting time $\mathcal{O}(|P|+|\textsf{Occ}_T(P)|)$ to support counting queries in time $\mathcal{O}(|P|)$, where $P$ is the queried pattern and $\textsf{Occ}_T(P)$ is the set of occurrences of $P$ in $T$. As applications, we show efficient indexes for consecutive occurrences, weighted sequences, strings with utilities, and non-overlapping occurrences.
B. Bals, P. Charalampopoulos, Oded Lachish et al.· Embedded Systems and Applica...· 0 citations
It is proved that, as long as $p=o(\log n /n)$, for any $\varepsilon>0$, one can construct a confidence set of vertices of size $K(\varepsilon)$ that depends only on $\varepsilon$ and not on $n$, such that it contains the root with probability at least $1-\varepsilon$.
Luc Devroye, Gábor Lugosi, Neeladri Maitra· 1 citation· ⚡1
This paper provides the first truly linear-time approximation scheme for the Densest Subgraph Problem, and uses assignments arising from a flow-based formulation together with a structural carving lemma to progressively carve "sparse" parts of the graph while nearly preserving the densest subgraph.
A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $\gamma^{+}$ (additions only) and the Cayley edit distance $\gamma_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $\gamma^{+}(G)\ge n\Delta^{*}/(2m)-1$, where $\Delta^{*}$ is the least $d\ge\Delta$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $\gamma^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $\gamma^{+}$, while $\gamma_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $\gamma^{+}(P_n)=\gamma^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $\gamma_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.
