cs.DM

83 posts

arXiv:1803.04660v3 Announce Type: replace Abstract: In the context of fine-grained complexity, we investigate the notion of certificate enabling faster polynomial-time algorithms. We specifically target radius (minimum eccentricity), diameter (maximum eccentricity), and all-eccentricity computations for which quadratic-time lower bounds are known under plausible conjectures. In each case, we introduce a notion of certificate as a specific set of nodes from which appropriate bounds on all eccentricities can be derived in subquadratic time when this set has sublinear size. The existence of small certificates is a barrier against SETH-based lower bounds for these problems. We indeed prove that for graph classes with small certificates, there exist randomized subquadratic-time algorithms for computing the radius, the diameter, and all eccentricities respectively.Moreover, these notions of certificates are tightly related to algorithms probing the graph through one-to-all distance queries and allow to explain the efficiency of practical radius and diameter algorithms from the literature. Our formalization enables a novel primal-dual analysis of a classical approach for diameter computation that leads to algorithms for radius, diameter and all eccentricities with theoretical guarantees with respect to certain graph parameters. This is complemented by experimental results on various types of real-world graphs showing that these parameters appear to be low in practice. Finally, we obtain refined results for several graph classes.

Feodor F. Dragan (UniBuc, ICI), Guillaume Ducoffe (UniBuc, ICI), Michel Habib (IRIF), Laurent Viennot (DI-ENS, ARGO)1/22/2025

arXiv:2403.09122v2 Announce Type: replace Abstract: A monitoring edge-geodetic set, or simply an MEG-set, of a graph $G$ is a vertex subset $M \subseteq V(G)$ such that given any edge $e$ of $G$, $e$ lies on every shortest $u$-$v$ path of $G$, for some $u,v \in M$. The monitoring edge-geodetic number of $G$, denoted by $meg(G)$, is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem. In this article, we compare $meg(G)$ with some other graph theoretic parameters stemming from the network monitoring problem and provide examples of graphs having prescribed values for each of these parameters. We also characterize graphs $G$ that have $V(G)$ as their minimum MEG-set, which settles an open problem due to Foucaud \textit{et al.} (CALDAM 2023), and prove that some classes of graphs fall within this characterization. We also provide a general upper bound for $meg(G)$ for sparse graphs in terms of their girth, and later refine the upper bound using the chromatic number of $G$. We examine the change in $meg(G)$ with respect to two fundamental graph operations: clique-sum and subdivisions. In both cases, we provide a lower and an upper bound of the possible amount of changes and provide (almost) tight examples.

Florent Foucaud, Clara Marcille, Zin Mar Myint, R. B. Sandeep, Sagnik Sen, S. Taruni1/22/2025

arXiv:2501.11386v1 Announce Type: cross Abstract: This paper proves the minimum size of a supersequence over a set of eight elements is 52. This disproves a conjecture that the lower bound of the supersequence is the partial sum of the geometric Connell sequence. By studying the internal distribution of individual elements within sub-strings of the supersequence called segments, the proof provides important results on the internal structure that could help to understand the general lower bound problem for finite sets.

Oliver Tan1/22/2025

arXiv:2501.11617v1 Announce Type: cross Abstract: For every positive integer $k$, we define the $k$-treedepth as the largest graph parameter $\mathrm{td}_k$ satisfying (i) $\mathrm{td}_k(\emptyset)=0$; (ii) $\mathrm{td}_k(G) \leq 1+ \mathrm{td}_k(G-u)$ for every graph $G$ and every vertex $u \in V(G)$; and (iii) if $G$ is a $(<k)$-clique-sum of $G_1$ and $G_2$, then $\mathrm{td}_k(G) \leq \max \{\mathrm{td}_k(G_1),\mathrm{td}_k(G_2)\}$, for all graphs $G_1,G_2$. This parameter coincides with treedepth if $k=1$, and with treewidth plus $1$ if $k \geq |V(G)|$. We prove that for every positive integer $k$, a class of graphs $\mathcal{C}$ has bounded $k$-treedepth if and only if there is a positive integer $\ell$ such that for every tree $T$ on $k$ vertices, no graph in $\mathcal{C}$ contains $T \square P_\ell$ as a minor. This implies for $k=1$ that a minor-closed class of graphs has bounded treedepth if and only if it excludes a path, for $k=2$ that a minor-closed class of graphs has bounded $2$-treedepth if and only if it excludes as a minor a ladder (Huynh, Joret, Micek, Seweryn, and Wollan; Combinatorica, 2021), and for large values of $k$ that a minor-closed class of graphs has bounded treewidth if and only if it excludes a grid (Grid-Minor Theorem, Robertson and Seymour; JCTB, 1986). As a corollary, we obtain the following qualitative strengthening of the Grid-Minor Theorem in the case of bounded-height grids. For all positive integers $k, \ell$, every graph that does not contain the $k \times \ell$ grid as a minor has $(2k-1)$-treedepth at most a function of $(k, \ell)$.

Cl\'ement Rambaud1/22/2025

arXiv:2111.08328v2 Announce Type: replace Abstract: We consider the problem of assigning appearing times to the edges of a digraph in order to maximize the (average) temporal reachability between pairs of nodes. Motivated by the application to public transit networks, where edges cannot be scheduled independently one of another, we consider the setting where the edges are grouped into certain walks (called trips) in the digraph and where assigning the appearing time to the first edge of a trip forces the appearing times of the subsequent edges. In this setting, we show that, quite surprisingly, it is NP-complete to decide whether there exists an assignment of times connecting a given pair of nodes. This result allows us to prove that the problem of maximising the temporal reachability cannot be approximated within a factor better than some polynomial term in the size of the graph. We thus focus on the case where, for each pair of nodes, there exists an assignment of times such that one node is reachable from the other. We call this property strong temporalisability. It is a very natural assumption for the application to public transit networks. On the negative side, the problem of maximising the temporal reachability remains hard to approximate within a factor $\sqrt$ n/12 in that setting. Moreover, we show the existence of collections of trips that are strongly temporalisable but for which any assignment of starting times to the trips connects at most an O(1/ $\sqrt$ n) fraction of all pairs of nodes. On the positive side, we show that there must exist an assignment of times that connects a constant fraction of all pairs in the strongly temporalisable and symmetric case, that is, when the set of trips to be scheduled is such that, for each trip, there is a symmetric trip visiting the same nodes in reverse order. Keywords:edge labeling edge scheduled network network optimisation temporal graph temporal path temporal reachability time assignment

Filippo Brunelli (UPCit\'e, IRIF), Pierluigi Crescenzi (GSSI), Laurent Viennot (ARGO)1/22/2025

arXiv:2210.14100v4 Announce Type: replace Abstract: The Additive-Multiplicative Matrix Channel (AMMC) was introduced by Silva, Kschischang and K\"otter in 2010 to model data transmission using random linear network coding. The input and output of the channel are $n\times m$ matrices over a finite field $\mathbb{F}_q$. On input the matrix $X$, the channel outputs $Y=A(X+W)$ where $A$ is a uniformly chosen $n\times n$ invertible matrix over $\mathbb{F}_q$ and where $W$ is a uniformly chosen $n\times m$ matrix over $\mathbb{F}_q$ of rank $t$. Silva \emph{et al} considered the case when $2n\leq m$. They determined the asymptotic capacity of the AMMC when $t$, $n$ and $m$ are fixed and $q\rightarrow\infty$. They also determined the leading term of the capacity when $q$ is fixed, and $t$, $n$ and $m$ grow linearly. We generalise these results, showing that the condition $2n\geq m$ can be removed. (Our formula for the capacity falls into two cases, one of which generalises the $2n\geq m$ case.) We also improve the error term in the case when $q$ is fixed.

Simon R. Blackburn, Jessica Claridge1/22/2025

arXiv:2501.12365v1 Announce Type: new Abstract: Computing the Fourier transform of a $q$-ary function $f:\mathbb{Z}_{q}^n\rightarrow \mathbb{R}$, which maps $q$-ary sequences to real numbers, is an important problem in mathematics with wide-ranging applications in biology, signal processing, and machine learning. Previous studies have shown that, under the sparsity assumption, the Fourier transform can be computed efficiently using fast and sample-efficient algorithms. However, in many practical settings, the function is defined over a more general space -- the space of generalized $q$-ary sequences $\mathbb{Z}_{q_1} \times \mathbb{Z}_{q_2} \times \cdots \times \mathbb{Z}_{q_n}$ -- where each $\mathbb{Z}_{q_i}$ corresponds to integers modulo $q_i$. A naive approach involves setting $q=\max_i{q_i}$ and treating the function as $q$-ary, which results in heavy computational overheads. Herein, we develop GFast, an algorithm that computes the $S$-sparse Fourier transform of $f$ with a sample complexity of $O(Sn)$, computational complexity of $O(Sn \log N)$, and a failure probability that approaches zero as $N=\prod_{i=1}^n q_i \rightarrow \infty$ with $S = N^\delta$ for some $0 \leq \delta 25\%$ smaller normalized mean-squared error compared to existing algorithms.

Darin Tsui, Kunal Talreja, Amirali Aghazadeh1/22/2025

arXiv:2501.11281v1 Announce Type: cross Abstract: A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiam\v{c}\'{\i}k conjectured that for a graph $G$ with maximum degree $\Delta$, $a'(G) \le \Delta+2$. A graph $G$ is said to be $3$-sparse if every edge in $G$ is incident on at least one vertex of degree at most $3$. We prove the conjecture for the class of $3$-sparse graphs. Further, we give a stronger bound of $\Delta +1$, if there exists an edge $xy$ in the graph with $d_G(x)+ d_G(y) 3$, the $3$-sparse graphs where no such edge exists is the set of bipartite graphs where one partition has vertices with degree exactly $3$ and the other partition has vertices with degree exactly $\Delta$.

Nevil Anto, Manu Basavaraju, Shashanka Kulamarva1/22/2025

arXiv:2501.11157v1 Announce Type: new Abstract: The study of structural graph width parameters like tree-width, clique-width and rank-width has been ongoing during the last five decades, and their algorithmic use has also been increasing [Cygan et al., 2015]. New width parameters continue to be defined, for example, MIM-width in 2012, twin-width in 2020, and mixed-thinness, a generalization of thinness, in 2022. The concept of thinness of a graph was introduced in 2007 by Mannino, Oriolo, Ricci and Chandran, and it can be seen as a generalization of interval graphs, which are exactly the graphs with thinness equal to one. This concept is interesting because if a representation of a graph as a $k$-thin graph is given for a constant value $k$, then several known NP-complete problems can be solved in polynomial time. Some examples are the maximum weighted independent set problem, solved in the seminal paper by Mannino et al., and the capacitated coloring with fixed number of colors [Bonomo, Mattia and Oriolo, 2011]. In this work we present a constructive $O(n\log(n))$-time algorithm to compute the thinness for any given $n$-vertex tree, along with a corresponding thin representation. We use intermediate results of this construction to improve known bounds of the thinness of some special families of trees.

Flavia Bonomo-Braberman, Eric Brandwein, Carolina Luc\'ia Gonz\'alez, Agust\'in Sansone1/22/2025

arXiv:2501.10824v1 Announce Type: new Abstract: A unified combinatorial definition of the information content and entropy of different types of patterns, compatible with the traditional concepts of information and entropy, going beyond the limitations of Shannon information interpretable for ergodic Markov processes. We compare the information content of various finite patterns and derive general properties of information quantity from these comparisons. Using these properties, we define normalized information estimation methods based on compression algorithms and Kolmogorov complexity. From a combinatorial point of view, we redefine the concept of entropy in a way that is asymptotically compatible with traditional entropy.

Zsolt Pocze1/22/2025

arXiv:2501.11419v1 Announce Type: new Abstract: Payment Channel Networks (PCNs) are a method for improving the scaling and latency of cryptocurrency transactions. For a payment to be made between two peers in a PCN, a feasible low-fee path in the network must be planned. Many PCN path planning algorithms use a search algorithm that is a variant of Dijkstra's algorithm. In this article, we prove the correctness and computational complexity of this algorithm. Specifically, we show that, if the PCN satisfies a consistency property relating to the fees charged by payment channels, the algorithm is correct and has polynomial computational complexity. However, in the general case, the algorithm is not correct and the path planning problem is NP-hard. These newly developed results can be used to inform the development of new or existing PCNs amenable to path planning. For example, we show that the Lightning Network, which is the most widely used PCN and is built on the Bitcoin cryptocurrency, currently satisfies the above consistency property. As a second contribution, we demonstrate that a small modification to the above path planning algorithm which, although having the same asymptotic computational complexity, empirically shows better performance. This modification involves the use of a bidirectional search and is empirically evaluated by simulating transactions on the Lightning Network.

Padraig Corcoran, Rhyd Lewis1/22/2025

arXiv:2501.11697v1 Announce Type: new Abstract: We present the first comprehensive analysis of temporal settings for directed temporal graphs, fully resolving their hierarchy with respect to support, reachability, and induced-reachability equivalence. These notions, introduced by Casteigts, Corsini, and Sarkar, capture different levels of equivalence between temporal graph classes. Their analysis focused on undirected graphs under three dimensions: strict vs. non-strict (whether times along paths strictly increase), proper vs. arbitrary (whether adjacent edges can appear simultaneously), and simple vs. multi-labeled (whether an edge can appear multiple times). In this work, we extend their framework by adding the fundamental distinction of directed vs. undirected. Our results reveal a single-strand hierarchy for directed graphs, with strict & simple being the most expressive class and proper & simple the least expressive. In contrast, undirected graphs form a two-strand hierarchy, with strict & multi-labeled being the most expressive and proper & simple the least expressive. The two strands are formed by the non-strict & simple and the strict & simple class, which we show to be incomparable. In addition to examining the internal hierarchies of directed and of undirected graph classes, we compare the two. We show that each undirected class can be transformed into its directed counterpart under reachability equivalence, while no directed class can be transformed into any undirected one. Our findings have significant implications for the study of computational problems on temporal graphs. Positive results in more expressive graph classes extend to weaker classes as long as the problem is independent under reachability equivalence. Conversely, hardness results for a less expressive class propagate to stronger classes. We hope these findings will inspire a unified approach for analyzing temporal graphs under the different settings.

Michelle D\"oring1/22/2025

arXiv:2501.11541v1 Announce Type: new Abstract: The problem of sampling edge-colorings of graphs with maximum degree $\Delta$ has received considerable attention and efficient algorithms are available when the number of colors is large enough with respect to $\Delta$. Vizing's theorem guarantees the existence of a $(\Delta+1)$-edge-coloring, raising the natural question of how to efficiently sample such edge-colorings. In this paper, we take an initial step toward addressing this question. Building on the approach of Dotan, Linial, and Peled, we analyze a randomized algorithm for generating random proper $(\Delta+1)$-edge-colorings, which in particular provides an algorithmic interpretation of Vizing's theorem. The idea is to start from an arbitrary non-proper edge-coloring with the desired number of colors and at each step, recolor one edge uniformly at random provided it does not increase the number of conflicting edges (a potential function will count the number of pairs of adjacent edges of the same color). We show that the algorithm almost surely produces a proper $(\Delta+1)$-edge-coloring and propose several conjectures regarding its efficiency and the uniformity of the sampled colorings.

Lucas De Meyer, Franti\v{s}ek Kardo\v{s}, Aur\'elie Lagoutte, Guillem Perarnau1/22/2025

arXiv:2501.11467v1 Announce Type: new Abstract: The possibility of errors in human-engineered formal verification software, such as model checkers, poses a serious threat to the purpose of these tools. An established approach to mitigate this problem are certificates -- lightweight, easy-to-check proofs of the verification results. In this paper, we develop novel certificates for model checking of Markov decision processes (MDPs) with quantitative reachability and expected reward properties. Our approach is conceptually simple and relies almost exclusively on elementary fixed point theory. Our certificates work for arbitrary finite MDPs and can be readily computed with little overhead using standard algorithms. We formalize the soundness of our certificates in Isabelle/HOL and provide a formally verified certificate checker. Moreover, we augment existing algorithms in the probabilistic model checker Storm with the ability to produce certificates and demonstrate practical applicability by conducting the first formal certification of the reference results in the Quantitative Verification Benchmark Set.

Krishnendu Chatterjee, Tim Quatmann, Maximilian Sch\"affeler, Maximilian Weininger, Tobias Winkler, Daniel Zilken1/22/2025

arXiv:2408.00933v2 Announce Type: replace-cross Abstract: The bad science matrix problem consists in finding, among all matrices $A \in \mathbb{R}^{n \times n}$ with rows having unit $\ell^2$ norm, one that maximizes $\beta(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$. Our main contribution is an explicit construction of an $n \times n$ matrix $A$ showing that $\beta(A) \geq \sqrt{\log_2(n+1)}$, which is only 18% smaller than the asymptotic rate. We prove that every entry of any optimal matrix is a square root of a rational number, and we find provably optimal matrices for $n \leq 4$.

Alex Albors, Hisham Bhatti, Lukshya Ganjoo, Raymond Guo, Dmitriy Kunisky, Rohan Mukherjee, Alicia Stepin, Tony Zeng1/22/2025

arXiv:2501.12062v1 Announce Type: new Abstract: Using the algebraic approach to promise constraint satisfaction problems, we establish complexity classifications of three natural variants of hypergraph colourings: standard nonmonochromatic colourings, conflict-free colourings, and linearly-ordered colourings. Firstly, we show that finding an $\ell$-colouring of a $k$-colourable $r$-uniform hypergraph is NP-hard for all constant $2\leq k\leq \ell$ and $r\geq 3$. This provides a shorter proof of a celebrated result by Dinur et al. [FOCS'02/Combinatorica'05]. Secondly, we show that finding an $\ell$-conflict-free colouring of an $r$-uniform hypergraph that admits a $k$-conflict-free colouring is NP-hard for all constant $3\leq k\leq\ell$ and $r\geq 4$, except for $r=4$ and $k=2$ (and any $\ell$); this case is solvable in polynomial time. The case of $r=3$ is the standard nonmonochromatic colouring, and the case of $r=2$ is the notoriously difficult open problem of approximate graph colouring. Thirdly, we show that finding an $\ell$-linearly-ordered colouring of an $r$-uniform hypergraph that admits a $k$-linearly-ordered colouring is NP-hard for all constant $3\leq k\leq\ell$ and $r\geq 4$, thus improving on the results of Nakajima and \v{Z}ivn\'y~[ICALP'22/ACM TocT'23].

Tamio-Vesa Nakajima, Zephyr Verwimp, Marcin Wrochna, Stanislav \v{Z}ivn\'y1/22/2025

arXiv:2501.11192v1 Announce Type: new Abstract: We prove new parameterized complexity results for the FO Model Checking problem and in particular for Independent Set, for two recently introduced subclasses of $H$-graphs, namely proper $H$-graphs and non-crossing $H$-graphs. It is known that proper $H$-graphs, and thus $H$-graphs, may have unbounded twin-width. However, we prove that for every connected multigraph $H$ with no self-loops, non-crossing $H$-graphs have bounded proper mixed-thinness, and thus bounded twin-width. Consequently, we can apply a well-known result of Bonnet, Kim, Thomass\'e, and Watrigant (2021) to find that the FO Model Checking problem is in $\mathsf{FPT}$ for non-crossing $H$-graphs when parameterized by $\Vert H \Vert+\ell$, where $\Vert H \Vert$ is the size of $H$ and $\ell$ is the size of a formula. In particular, this implies that Independent Set is in $\mathsf{FPT}$ on non-crossing $H$-graphs when parameterized by $\Vert H \Vert+k$, where $k$ is the solution size. In contrast, Independent Set for general $H$-graphs is $\mathsf{W[1]}$-hard when parameterized by $\Vert H \Vert +k$. We strengthen the latter result by proving thatIndependent Set is $\mathsf{W[1]}$-hard even on proper $H$-graphs when parameterized by $\Vert H \Vert+k$. In this way, we solve, subject to $\mathsf{W[1]}\neq \mathsf{FPT}$, an open problem of Chaplick (2023), who asked whether there exist problems that can be solved faster for non-crossing $H$-graphs than for proper $H$-graphs.

Flavia Bonomo-Braberman, Nick Brettell, Andrea Munaro, Dani\"el Paulusma1/22/2025

arXiv:2501.10828v1 Announce Type: cross Abstract: The (strong) isometric path complexity is a recently introduced graph invariant that captures how arbitrary isometric paths (i.e. shortest paths) of a graph can be viewed as a union of few ``rooted" isometric paths (i.e. isometric paths with a common end-vertex). It is known that this parameter can be computed optimally in polynomial time. Seemingly unrelated graph classes studied in metric graph theory (e.g. hyperbolic graphs), geometric intersection graph theory (e.g. outerstring graphs), and structural graph theory (e.g. (theta, prism, pyramid)-free graphs) have been shown to have bounded strong isometric path complexity [Chakraborty et al., MFCS '23]. We show that important graph classes studied in \emph{coarse graph theory} (as introduced by [Georgakopoulos & Papasoglu '23]) have bounded strong isometric path complexity. We show that the strong isometric path complexity of $K_{2,t}$-asymptotic minor-free graphs is bounded. Let $U_t$ denote the graph obtained by adding a universal vertex to a path of $t-1$ edges. We show that the strong isometric path complexity of $U_t$-asymptotic minor-free graphs is bounded. This implies $K_4^-$-asymptotic minor-free graphs, i.e. graphs that are quasi-isometric to a cactus [Fujiwara & Papasoglu '23] have bounded strong isometric path complexity. On the other hand, $K_4$-minor-free graphs have unbounded strong isometric path complexity. We show that graphs whose all induced cycles of length at least 4 have the same length (also known as monoholed graphs as defined by [Cook et al., JCTB '24]) form a subclass of $U_4$-asymptotic minor-free graphs. Hence, the strong isometric path complexity of monoholed graphs is bounded. We show that even-hole free graphs have unbounded strong isometric path complexity. We show that the strong isometric path complexity is preserved under the fixed power, line graph, and clique-sums operators.

Dibyayan Chakraborty, Florent Foucaud1/22/2025

arXiv:2501.10633v1 Announce Type: new Abstract: We introduce the meta-problem Sidestep$(\Pi, \mathsf{dist}, d)$ for a problem $\Pi$, a metric $\mathsf{dist}$ over its inputs, and a map $d: \mathbb N \to \mathbb R_+ \cup \{\infty\}$. A solution to Sidestep$(\Pi, \mathsf{dist}, d)$ on an input $I$ of $\Pi$ is a pair $(J, \Pi(J))$ such that $\mathsf{dist}(I,J) \leqslant d(|I|)$ and $\Pi(J)$ is a correct answer to $\Pi$ on input $J$. This formalizes the notion of answering a related question (or sidestepping the question), for which we give some practical and theoretical motivations, and compare it to the neighboring concepts of smoothed analysis, planted problems, and edition problems. Informally, we call hardness radius the ``largest'' $d$ such that Sidestep$(\Pi, \mathsf{dist}, d)$ is NP-hard. This framework calls for establishing the hardness radius of problems $\Pi$ of interest for the relevant distances $\mathsf{dist}$. We exemplify it with graph problems and two distances $\mathsf{dist}_\Delta$ and $\mathsf{dist}_e$ (the edge edit distance) such that $\mathsf{dist}_\Delta(G,H)$ (resp. $\mathsf{dist}_e(G,H)$) is the maximum degree (resp. number of edges) of the symmetric difference of $G$ and $H$ if these graphs are on the same vertex set, and $+\infty$ otherwise. We show that the decision problems Independent Set, Clique, Vertex Cover, Coloring, Clique Cover have hardness radius $n^{\frac{1}{2}-o(1)}$ for $\mathsf{dist}_\Delta$, and $n^{\frac{4}{3}-o(1)}$ for $\mathsf{dist}_e$, that Hamiltonian Cycle has hardness radius 0 for $\mathsf{dist}_\Delta$, and somewhere between $n^{\frac{1}{2}-o(1)}$ and $n/3$ for $\mathsf{dist}_e$, and that Dominating Set has hardness radius $n^{1-o(1)}$ for $\mathsf{dist}_e$. We leave several open questions.

\'Edouard Bonnet1/22/2025

arXiv:2411.19351v2 Announce Type: replace-cross Abstract: This article presents examples of an application of the finite field method for the computation of the characteristic polynomial of the matching arrangement of a graph. Weight functions on edges of a graph with weights from a finite field are divided into proper and improper functions in connection with proper colorings of vertices of the matching polytope of a graph. An improper weight function problem is introduced, a proof of its NP-completeness is presented, and a knapsack-like public key cryptosystem is constructed based on the improper weight function problem.

Aleksey Bolotnikov, Anvar Irmatov1/22/2025