arXiv subjects

Alexandru Popa

Publications and source records attributed to Alexandru Popa.

At least 19 recordsLinked to original sources

arXiv2026

Maximum Matching-Match: Hardness and Approximation

In this paper, we study \textsc{MaxMMP}, an optimization variant of the Matching-Match Puzzle introduced by Iburi and Uehara (FUN 2024). Given a graph, a partial vertex coloring, and a multiset of colored sticks, the goal is to complete the coloring and assign the sticks to graph edges so as to maximize the number of satisfied edges. We first prove that \textsc{MaxMMP} is APX-hard by an reduction from \textsc{Max-Cut}. The hardness already holds with two colors, no precolored vertices, and only bichromatic sticks. We then give a simple deterministic $\frac{2}{c(c+1)}$-approximation for completely uncolored instances, improving to $\frac{2}{c(c-1)}$ when all sticks are bichromatic. Next, we obtain a randomized $\frac{1-\frac{1}{e}}{2c}$-approximation for arbitrary instances with $c$ colors by reducing the remaining coloring choices to monotone submodular maximization under a partition matroid. On bipartite graphs, the approximation ratio improves to $\frac{1-\frac{1}{e}}{c}$. For every fixed $c$, we further obtain deterministic $\frac{1}{2c}$ and $\frac{1}{c}$-approximations on general and bipartite graphs, respectively, in time $n^{O(c^2)}$. Finally, for every fixed number of colors, we show that \textsc{MaxMMP} can be solved exactly in time $n^{O(c^2)}$ on trees and on cographs.

cs.DS↗

arXiv2026

Complexity and Algorithms for Unary Translocation Distance

Given a finite set of integers $A$, a \emph{unary translocation} produces a new set $A' = A \cup \{u,v\}$, where $u$ and $v$ are nonnegative integers satisfying $x+y=u+v$ for some $x,y\in A$. For an input set $A$ and a target set $B$, the \emph{unary translocation distance} is the minimum number of unary translocations required to obtain a superset containing $B$. In this paper, we study this problem from both theoretical and computational perspectives. We prove that computing the unary translocation distance is strongly NP-hard, thereby answering an open question raised by \citet{ConstantinMiclausPopa2026UnaryTranslocation}. On the positive side, we give an exact pseudo-polynomial algorithm for every fixed constant value of $|B|$, extending our previous results for $|B|\leq 2$. For arbitrary target sets, we present a $2$-approximation algorithm, an additive $(|B|-1)$-approximation algorithm, and show that the additive algorithm also yields a $3$-approximation. We also propose parameterized algorithms, including algorithms parameterized by the maximum value in the input set together with the optimum distance, and by the maximum value in the target set together with $|B|$. In addition, we propose an integer linear programming formulation that gives an exact mathematical model for the problem, analyze its size, and show that the LP relaxation has integrality gap at least $\frac{4}{3}$. Finally, we report computational experiments comparing the $2$-approximation algorithm, beam search, and simulated annealing. The results show that the approximation algorithm is highly effective in practice and often outperforms the heuristic baselines.

cs.DS↗

arXiv2026

Optimal Portfolio Compression for Priority-Proportional Clearing with Defaulting Costs

We study financial networks where banks are connected through bilateral liabilities and may default when resources are insufficient to meet obligations. We consider both the standard proportional clearing model and a priority-proportional clearing model in which banks repay creditors according to exogenously given priority classes. In such markets, portfolio compression is a process where several banks come to a netting arrangement which reduces liabilities without changing any bank's net exposure, essentially removing cycles of debt. Our goal is to understand whether portfolio compression schemes can be designed to improve clearing outcomes for a large fraction of banks. We provide a computational characterization of the benefits and limitations of compression. On the positive side, we give a polynomial-time algorithm to compute a maximal clearing outcome under priority-proportional clearing, and we show that it is possible to decide in polynomial time whether there exists a compression that limits defaults to at most one bank. On the negative side, we show that several natural optimization and decision problems are computationally intractable: deciding whether some compression can reduce the number of defaulting banks below a given threshold, or whether a specific bank can be saved from defaulting, is $\NP$-hard even in restricted settings and under proportional clearing. We further present a mixed integer linear programming (MILP) formulation that computes a compression maximizing the number of non-defaulting banks, providing a practical approach to this hard problem. Using our MILP formulation, we perform simulations on both synthetic and real-world datasets to analyze the effects of portfolio compression.

cs.GT↗

arXiv2024

Approximating Maximum Edge 2-Coloring by Normalizing Graphs

In a simple, undirected graph G, an edge 2-coloring is a coloring of the edges such that no vertex is incident to edges with more than 2 distinct colors. The problem maximum edge 2-coloring (ME2C) is to find an edge 2-coloring in a graph G with the goal to maximize the number of colors. For a relevant graph class, ME2C models anti-Ramsey numbers and it was considered in network applications. For the problem a 2-approximation algorithm is known, and if the input graph has a perfect matching, the same algorithm has been shown to have a performance guarantee of 5/3. It is known that ME2C is APX-hard and that it is UG-hard to obtain an approximation ratio better than 1.5. We show that if the input graph has a perfect matching, there is a polynomial time 1.625-approximation and if the graph is claw-free or if the maximum degree of the input graph is at most three (i.e., the graph is subcubic), there is a polynomial time 1.5-approximation algorithm for ME2C

cs.DM↗

arXiv2021

Approximate and exact results for the harmonious chromatic number

Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring such that for every two distinct colors i, j at most one pair of adjacent vertices are colored with i and j. This type of coloring is edge-distinguishing and has potential applications in transportation network, computer network, airway network system. The results presented in this paper fall into two categories: in the first part of the paper we are concerned with the computational aspects of finding a minimum harmonious coloring and in the second part we determine the exact value of the harmonious chromatic number for some particular graphs and classes of graphs. More precisely, in the first part we show that finding a minimum harmonious coloring for arbitrary graphs is APX-hard, the natural greedy algorithm is a $Ω(\sqrt{n})$-approximation, and, moreover, we show a relationship between the vertex cover and the harmonious chromatic number. In the second part we determine the exact value of the harmonious chromatic number for all 3-regular planar graphs of diameter 3, some non-planar regular graphs and cycle-related graphs.

cs.DM↗

arXiv2020

A novel algorithm for clearing financial obligations between companies -- an application within the Romanian Ministry of Economy

The concept of clearing or netting, as defined in the glossaries of European Central Bank, has a great impact on the economy of a country influencing the exchanges and the interactions between companies. On short, netting refers to an alternative to the usual way in which the companies make the payments to each other: it is an agreement in which each party sets off amounts it owes against amounts owed to it. Based on the amounts two or more parties owe between them, the payment is substituted by a direct settlement. In this paper we introduce a set of graph algorithms which provide optimal netting solutions for the scale of a country economy. The set of algorithms computes results in an efficient time and is tested on invoice data provided by the Romanian Ministry of Economy. Our results show that classical graph algorithms are still capable of solving very important modern problems.

cs.DS↗

arXiv2020

Complexity of Computing the Anti-Ramsey Numbers for Paths

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erd\" os, Simonovits and S\' os. For given graphs $G$ and $H$ the \emph{anti-Ramsey number} $\textrm{ar}(G,H)$ is defined to be the maximum number $k$ such that there exists an assignment of $k$ colors to the edges of $G$ in which every copy of $H$ in $G$ has at least two edges with the same color. There are works on the computational complexity of the problem when $H$ is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number $\textrm{ar}(G,P_k)$, where $P_k$ is a path of length $k$. First, we observe that when $k = Ω(n)$, the problem is hard; hence, the challenging part is the computational complexity of the problem when $k$ is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing $\textrm{ar}(G,P_k)$ for every integer $k>2$ is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating $\textrm{ar}(G,P_3)$ to a factor of $n^{-1/2 - ε}$ is hard already in $3$-partite graphs, unless P=NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant $k$. Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. We introduce the notion of color connected coloring and employing this structural property. We obtain a linear time algorithm to compute $\textrm{ar}(G,P_k)$, for every integer $k$, when the host graph, $G$, is a tree.

cs.CC↗

arXiv2020

A decision support system for optimizing the cost of social distancing in order to stop the spread of COVID-19

Currently there are many attempts around the world to use computers, smartphones, tablets and other electronic devices in order to stop the spread of COVID-19. Most of these attempts focus on collecting information about infected people, in order to help healthy people avoid contact with them. However, social distancing decisions are still taken by the governments empirically. That is, the authorities do not have an automated tool to recommend which decisions to make in order to maximize social distancing and to minimize the impact for the economy. In this paper we address the aforementioned problem and we design an algorithm that provides social distancing methods (i.e., what schools, shops, factories, etc. to close) that are efficient (i.e., that help reduce the spread of the virus) and have low impact on the economy. On short: a) we propose several models (i.e., combinatorial optimization problems); b) we show some theoretical results regarding the computational complexity of the formulated problems; c) we give an algorithm for the most complex of the previously formulated problems; d) we implement and test our algorithm; and e) we show an integer linear program formulation for our problem.

cs.SI↗

arXiv2019

An output-sensitive algorithm for the minimization of 2-dimensional String Covers

String covers are a powerful tool for analyzing the quasi-periodicity of 1-dimensional data and find applications in automata theory, computational biology, coding and the analysis of transactional data. A \emph{cover} of a string $T$ is a string $C$ for which every letter of $T$ lies within some occurrence of $C$. String covers have been generalized in many ways, leading to \emph{k-covers}, \emph{$λ$-covers}, \emph{approximate covers} and were studied in different contexts such as \emph{indeterminate strings}. In this paper we generalize string covers to the context of 2-dimensional data, such as images. We show how they can be used for the extraction of textures from images and identification of primitive cells in lattice data. This has interesting applications in image compression, procedural terrain generation and crystallography.

cs.DS↗

arXiv2019

IP Solutions for International Kidney Exchange Programmes

In kidney exchange programmes patients with end-stage renal failure may exchange their willing, but incompatible living donors among each other. National kidney exchange programmes are in operation in ten European countries, and some of them have already conducted international exchanges through regulated collaborations. The exchanges are selected by conducting regular matching runs (typically every three months) according to well-defined constraints and optimisation criteria, which may differ across countries. In this work we give integer programming formulations for solving international kidney exchange problems, where the optimisation goals and constraints may be different in the participating countries and various feasibility criteria may apply for the international cycles and chains. We also conduct simulations showing the long-run effects of international collaborations for different pools and under various national restrictions and objectives.

math.OC↗

arXiv2019

Heuristic algorithms for the Longest Filled Common Subsequence Problem

At CPM 2017, Castelli et al. define and study a new variant of the Longest Common Subsequence Problem, termed the Longest Filled Common Subsequence Problem (LFCS). For the LFCS problem, the input consists of two strings $A$ and $B$ and a multiset of characters $\mathcal{M}$. The goal is to insert the characters from $\mathcal{M}$ into the string $B$, thus obtaining a new string $B^*$, such that the Longest Common Subsequence (LCS) between $A$ and $B^*$ is maximized. Casteli et al. show that the problem is NP-hard and provide a 3/5-approximation algorithm for the problem. In this paper we study the problem from the experimental point of view. We introduce, implement and test new heuristic algorithms and compare them with the approximation algorithm of Casteli et al. Moreover, we introduce an Integer Linear Program (ILP) model for the problem and we use the state of the art ILP solver, Gurobi, to obtain exact solution for moderate sized instances.

cs.DS↗

arXiv2019

NewsCompare - a novel application for detecting news influence in a country

The concept of `fake news' has been referenced and thrown around in news reports so much in recent years that it has become a news topic in its own right. At its core, it poses a chilling question -- what do we do if our worldview is fundamentally wrong? Even if internally consistent, what if it does not match the real world? Are our beliefs justified, or could we become indoctrinated from living in a `bubble'? If the latter is true, how could we even test the limits of said bubble from within its confines? We propose a new method to augment the process of identifying fake news, by speeding up and automating the more cumbersome and time-consuming tasks involved. Our application, NewsCompare takes any list of target websites as input (news-related in our use case, but otherwise not restricted), visits them in parallel and retrieves any text content found within. Web pages are subsequently compared to each other, and similarities are tentatively pointed out. These results can be manually verified in order to determine which websites tend to draw inspiration from one another. The data gathered on every intermediate step can be queried and analyzed separately, and most notably we already use the set of hyperlinks to and from the various websites we encounter to paint a sort of `map' of that particular slice of the web. This map can then be cross-referenced and further strengthen the conclusion that a particular grouping of sites with strong links to each other, and posting similar content, are likely to share the same allegiance. We run our application on the Romanian news websites and we draw several interesting observations.

cs.SI↗

arXiv2018

Heuristic algorithms for the min-max edge 2-coloring problem

In multi-channel Wireless Mesh Networks (WMN), each node is able to use multiple non-overlapping frequency channels. Raniwala et al. (MC2R 2004, INFOCOM 2005) propose and study several such architectures in which a computer can have multiple network interface cards. These architectures are modeled as a graph problem named \emph{maximum edge $q$-coloring} and studied in several papers by Feng et. al (TAMC 2007), Adamaszek and Popa (ISAAC 2010, JDA 2016). Later on Larjomaa and Popa (IWOCA 2014, JGAA 2015) define and study an alternative variant, named the \emph{min-max edge $q$-coloring}. The above mentioned graph problems, namely the maximum edge $q$-coloring and the min-max edge $q$-coloring are studied mainly from the theoretical perspective. In this paper, we study the min-max edge 2-coloring problem from a practical perspective. More precisely, we introduce, implement and test four heuristic approximation algorithms for the min-max edge $2$-coloring problem. These algorithms are based on a \emph{Breadth First Search} (BFS)-based heuristic and on \emph{local search} methods like basic \emph{hill climbing}, \emph{simulated annealing} and \emph{tabu search} techniques, respectively. Although several algorithms for particular graph classes were proposed by Larjomaa and Popa (e.g., trees, planar graphs, cliques, bi-cliques, hypergraphs), we design the first algorithms for general graphs. We study and compare the running data for all algorithms on Unit Disk Graphs, as well as some graphs from the DIMACS vertex coloring benchmark dataset.

cs.DS↗

arXiv2018

Analytic Geometry of Homogeneous Spaces

The theory uses methods and language of linear algebra to study nonlinear spaces. These techniques can be used particularly to describe analytic geometry of non-linear elliptic, hyperbolic, De Sitter and Anti de Sitter spaces. The main innovation of elaborated theory is space parameterization by introduction of space signature. This parameterization allows studying of different homogeneous spaces in one global framework. When the parameters are used as variables in definitions, axioms, equations, theorems, proofs, all these have exactly the same form that describes the reality of all homogeneous spaces simultaneously. When it is necessary to describe some space particularities or to see the difference between two concrete spaces, the concrete values can be put in parameters of each definition, axiom, equation, theorem and proof.

math.HO↗

arXiv2018

Hardness and algorithmic results for the approximate cover problem

In CPM 2017, Amir et al. introduce a problem, named \emph{approximate string cover} (\textbf{ACP}), motivated by many aplications including coding and automata theory, formal language theory, combinatorics and molecular biology. A \emph{cover} of a string $T$ is a string $C$ for which every letter of $T$ lies within some occurrence of $C$. The input of the \textbf{ACP} problem consists of a string $T$ and an integer $m$ (less than the length of $T$), and the goal is to find a string $C$ of length $m$ that covers a string $T'$ which is as close to $T$ as possible (under some predefined distance). Amir et al. study the problem for the Hamming distance. In this paper we continue the work of Amir et al. and show the following results: We show an approximation algorithm for the \textbf{ACP} with an approximation ratio of $\sqrt{OPT}$, where OPT is the size of the optimal solution. We provide an FPT algorithm with respect to the alphabet size. \item The \textbf{ACP} problem naturally extends to pseudometrics. Moreover, we show that for some family of pseudometrics, that we term \emph{homogenous additive pseudometrics}, the complexity of \textbf{ACP} remains unchanged. We partially give an answer to an open problem of Amir et al. and show that the Hamming distance over an unbounded alphabet is equivalent to an extended metric over a fixed sized alphabet.

cs.DS↗

arXiv2018

A connection between String Covers and Cover Deterministic Finite Tree Automata Minimization

Data compression plays a crucial part in the cloud based systems of today. One the fundaments of compression is quasi-periodicity, for which there are several models. We build upon the most popular quasi-periodicity model for strings, i.e., covers, generalizing it to trees. We introduce a new type of cover automata, which we call \textbf{D}eterministic \textbf{T}ree \textbf{A}utomata. Then, we formulate a cover problem on these DTA and study its complexity, in both sequential and parallel settings. We obtain bounds for the Cover Minimization Problem. Along the way, we uncover an interesting application, the Shortest Common Cover Problem, for which we give an optimal solution.

cs.DS↗

arXiv2018

A quasi linear-time b-Matching algorithm on distance-hereditary graphs and bounded split-width graphs

We present a quasi linear-time algorithm for Maximum Matching on distance-hereditary graphs and some of their generalizations. This improves on [Dragan, WG'97], who proposed such an algorithm for the subclass of (tent,hexahedron)-free distance-hereditary graphs. Furthermore, our result is derived from a more general one that is obtained for b-Matching. In the (unit cost) b-Matching problem, we are given a graph G = (V, E) together with a nonnegative integer capacity b v for every vertex v $\in$ V. The objective is to assign nonnegative integer weights (x e) e$\in$E so that: for every v $\in$ V the sum of the weights of its incident edges does not exceed b v , and e$\in$E x e is maximized. We present the first algorithm for solving b-Matching on cographs, distance-hereditary graphs and some of their generalizations in quasi linear time. For that, we use a decomposition algorithm that outputs for any graph G a collection of subgraphs of G with no edge-cutsets inducing a complete bipartite subgraph (a.k.a., splits). The latter collection is sometimes called a split decomposition of G. Furthermore, there exists a generic method in order to design graph algorithms based on split decomposition [Rao, DAM'08]. However, this technique only applies to "localized" problems: for which a "best" partial solution for any given subgraph in a split decomposition can be computed almost independently from the remaining of the graph. Such framework does not apply to matching problems since an augmenting path may cross the subgraphs arbitrarily. We introduce a new technique that somehow captures all the partial solutions for a given union of subgraphs in a split decomposition, in a compact and amenable way for algorithms - assuming some piecewise linear assumption holds on the value of such solutions. The latter assumption is shown to hold for b-Matching. Doing so, we prove that solving b-Matching on any pair G, b can be reduced in quasi linear-time to solving this problem on a collection of smaller graphs: that are obtained from the subgraphs in any split decomposition of G by replacing every vertex with a constant-size module. In particular, if G has a split decomposition where all subgraphs have order at most a fixed k, then we can solve b-Matching for G, b in O((k log 2 k)$\times$(m+n)$\times$log ||b|| 1)-time. This answers an open question of [Coudert et al., SODA'18].

cs.DS↗

arXiv2018

The use of a pruned modular decomposition for Maximum Matching algorithms on some graph classes

We address the following general question: given a graph class C on which we can solve Maximum Matching in (quasi) linear time, does the same hold true for the class of graphs that can be modularly decomposed into C ? A major difficulty in this task is that the Maximum Matching problem is not preserved by quotient, thereby making difficult to exploit the structural properties of the quotient subgraphs of the modular decomposition. So far, we are only aware of a recent framework in [Coudert et al., SODA'18] that only applies when the quotient subgraphs have bounded order and/or under additional assumptions on the nontriv-ial modules in the graph. As a first attempt toward improving this framework we study the combined effect of modular decomposition with a pruning process over the quotient subgraphs. More precisely, we remove sequentially from all such subgraphs their so-called one-vertex extensions (i.e., pendant, anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a "pruned modular decomposition", that can be computed in O(m log n)-time. Our main result is that if all the pruned quotient subgraphs have bounded order then a maximum matching can be computed in linear time. This result is mostly based on two pruning rules on pendant and anti-pendant modules -- that are adjacent, respectively, to one or all but one other modules in the graph. Furthermore, these two latter rules are surprisingly intricate and we consider them as our main technical contribution in the paper. We stress that the class of graphs that can be totally decomposed by the pruned modular decomposition contains all the distance-hereditary graphs, and so, it is larger than cographs. In particular, as a byproduct of our approach we also obtain the first known linear-time algorithms for Maximum Matching on distance-hereditary graphs and graphs with modular-treewidth at most one. Finally, we can use an extended version of our framework in order to compute a maximum matching, in linear-time, for all graph classes that can be modularly decomposed into cycles. Our work is the first to explain why the existence of some nice ordering over the modules of a graph, instead of just over its vertices, can help to speed up the computation of maximum matchings on some graph classes.

cs.DS↗