Search

This paper deals with decomposition of complete graphs on $n$ vertices into circulant graphs with reduced degree $r< n-1$ . They are denoted as $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ , where $a_{1}$ to $a_{m}$ are generators. Mathematical labeling for such bigger (higher order and huge size) and complex (strictly regular with so many triangles) graphs is very difficult. That is why after decomposition, an edge irregular $k$ -labeling for these subgraphs is computed with the help of algorithmic approach. Results of $k$ are computed by implementing this iterative algorithm in computer. Using the values of $k$ , an upper bound for edge irregularity strength is suggested for $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ that is ${\vert E\vert }/{2}\log _{2} \vert V\vert $ .

The Minimum Maximum Outdegree Problem (MMO) is to assign a direction to every edge in an input undirected, edge-weighted graph so that the maximum weighted outdegree taken over all vertices becomes as small as possible. In this paper, we introduce a new variant of MMO called the p-Split Minimum Maximum Outdegree Problem (p-Split-MMO) in which one is allowed to perform a sequence of p split operations on the vertices before orienting the edges, for some specified non-negative integer p, and study its computational complexity.

11th International Conference on Algorithms and Complexity ( 2019: Rome; Italy ), Hedonic games are a prominent model of coalition formation, in which each agent’s utility only depends on the coalition she resides. The subclass of hedonic games that models the formation of general partnerships [21], where output is shared equally among affiliates, is called hedonic games with common ranking property (HGCRP). Aside from their economic motivation, HGCRP came into prominence since they are guaranteed to have core stable solutions that can be found efficiently [2]. Nonetheless, a core stable solution is not necessarily a socially desirable (Pareto optimal) outcome. We improve upon existing results by proving that every instance of HGCRP has a solution that is both Pareto optimal and core stable. We establish that finding such a solution is, however, - by proving the stronger statement that finding any Pareto optimal solution is. We show that the gap between the total utility of a core stable solution and that of the socially optimal solution (OPT) is bounded by |N|, where N is the set of agents, and that this bound is tight. Our investigations reveal that finding a solution, whose total utility is within a constant factor of that of OPT, is intractable. © 2019, Springer Nature Switzerland AG.

We study the computational complexity of various problems related to synchronization of weakly acyclic automata, a subclass of widely studied aperiodic automata. We provide upper and lower bounds on the length of a shortest word synchronizing a weakly acyclic automaton or, more generally, a subset of its states, and show that the problem of approximating this length is hard. We investigate the complexity of finding a synchronizing set of states of maximum size. We also show inapproximability of the problem of computing the rank of a subset of states in a binary weakly acyclic automaton and prove that several problems related to recognizing a synchronizing subset of states in such automata are NP-complete., Comment: Extended and corrected version, including arXiv:1608.00889. Conference version was published at CIAA 2017, LNCS vol. 10329, pages 188-200, 2017

The $k$-dimensional Weisfeiler-Leman procedure ($k$-WL), which colors $k$-tuples of vertices in rounds based on the neighborhood structure in the graph, has proven to be immensely fruitful in the algorithmic study of Graph Isomorphism. More generally, it is of fundamental importance in understanding and exploiting symmetries in graphs in various settings. Two graphs are $k$-WL-equivalent if the $k$-dimensional Weisfeiler-Leman procedure produces the same final coloring on both graphs. 1-WL-equivalence is known as fractional isomorphism of graphs, and the $k$-WL-equivalence relation becomes finer as $k$ increases. We investigate to what extent standard graph parameters are preserved by $k$-WL-equivalence, focusing on fractional graph packing numbers. The integral packing numbers are typically NP-hard to compute, and we discuss applicability of $k$-WL-invariance for estimating the integrality gap of the LP relaxation provided by their fractional counterparts., Comment: 26 pages, 1 figure, major revision of the previous version

Given an undirected graph, a star partition is a partition of the nodes into subsets with at least two nodes so that the subgraph induced by each subset has a spanning star. Star partitions are related to well-known problems concerning domination in graphs and edge covering. We focus on the Constrained Star Partition Problem (CSP) that asks for finding a star partition of given cardinality. The problem is new and presents interesting peculiarities. We explore the relation between the cardinalities of star partitions and domatic bipartitions, showing that there are star partitions of any cardinality between minimum and maximum values, and that a similar but weaker result holds for domatic bipartitions. We study the computational complexity of different versions of star partition and domatic bipartition problems, proving that most of them, in particular CSP, constrained domatic bipartition and balanced domatic bipartition, are NP-complete. We also show that star partition problems are polynomial on trees and, more generally, on bounded treewidth graphs. We introduce an integer linear programming formulation that defines a polytope containing all the star partitions of a graph, showing that its vertices have only integral components for trees, which implies that linear programming can be used to solve weighted star partition problems on trees.

We consider two variations of graph total domination, namely, k-tuple total domination and total {k}-domination (for a fixed positive integer k). Their related decision problems are both NP-complete even for bipartite graphs. In this work, we study some subclasses of bipartite graphs. We prove the NP-completeness of both problems (for every fixed k) for bipartite planar graphs and we provide an APX-hardness result for the total domination problem for bipartite subcubic graphs. In addition, we introduce a more general variation of total domination (total (r,m)-domination) that allows us to design a specific linear time algorithm for bipartite distance-hereditary graphs. In particular, it returns a minimum weight total {k}-dominating function for bipartite distance-hereditary graphs. Fil: Argiroffo, Gabriela Rut. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina Fil: Leoni, Valeria Alejandra. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Torres, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura. Escuela de Ciencias Básicas; Argentina

The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of ${\rm gp}_d(G)$ with respect to the suitable values of $d$. We show that the decision problem concerning finding ${\rm gp}_d(G)$ is NP-complete for any value of $d$. The value of ${\rm gp}_d(G)$ when $G$ is a path or a cycle is computed and a structural characterization of general $d$-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that ${\rm gp}_d(G)$ is infinite whenever $G$ is an infinite graph and $d$ is a finite integer., Comment: 16 pages

© 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works., Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): the problem of finding a mapping of a request graph (describing a workload) onto a substrate graph (describing the physical infrastructure). Applications range from mapping testbeds, over the embedding of batch-processing workloads to the embedding of service function chains and come with different mapping restrictions for nodes and edges. The restrictions studied most often are node and edge capacities, node mapping and edge routing policies, and latency restrictions. While the VNEP has been studied intensively, complexity results are only known for specific models and this paper provides a first comprehensive study of the computational complexity of the VNEP by systematically analyzing its hardness for any combination of the above stated node and edge mapping restrictions. For all studied variants the NPcompleteness of their respective decision problems is shown. Furthermore, NP-completeness results for finding approximate embeddings, which may, e.g., violate capacity constraints by certain factors, are derived. Lastly, it is also shown that all these results pertain when restricting the request graphs to planar and degree-bounded graphs. While theoretic in nature, our results have severe practical implications. Firstly, any optimization variant of the VNEP is NP-hard and cannot be approximated for any of the studied restrictions, unless P=NP holds. Secondly, we uncover structural hardness properties: the VNEP is NP-hard and inapproximable even if, e.g., only node placement and edge routing restrictions are considered.

The problem of matching a query string to a directed graph, whose vertices are labeled by strings, has application in different fields, from data mining to computational biology. Several variants of the problem have been considered, depending on the fact that the match is exact or approximate and, in this latter case, which edit operations are considered and where are allowed. In this paper we present results on the complexity of the approximate matching problem, where edit operations are symbol substitutions and are allowed only on the graph labels or both on the graph labels and the query string. We introduce a variant of the problem that asks whether there exists a path in a graph that represents a query string with any number of edit operations and we show that is is NP-complete, even when labels have length one and in the case the alphabet is binary. Moreover, when it is parameterized by the length of the input string and graph labels have length one, we show that the problem is fixed-parameter tractable and it is unlikely to admit a polynomial kernel. The NP-completeness of this problem leads to the inapproximability (within any factor) of the approximate matching when edit operations are allowed only on the graph labels. Moreover, we show that the variants of approximate string matching to graph we consider are not fixed-parameter tractable, when the parameter is the number of edit operations, even for graphs that have distance one from a DAG. The reduction for this latter result allows us to prove the inapproximability of the variant where edit operations can be applied both on the query string and on graph labels., Comment: Extended version of a paper accepted to LATA 2020

Let k be a positive integer and let G be a graph with vertex set V(G). A subset D subset of V(G) is a k-dominating set if every vertex outside D is adjacent to at least k vertices in D. the k-domination number gamma(k)(G) is the minimum cardinality of a k-dominating set in G. For any graph G, we know that gamma(k)(G) >= gamma(G) + k - 2 where Delta(G) >= k >= 2 and this bound is sharp for every k >= 2. in this paper, we characterize bipartite graphs satisfying the equality for k >= 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k = 3. We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general., TUBITAK under the grant BIDEB 2221Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [1059B211800686]; Slovenian Research AgencySlovenian Research Agency - Slovenia [N1-0108], This research was started when Csilla Bujtas visited the Ege University in Izmir, Turkey; the authors thank the financial support of TUBITAK under the grant BIDEB 2221 (1059B211800686). Csilla Bujtas also acknowledges the financial support from the Slovenian Research Agency under the project N1-0108.

Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. [Math. Comput. Modelling, 1993, 17, 89-95]. Here we introduce a variation of the geodetic problem and call it strong edge geodetic problem. We illustrate how this problem is evolved from social transport networks. It is shown that the strong edge geodetic problem is NP-complete. We derive lower and upper bounds for the strong edge geodetic number and demonstrate that these bounds are sharp. We produce exact solutions for trees, block graphs, silicate networks and glued binary trees without randomization.

For a metric μ on a finite set T , the minimum 0-extension problem 0 - Ext [ μ ] is defined as follows: Given V ⊇ T and c : V 2 → Q + , minimize ∑ c ( x y ) μ ( γ ( x ) , γ ( y ) ) subject to γ : V → T , γ ( t ) = t ( ∀ t ∈ T ) , where the sum is taken over all unordered pairs in V . This problem generalizes several classical combinatorial optimization problems such as the minimum cut problem or the multiterminal cut problem. Karzanov and Hirai established a complete classification of metrics μ for which 0 - Ext [ μ ] is polynomial time solvable or NP-hard. This result can also be viewed as a sharpening of the general dichotomy theorem for finite-valued CSPs (Thapper and Živný 2016) specialized to 0 - Ext [ μ ] . In this paper, we consider a directed version 0 - Ext [ μ ] of the minimum 0-extension problem, where μ and c are not assumed to be symmetric. We extend the NP-hardness condition of 0 - Ext [ μ ] to 0 - Ext [ μ ] : If μ cannot be represented as the shortest path metric of an orientable modular graph with an orbit-invariant “directed” edge-length, then 0 - Ext [ μ ] is NP-hard. We also show a partial converse: If μ is a directed metric of a modular lattice with an orbit-invariant directed edge-length, then 0 - Ext [ μ ] is tractable. We further provide a new NP-hardness condition characteristic of 0 - Ext [ μ ] , and establish a dichotomy for the case where μ is a directed metric of a star.

Classical clustering problems search for a partition of objects into a fixed number of clusters. In many scenarios, however, the number of clusters is not known or necessarily fixed. Further, clusters are sometimes only considered to be of significance if they have a certain size. We discuss clustering into sets of minimum cardinality k without a fixed number of sets and present a general model for these types of problems. This general framework allows the comparison of different measures to assess the quality of a clustering. We specifically consider nine quality-measures and classify the complexity of the resulting problems with respect to k. Further, we derive some polynomial-time solvable cases for $$k=2$$k=2 with connections to matching-type problems which, among other graph problems, then are used to compute approximations for larger values of k.

We study the complexity of the problem of searching for a set of patterns that separate two given sets of strings. This problem has applications in a wide variety of areas, most notably in data mining, computational biology, and in understanding the complexity of genetic algorithms. We show that the basic problem of finding a small set of patterns that match one set of strings but do not match any string in a second set is difficult ( NP -complete, W [ 2 ] -hard when parameterized by the size of the pattern set, and APX -hard). We then perform a detailed parameterized analysis of the problem, separating tractable and intractable variants. In particular we show that parameterizing by the size of pattern set and the number of strings, and the size of the alphabet and the number of strings give FPT results, amongst others.

We study the computational complexity and approximability for the problem of partitioning a vertex-weighted undirected graph into p connected subgraphs with minimum gap between the largest and the smallest vertex weights.

Properties of graphs that can be characterized by the spectrum of the adjacency matrix of the graph have been studied systematically recently. Motivated by the complexity of these properties, we show that there are such properties for which testing whether a graph has that property can be NP-hard (or belong to other computational complexity classes consisting of even harder problems). In addition, we discuss a possible spectral characterization of some well-known NP-hard properties. In particular, for every integer k ≥ 6 we construct a pair of k -regular cospectral graphs, where one graph is Hamiltonian and the other one not.

Among $\mathbf{PSPACE}$-complete problems, QSAT, or quantified SAT, is one of the most used to show that the class of problems solvable in polynomial time by families of a given variant of P systems includes the whole $\mathbf{PSPACE}$. However, most solutions require a membrane nesting depth that is linear with respect to the number of variables of the QSAT instance under consideration. While a system of a certain depth is needed, since depth 1 systems only allows to solve problems in $\mathbf{P^{\#P}}$, it was until now unclear if a linear depth was, in fact, necessary. Here we use P systems with active membranes with charges, and we provide a construction that proves that QSAT can be solved with a sublinear nesting depth of order $\frac{n}{\log n}$, where $n$ is the number of variables in the quantified formula given as input., Comment: 19th International Conference on Membrane Computing (CMC19)

Given a graph G, a q-open neighborhood conflict-free coloring or q-ONCF-coloring is a vertex coloring $c:V(G) \rightarrow \{1,2,\ldots,q\}$ such that for each vertex $v \in V(G)$ there is a vertex in $N(v)$ that is uniquely colored from the rest of the vertices in $N(v)$. When we replace $N(v)$ by the closed neighborhood $N[v]$, then we call such a coloring a q-closed neighborhood conflict-free coloring or simply q-CNCF-coloring. In this paper, we study the NP-hard decision questions of whether for a constant q an input graph has a q-ONCF-coloring or a q-CNCF-coloring. We will study these two problems in the parameterized setting. First of all, we study running time bounds on FPT-algorithms for these problems, when parameterized by treewidth. We improve the existing upper bounds, and also provide lower bounds on the running time under ETH and SETH. Secondly, we study the kernelization complexity of both problems, using vertex cover as the parameter. We show that both $(q \geq 2)$-ONCF-coloring and $(q \geq 3)$-CNCF-coloring cannot have polynomial kernels when parameterized by the size of a vertex cover unless $NP \in coNP/poly$. However, we obtain a polynomial kernel for 2-CNCF-coloring parameterized by vertex cover. We conclude with some combinatorial results. Denote $\chi_{ON}(G)$ and $\chi_{CN}(G)$ to be the minimum number of colors required to ONCF-color and CNCF-color G, respectively. Upper bounds on $\chi_{CN}(G)$ with respect to structural parameters like minimum vertex cover size, minimum feedback vertex set size and treewidth are known. To the best of our knowledge only an upper bound on $\chi_{ON}(G)$ with respect to minimum vertex cover size was known. We provide tight bounds for $\chi_{ON}(G)$ with respect to minimum vertex cover size. Also, we provide the first upper bounds on $\chi_{ON}(G)$ with respect to minimum feedback vertex set size and treewidth., Comment: Accepted to WADS 2019

A $k$-subcoloring of a graph is a partition of the vertex set into at most $k$ cluster graphs, that is, graphs with no induced $P_3$. 2-subcoloring is known to be NP-complete for comparability graphs and three subclasses of planar graphs, namely triangle-free planar graphs with maximum degree 4, planar perfect graphs with maximum degree 4, and planar graphs with girth 5. We show that 2-subcoloring is also NP-complete for planar comparability graphs with maximum degree 4.

International audience; We study a 0-1 knapsack problem, in which the objective value is forbidden to take some values. We call gaps related forbidden intervals. The problem is NP-hard and pseudo-polynomially solvable independently on the measure of gaps. If the gaps are large, then the problem is polynomially non-approximable. A non-trivial special case with respect to the approximate solution appears when the gaps are small and polynomially close to zero. For this case, two fully polynomial time approximation schemes are proposed. The results can be extended for the constrained longest path problem and other combinatorial problems.

Gene tree correction has recently gained interest in phylogenomics, as it gives insights in understanding the evolution of gene families. Following some recent approaches based on leaf edit operations, we consider a variant of the problem where a gene tree is corrected by inserting leaves with labels in a multiset M. We show that the problem of deciding whether a gene tree can be corrected by inserting leaves with labels in M is NP-complete. Then, we consider an optimization variant of the problem that asks for the correction of a gene tree with leaves labeled by a multiset M ′ , with M ′ ⊇ M , having minimum size. For this optimization variant of the problem, we present a factor 2 approximation algorithm.

We apply the well-known Condorcet criterion from voting theory outside of its classical framework and link it with spanning trees of an undirected graph. In situations in which a network, represented by a spanning tree of an undirected graph, needs to be installed, decision-makers typically do not agree on the network to be implemented. Instead, each of these decision-makers has her own ideal conception of the network. In order to derive a group decision, i.e., a single spanning tree for the entire group of decision-makers, the goal would be a spanning tree that beats each other spanning tree in a simple majority comparison. When comparing two dedicated spanning trees, a decision-maker will be considered to be more satisfied with the one that is "closer" to her proposal. In this context, the most basic and natural measure of distance is the usual set difference: we simply count the number of edges the spanning tree has in common with the proposal of the decision-maker. In this work, we show that it is computationally intractable to decide (1) if such a spanning tree exists, and (2) if a given spanning tree satisfies the Condorcet criterion.

We study the Min-k-SCCP on the partition of a complete weighted digraph by k vertex-disjoint cycles of minimum total weight. This problem is the generalization of the well-known traveling salesman problem (TSP) and the special case of the classical vehicle routing problem (VRP). It is known that the problem Min-k-SCCP is strongly NP-hard and remains intractable even in the geometric statement. For the Euclidean Min-k-SCCP in Rd, we construct a polynomial-time approximation scheme, which generalizes the approach proposed earlier for the planar Min-2-SCCP. For any fixed c > 1, the scheme finds a (1 + 1/c)-approximate solution in time of O(nd+1(k log n)(O (√dc))d-1 2k).

Holant is a framework of counting characterized by local constraints. It is closely related to other well-studied frameworks such as #CSP and Graph Homomorphism. An effective dichotomy for such frameworks can immediately settle the complexity of all combinatorial problems expressible in that framework. Both #CSP and Graph Homomorphism can be viewed as sub-families of Holant with the additional assumption that the equality constraints are always available. Other sub-families of Holant such as Holant^* and Holant^c problems, in which we assume some specific sets of constraints to be freely available, were also studied. The Holant framework becomes more expressive and contains more interesting tractable cases with less or no freely available constraint functions, while, on the other hand, it also becomes more challenging to obtain a complete characterization of its time complexity. Recently, complexity dichotomy for a variety of sub-families of Holant such as #CSP, Graph Homomorphism, Holant^* and Holant^c were proved. The dichotomy for the general Holant framework, which is the most desirable, still remains open. In this paper, we prove a dichotomy for the general Holant framework where all the constraints are real symmetric functions. This setting already captures most of the interesting combinatorial problems defined by local constraints, such as (perfect) matching, independent set, vertex cover and so on. This is the first time a dichotomy is obtained for general Holant Problems without any auxiliary functions. One benefit of working with Holant framework is some powerful new reduction techniques such as Holographic reduction. Along the proof of our dichotomy, we introduce a new reduction technique, namely realizing a constraint function by approximating it. This new technique is employed in our proof in a situation where it seems that all previous reduction techniques fail, thus this new idea of reduction might also be of independent interest. Besides proving dichotomy and developing new technique, we also obtained some interesting by-products. We prove a dichotomy for #CSP restricting to instances where each variable appears a multiple of d times for any d. We also prove that counting the number of Eulerian-Orientations on 2k-regular graphs is #P-hard for any k>=2.

An impurity measures $I: \mathbb {R}^{d} \mapsto \mathbb {R}^{+}$ is a function that assigns a $d$ -dimensional vector $\mathbf {v}$ to a non-negative value $I(\mathbf {v})$ so that the more homogeneous $\mathbf {v}$ , with respect to the values of its coordinates, the larger its impurity. A well known example of impurity measures is the entropy impurity. We study the problem of clustering based on the entropy impurity measures. Let $V$ be a collection of $n$ many $d$ -dimensional vectors with non-negative components. Given $V$ and an impurity measure $I$ , the goal is to find a partition ${\mathcal{ P}}$ of $V$ into $k$ groups $V_{1},\ldots,V_{k}$ so as to minimize the sum of the impurities of the groups in ${\mathcal{ P}}$ , i.e., $I({\mathcal{ P}})= \sum _{i=1}^{k} I\left({\sum _{ \mathbf {v}\in V_{i}} \mathbf {v}}\right)$ . Impurity minimization has been widely used as quality assessment measure in probability distribution clustering (KL-divergence) as well as in categorical clustering. However, in contrast to the case of metric based clustering, the current knowledge of impurity measure based clustering in terms of approximation and inapproximability results is very limited. Here, we contribute to change this scenario by proving that the problem of finding a clustering that minimizes the Entropy impurity measure is APX-hard, i.e., there exists a constant $\epsilon > 0$ such that no polynomial time algorithm can guarantee $(1+\epsilon)$ -approximation under the standard complexity hypothesis $P \neq NP$ . The inapproximability holds even when all vectors have the same $\ell _{1}$ norm. This result provides theoretical limitations on the computational efficiency that can be achievable in the quantization of discrete memoryless channels, a problem that has recently attracted significant attention in the signal processing community. In addition, it also solve a question that remained open in previous work on this topic [Chaudhuri and McGregor COLT 08; Ackermann et. al. ECCC 11].

In this paper we investigate the computational complexity of motivating time-inconsistent agents to complete long term projects. We resort to an elegant graph-theoretic model, introduced by Kleinberg and Oren, which consists of a task graph $G$ with $n$ vertices, including a source $s$ and target $t$, and an agent that incrementally constructs a path from $s$ to $t$ in order to collect rewards. The twist is that the agent is present-biased and discounts future costs and rewards by a factor $\beta\in [0,1]$. Our design objective is to ensure that the agent reaches $t$ i.e.\ completes the project, for as little reward as possible. Such graphs are called motivating. We consider two strategies. First, we place a single reward $r$ at $t$ and try to guide the agent by removing edges from $G$. We prove that deciding the existence of such motivating subgraphs is NP-complete if $r$ is fixed. More importantly, we generalize our reduction to a hardness of approximation result for computing the minimum $r$ that admits a motivating subgraph. In particular, we show that no polynomial-time approximation to within a ratio of $\sqrt{n}/4$ or less is possible, unless ${\rm P}={\rm NP}$. Furthermore, we develop a $(1+\sqrt{n})$-approximation algorithm and thus settle the approximability of computing motivating subgraphs. Secondly, we study motivating reward configurations, where non-negative rewards $r(v)$ may be placed on arbitrary vertices $v$ of $G$. The agent only receives the rewards of visited vertices. Again we give an NP-completeness result for deciding the existence of a motivating reward configuration within a fixed budget $b$. This result even holds if $b=0$, which in turn implies that no efficient approximation of a minimum $b$ within a ration grater or equal to $1$ is possible, unless ${\rm P}={\rm NP}$.

In this work we investigate the complexity of some problems related to the {\em Simultaneous Embedding with Fixed Edges} (SEFE) of $k$ planar graphs and the PARTITIONED $k$-PAGE BOOK EMBEDDING (PBE-$k$) problems, which are known to be equivalent under certain conditions. While the computational complexity of SEFE for $k=2$ is still a central open question in Graph Drawing, the problem is NP-complete for $k \geq 3$ [Gassner {\em et al.}, WG '06], even if the intersection graph is the same for each pair of graphs ({\em sunflower intersection}) [Schaefer, JGAA (2013)]. We improve on these results by proving that SEFE with $k \geq 3$ and sunflower intersection is NP-complete even when the intersection graph is a tree and all the input graphs are biconnected. Also, we prove NP-completeness for $k \geq 3$ of problem PBE-$k$ and of problem PARTITIONED T-COHERENT $k$-PAGE BOOK EMBEDDING (PTBE-$k$) - that is the generalization of PBE-$k$ in which the ordering of the vertices on the spine is constrained by a tree $T$ - even when two input graphs are biconnected. Further, we provide a linear-time algorithm for PTBE-$k$ when $k-1$ pages are assigned a connected graph. Finally, we prove that the problem of maximizing the number of edges that are drawn the same in a SEFE of two graphs is NP-complete in several restricted settings ({\em optimization version of SEFE}, Open Problem $9$, Chapter $11$ of the Handbook of Graph Drawing and Visualization)., 29 pages, 10 figures, extended version of 'On Some NP-complete SEFE Problems' (Eighth International Workshop on Algorithms and Computation, 2014)

We study some problems of nodes localization in a Delaunay triangulation and problem-solving procedures. For the problem of the set of nodes the computationally efficient approach that uses Euclidean minimum spanning tree of Delaunay triangulation is proposed. Efficient estimations for computational comlexity of the proposed methods in the average and in the worst cases are proved.computational geometry, geometric search, Delaunay triangulation, merging of overlapping triangulations, unregular discrete mesh, computational complexity

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let D be a digraph and f a labeling of its vertices with positive integers; denote by S ( v ) the sum of labels over all neighbors of each vertex v. The labeling f is called topological additive numbering if S ( u ) < S ( v ) for each arc ( u , v ) of the digraph. The problem asks to find the minimum number k for which D has a topological additive numbering with labels belonging to { 1 , ? , k } , denoted by ? t ( D ) .We characterize when a digraph has topological additive numberings, give a lower bound for ? t ( D ) and provide an integer programming formulation for our problem. We also present some families for which ? t ( D ) can be computed in polynomial time. Finally, we prove that this problem is NP -Hard even when its input is restricted to planar bipartite digraphs. We define and study a problem which we call Topological Additive Numbering of Directed Acyclic Graphs.We prove that the existence of a topological additive numbering in a digraph is polynomial.We give a lower bound of the parameter in terms of a clique of the digraph.We give the exact value of the parameter for certain families of digraphs.We prove that the problem is NP-Hard even when the digraph is planar and bipartite.

The interplay between computational efficiency and statistical accuracy in high-dimensional inference has drawn increasing attention in the literature. In this paper, we study computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple non-overlapping) signal submatrix (of magnitude $\lambda$ and size $k_m \times k_n$) contaminated with a noise matrix (of size $m \times n$), we establish two transition thresholds for the signal to noise $\lambda/\sigma$ ratio in terms of $m$, $n$, $k_m$, and $k_n$. The first threshold, $\sf SNR_c$, corresponds to the computational boundary. Below this threshold, it is shown that no polynomial time algorithm can succeed in identifying the submatrix, under the \textit{hidden clique hypothesis}. We introduce adaptive linear time spectral algorithms that identify the submatrix with high probability when the signal strength is above the threshold $\sf SNR_c$. The second threshold, $\sf SNR_s$, captures the statistical boundary, below which no method can succeed with probability going to one in the minimax sense. The exhaustive search method successfully finds the submatrix above this threshold. The results show an interesting phenomenon that $\sf SNR_c$ is always significantly larger than $\sf SNR_s$, which implies an essential gap between statistical optimality and computational efficiency for submatrix localization., Comment: 37 pages, 1 figure

We consider the problem of finding a $k\times k$ submatrix of an $n\times n$ matrix with i.i.d. standard Gaussian entries, which has a large average entry. It was shown earlier by Bhamidi et al. that the largest average value of such a matrix is $2\sqrt{\log n/k}$ with high probability. In the same paper an evidence was provided that a natural greedy algorithm called Largest Average Submatrix ($\LAS$) should produce a matrix with average entry approximately $\sqrt{2}$ smaller. In this paper we show that the matrix produced by the $\LAS$ algorithm is indeed $\sqrt{2\log n/k}$ w.h.p. Then by drawing an analogy with the problem of finding cliques in random graphs, we propose a simple greedy algorithm which produces a $k\times k$ matrix with asymptotically the same average value. Since the greedy algorithm is the best known algorithm for finding cliques in random graphs, it is tempting to believe that beating the factor $\sqrt{2}$ performance gap suffered by both algorithms might be very challenging. Surprisingly, we show the existence of a very simple algorithm which produces a matrix with average value $(4/3)\sqrt{2\log n/k}$. To get an insight into the algorithmic hardness of this problem, and motivated by methods originating in the theory of spin glasses, we conduct the so-called expected overlap analysis of matrices with average value asymptotically $\alpha\sqrt{2\log n/k}$. The overlap corresponds to the number of common rows and common columns for pairs of matrices achieving this value. We discover numerically an intriguing phase transition at $\alpha^*\approx 1.3608..$: when $\alpha\alpha^*$. We conjecture that $\alpha>\alpha^*$ marks the onset of the algorithmic hardness., Comment: 38 pages 6 figures

For a fixed graph H, the H-Minor Edit problem takes as input a graph G and an integer k and asks whether G can be modified into H by a total of at most k edge contractions, edge deletions and vertex deletions. Replacing edge contractions by vertex dissolutions yields the H-Topological Minor Edit problem. For each problem we show polynomial-time solvable and NP-complete cases depending on the choice of H. Moreover, when G is AT-free, chordal or planar, we show that H-Minor Edit is polynomial-time solvable for all graphs H.

LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of $O(1)$, $\Theta(\log^* n)$, or $\Theta(n)$, and the design of optimal algorithms can be fully automated. This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: $O(1)$, $\Theta(\log^* n)$, and $\Theta(n)$. However, given an LCL problem it is undecidable whether its complexity is $\Theta(\log^* n)$ or $\Theta(n)$ in 2-dimensional grids. Nevertheless, if we correctly guess that the complexity of a problem is $\Theta(\log^* n)$, we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form $A' \circ S_k$, where $A'$ is a finite function, $S_k$ is an algorithm for finding a maximal independent set in $k$th power of the grid, and $k$ is a constant. Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations.

International audience; The problems studied in this paper originate from Graph Motif, a problem introduced in 2006 in the context of biological networks. Informally speaking, it consists in deciding if a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Due to the high rate of noise in the biological data, more flexible definitions of the problem have been outlined. We present in this paper two inapproximability results for two different optimization variants of Graph Motif: one where the size of the solution is maximized, the other when the number of substitutions of colors to obtain the motif from the solution is minimized. We also study a decision version of Graph Motif where the connectivity constraint is replaced by the well known notion of graph modularity. While the problem remains N P-complete, it allows algorithms in F P T for biologically relevant parameterizations.

International audience; In this paper, we consider the selective graph coloring problem. Given an integer k≥1k≥1 and a graph G=(V,E)G=(V,E) with a partition V1,…,VpV1,…,Vp of VV, it consists in deciding whether there exists a set V∗V∗ in GG such that ∣V∗∩Vi∣=1∣V∗∩Vi∣=1 for all i∈{1,…,p}i∈{1,…,p}, and such that the graph induced by V∗V∗ is kk-colorable. We investigate the complexity status of this problem in various classes of graphs.

In this work we obtain a new graph class where the {. k}-dominating function problem ({. k}-DOM) is NP-complete: the class of chordal graphs. We also identify some maximal subclasses for which it is polynomial time solvable. Firstly, by relating this problem with the k-dominating function problem (k-DOM), we prove that {. k}-DOM is polynomial time solvable for strongly chordal graphs. Besides, by expressing the property involved in k-DOM in Counting Monadic Second-order Logic, we obtain that both problems are linear time solvable for bounded tree-width graphs. Finally, we show that {. k}-DOM is linear time solvable for spider graphs. Fil: Argiroffo, Gabriela Rut. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina Fil: Leoni, Valeria Alejandra. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina Fil: Torres, Pablo Daniel. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina

In the edge searching problem searchers move from vertex to vertex in a graph to capture an invisible fast intruder that may occupy either vertices or edges. Fast searching is a monotonic internal model in which at every move a new edge of the graph G must be guaranteed to be free of the intruder. That is once all searchers are placed the graph G is cleared in exactly vertical bar E(G)vertical bar moves. Such a restriction obviously necessitates a larger number of searchers. We examine this model and characterize graphs for which 2 or 3 searchers are sufficient. We prove that the corresponding decision problem is NP-complete. (C) 2013 Elsevier B.V. All rights reserved.

We show that testing if an undirected graph contains a bridgeless spanning cactus is NP-hard. As a consequence, the minimum spanning cactus problem (MSCP) on an undirected graph with 0–1 edge weights is NP-hard. For any subgraph S of Kn, we give polynomially testable necessary and sufficient conditions for S to be extendable to a cactus in Kn and the weighted version of this problem is shown to be NP-hard. A spanning tree is shown to be extendable to a cactus in Kn if and only if it has at least one node of even degree. When S is a spanning tree, we show that the weighted version can also be solved in polynomial time. Further, we give an O(n3) algorithm for computing a minimum cost spanning tree with at least one vertex of even degree on a graph on n nodes. Finally, we show that for a complete graph with edge-costs satisfying the triangle inequality, the MSCP is equivalent to a general class of optimization problems that properly includes the traveling salesman problem and they all have the same approximation hardness.

The Induced Minor problem is to test whether a graph G contains a graph H as an induced minor, i.e., if G can be modified into H by a sequence of vertex deletions and edge contractions. When H is fixed, i.e., not part of the input, this problem is denoted H-Induced Minor. We provide polynomial-time algorithms for this problem in the case that the fixed target graph has a star-like structure. In particular, we show polynomial-time solvability for all forests H on at most seven vertices except for one such case.

Motivated by problems in comparative genomics and paleogenomics, we study the computational complexity of the Gapped Consecutive-Ones Property ((k,δ)-C1P) Problem: given a binary matrix M and two integers k and δ, decide if the columns of M can be permuted such that each row contains at most k blocks of ones and no two neighboring blocks of ones are separated by a gap of more than δ zeros. The classical C1P decision problem, which is known to be polynomial-time solvable is equivalent to the (1,0)-C1P problem. We extend our earlier results on this problem [C. Chauve, J. Mauch, M. Patterson, On the gapped consecutive-ones property, in: Proceedings of the European Conference on Combinatorics, Graphs Theory and Applications (EuroComb), in: Electronic Notes in Discrete Mathematics, vol. 34, 2009, pp. 121-125] to show that for every k

The Cluster Editing problem asks to transform a graph by at most k edge modifications into a disjoint union of cliques. The problem is NP-complete, but several parameterized algorithms are known. We present a novel search tree algorithm for the problem, which improves running time from O(1.76k+m+n) to O(1.62k+m+n) for m edges and n vertices. In detail, we can show that we can always branch with branching vector (2,1) or better, resulting in the golden ratio as the base of the search tree size. Our algorithm uses a well-known transformation to the integer-weighted counterpart of the problem. To achieve our result, we combine three techniques: First, we show that zero-edges in the graph enforce structural features that allow us to branch more efficiently. This is achieved by keeping track of the parity of merged vertices. Second, by repeatedly branching we can isolate vertices, releasing cost. Third, we use a known characterization of graphs with few conflicts. We then show that Integer-Weighted Cluster Editing remains NP-hard for graphs that have a particularly simple structure: namely, a clique minus the edges of a triangle.

A homomorphism from a graph GG to a graph HH is a vertex mapping f:VG→VHf:VG→VH such that f(u)f(u) and f(v)f(v) form an edge in HH whenever uu and vv form an edge in GG. The HH-Coloring problem is that of testing whether a graph GG allows a homomorphism to a given graph HH. A well-known result of Hell and Nešetřil determines the computational complexity of this problem for any fixed graph HH. We study a natural variant of this problem, namely the SurjectiveHH-Coloring problem, which is that of testing whether a graph GG allows a homomorphism to a graph HH that is (vertex-)surjective. We classify the computational complexity of this problem for when HH is any fixed partially reflexive tree. Thus we identify the first class of target graphs HH for which the computational complexity of SurjectiveHH-Coloring can be determined. For the polynomial-time solvable cases we show a number of parameterized complexity results, including in particular ones on graph classes with (locally) bounded expansion.

International audience; We study the problem of computing possible and necessary winners for partially specified weighted and unweighted tournaments. This problem arises naturally in elections with incompletely specified votes, partially completed sports competitions, and more generally in any scenario where the outcome of some pairwise comparisons is not yet fully known. We specifically consider a number of well-known solution concepts---including the uncovered set, Borda, ranked pairs, and maximin---and show that for most of them, possible and necessary winners can be identified in polynomial time. These positive algorithmic results stand in sharp contrast to earlier results concerning possible and necessary winners given partially specified preference profiles.

LNCS, vol. 10043; International audience; The problem of finding the maximum number of vertex-disjoint uni-color paths in an edge-colored graph (MaxCDP) has been recently introduced in literature, motivated by applications in social network analysis. In this paper we investigate how the complexity of the problem depends on graph parameters (distance from disjoint paths and size of vertex cover), and that is not FPT-approximable. Moreover, we introduce a new variant of the problem, called MaxCDDP, whose goal is to find the maximum number of vertex-disjoint and color-disjoint uni-color paths. We extend some of the results of MaxCDP to this new variant, and we prove that unlike MaxCDP, MaxCDDP is already hard on graphs at distance two from disjoint paths.

The spectrum of a first-order sentence is the set of natural numbers occurring as the cardinalities of finite models of the sentence. In a recent survey, Durand et al. introduce a new class of real numbers, the spectral reals, induced by spectra and pose two open problems associated to this class. In the present note, we answer these open problems as well as other open problems from an earlier, unpublished version of the survey. ¶ Specifically, we prove that (i) every algebraic real is spectral, (ii) every automatic real is spectral, (iii) the subword density of a spectral real is either 0 or 1, and both may occur, and (iv) every right-computable real number between 0 and 1 occurs as the subword entropy of a spectral real. ¶ In addition, Durand et al. note that the set of spectral reals is not closed under addition or multiplication. We extend this result by showing that the class of spectral reals is not closed under any computable operation satisfying some mild conditions.

Let $S=\{K_{1,3},K_3,P_4\}$ be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph $G$ into graphs taken from any non-empty $S'\subseteq S$. The problem is known to be NP-complete for any possible choice of $S'$ in general graphs. In this paper, we assume that the input graph is cubic, and study the computational complexity of the problem of partitioning its edge set for any choice of $S'$. We identify all polynomial and NP-complete problems in that setting, and give graph-theoretic characterisations of $S'$-decomposable cubic graphs in some cases., Comment: to appear in the proceedings of COCOON 2016

Computational complexity and approximation algorithms are reported for a problem of stabbing a set of straight line segments with the least cardinality set of disks of fixed radii $r>0$ where the set of segments forms a straight line drawing $G=(V,E)$ of a planar graph without edge crossings. Close geometric problems arise in network security applications. We give strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs and other subgraphs (which are often used in network design) for $r\in [d_{\min},\eta d_{\max}]$ and some constant $\eta$ where $d_{\max}$ and $d_{\min}$ are Euclidean lengths of the longest and shortest graph edges respectively. Fast $O(|E|\log|E|)$-time $O(1)$-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality $r\geq \eta d_{\max}$ holds uniformly for some constant $\eta>0,$ i.e. when lengths of edges of $G$ are uniformly bounded from above by some linear function of $r.$, Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International Conference on Analysis of Images, Social Networks and Texts (AIST-2017)

For any integers $d, n \geq 2$ and $1/({\min\{n,d\}})^{0.4999} < \varepsilon, Comment: v2: simplified proof, also added reference to Lev83

We prove several complexity results about the gap inequalities for the max-cut problem, including (i) the gap- 1 inequalities do not imply the other gap inequalities, unless NP = Co NP ; (ii) there must exist non-redundant gap inequalities with exponentially large coefficients, unless NP = Co NP ; (iii) the associated separation problem can be solved in finite (doubly exponential) time.