Minimum-size Set Research Articles

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems unlikely to avoid algebraic methods, without additional assumptions. In this paper, we introduce the notion of vision stability. In order to describe vision stability consider an enhanced guard that can see "around the corner" by an angle of $\delta$ or a diminished guard whose vision is by an angle of $\delta$ "blocked" by reflex vertices. A polygon $P$ has vision stability $\delta$ if the optimal number of enhanced guards to guard $P$ is the same as the optimal number of diminished guards to guard $P$. We will argue that most relevant polygons are vision stable. We describe a one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision stable polygon. We implemented an iterative visionstable algorithm and show its practical performance is slower, but comparable with other state of the art algorithms. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord $c$ of a polygon, we denote by $n(c)$ the number of vertices visible from $c$. The chord-width of a polygon is the maximum $n(c)$ over all possible chords $c$. The set of vision stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.

Independent domination in the graph defined by two consecutive levels of the [formula omitted]-cube

Fix a positive integer n and consider the bipartite graph whose vertices are the 3-element subsets and the 2-element subsets of [n]={1,2,…,n}, and there is an edge between A and B if A⊂B. We prove that the domination number of this graph is n2−⌊(n+1)28⌋, we characterize the dominating sets of minimum size, and we observe that the minimum size dominating set can be chosen as an independent set. This is an exact version of an asymptotic result by Balogh, Katona, Linz and Tuza (2021). For the corresponding bipartite graph between the (k+1)-element subsets and the k-element subsets of [n] (k⩾3), we provide a new construction for small independent dominating sets. This improves on a construction by Gerbner, Kezegh, Lemons, Palmer, Pálvölgyi and Patkós (2012), who studied these independent dominating sets under the name saturating flat antichains.

Efficient approaches for attaining epidemic-free networks with minimum edge removal set

Spreads can be contained through tuning the underlying contact networks, such as the social distance policy reducing the average degree of social contacts and the suspension of flights increasing the average length of travel distances. This paper studies how to find the optimal removal edge set of minimum size, such that the resulted network can survive from varied outbreaks. Specifically, we investigate the containment strategies from network epidemiology and immunization, and propose three novel methods that can well balance the transmission characteristics and topology of the remaining network and can thus effectively suppress varied spreads. In particular, the developed methods use the epidemic threshold to characterize the transmission characteristics and the largest connected component to measure the topology, and obtain the edge set by simultaneously optimizing them. We further introduce a bound strategy to scale up our methods, providing a time complexity of O(mlogω(n/ℓ)). We also conduct extensive experiments to evaluate the proposed methods. Results show that the developed approaches outperform the state-of-the-art by a large margin. Meanwhile, our methods are also much faster than those compared strategies. We are convinced that the proposed containment approaches promise to be effective tools to suppress spread on large-scale networks.

Classifications and constructions of minimum size linear sets on the projective line

This paper aims to study linear sets of minimum size on the projective line, that is Fq-linear sets of rank k in PG(1,qn) admitting one point of weight one and having size qk−1+1. Examples of these linear sets have been found by Lunardon and the second author (2000) and, more recently, by Jena and Van de Voorde (2021). However, classification results for minimum size linear sets are known only for k≤5. In this paper we provide classification results for minimum size linear sets admitting two points with complementary weights. We construct new examples (not necessarily with complementary weights) and also study the related ΓL(2,qn)-equivalence issue. These results solve an open problem posed by Jena and Van de Voorde. The main tool relies on two results by Bachoc, Serra and Zémor (2017 and 2018) on the linear analogues of Kneser's and Vosper's theorems. We then conclude the paper pointing out a connection between critical pairs and linear sets, obtaining also some classification results for critical pairs.

Minimal Leader Selection in General Linear Multi-Agent Systems With Switching Topologies: Leveraging Submodularity Ratio

In multi-agent systems with leader-follower dynamics, choosing a subset of agents as leaders is a critical step in achieving the desired coordination performance. In this study, by considering consensus tracking for general linear multi-agent systems under switching topologies, we address the problem of selecting a minimum-size set of leaders by leveraging the submodularity ratio. First, using the dwell time technique, a criterion is derived to ensure that the states of all agents can converge to a reference trajectory that is directly tracked by each leader. Second, exploiting the derived consensus tracking criterion, the metrics with a structure of the Euclidean distance between specific vectors and the space spanned by an iteratively updated matrix are established to identify a set of leaders, and then the corresponding bound of the submodularity ratio is proposed. Third, combining the derived criterion and the constructed metrics, a leader selection scheme is presented together with three polynomial-time algorithms, and the related provable optimality bound of each algorithm can be obtained by leveraging the proposed bound of the submodularity ratio. Finally, illustrative examples are provided to verify the effectiveness of the proposed leader selection scheme.

Approximation algorithms for some Minimum Postmen Cover Problems

In this work we introduce the Minimum Rural Postmen Cover Problem (MRPCP) and the Minimum Chinese Postmen Cover Problem (MCPCP). The MRPCP aims to cover a given subset R of edges of an undirected weighted graph G = ( V , E ) by a minimum size set of closed walks of bounded length λ . The MCPCP is a special case of the MRPCP with R = E . We give the first approximation algorithms for these two problems, which have constant approximation ratios of 5 and 4, respectively.

Simultaneously dominating all spanning trees of a graph

We investigate the problem of simultaneously dominating all spanning trees of a given graph. We prove that on 2-connected graphs, a subset of the vertices dominates all spanning trees of the graph if and only if it is a vertex cover. Using this fact we present an exact algorithm that finds a simultaneous dominating set of minimum size using an oracle for finding a minimum vertex cover. The algorithm can be implemented to run in polynomial time on several graph classes, such as bipartite or chordal graphs. We prove that there is no polynomial time algorithm that finds a minimum simultaneous dominating set on perfect graphs unless P=NP. Finally, we provide a 2-approximation algorithm for finding a minimum simultaneous dominating set.

Computational Complexity and ILP Models for Pattern Problems in the Logical Analysis of Data

Logical Analysis of Data is a procedure aimed at identifying relevant features in data sets with both positive and negative samples. The goal is to build Boolean formulas, represented by strings over {0,1,-} called patterns, which can be used to classify new samples as positive or negative. Since a data set can be explained in alternative ways, many computational problems arise related to the choice of a particular set of patterns. In this paper we study the computational complexity of several of these pattern problems (showing that they are, in general, computationally hard) and we propose some integer programming models that appear to be effective. We describe an ILP model for finding the minimum-size set of patterns explaining a given set of samples and another one for the problem of determining whether two sets of patterns are equivalent, i.e., they explain exactly the same samples. We base our first model on a polynomial procedure that computes all patterns compatible with a given set of samples. Computational experiments substantiate the effectiveness of our models on fairly large instances. Finally, we conjecture that the existence of an effective ILP model for finding a minimum-size set of patterns equivalent to a given set of patterns is unlikely, due to the problem being NP-hard and co-NP-hard at the same time.

Extremal absorbing sets in low-density parity-check codes

Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of $ b $ for a given $ a $ for which an $ (a,b) $-absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.

Metric Dimension Parameterized By Treewidth

A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time f(text {pw})n^{o(text {pw})} on n-vertex graphs of constant degree, with text {pw} the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter text {tl}+Delta, where text {tl} is the tree-length and Delta the maximum-degree of the input graph.

Algorithmic aspects of secure connected domination in graphs

Let $G = (V,E)$ be a simple, undirected and connected graph. A connected dominating set $S \subseteq V$ is a secure connected dominating set of $G$, if for each $ u \in V\setminus S$, there exists $v\in S$ such that $(u,v) \in E$ and the set $(S \setminus \{ v \}) \cup \{ u \} $ is a connected dominating set of $G$. The minimum size of a secure connected dominating set of $G$ denoted by $ \gamma_{sc} (G)$, is called the secure connected domination number of $G$. Given a graph $ G$ and a positive integer $ k,$ the Secure Connected Domination (SCDM) problem is to check whether $ G $ has a secure connected dominating set of size at most $ k.$ In this paper, we prove that the SCDM problem is NP-complete for doubly chordal graphs, a subclass of chordal graphs. We investigate the complexity of this problem for some subclasses of bipartite graphs namely, star convex bipartite, comb convex bipartite, chordal bipartite and chain graphs. The Minimum Secure Connected Dominating Set (MSCDS) problem is to find a secure connected dominating set of minimum size in the input graph. We propose a $ (\Delta(G)+1) $ - approximation algorithm for MSCDS, where $ \Delta(G) $ is the maximum degree of the input graph $ G $ and prove that MSCDS cannot be approximated within $ (1 -\epsilon) ln(| V |)$ for any $ \epsilon > 0 $ unless $ NP \subseteq DTIME(| V |^{O(log log | V |)})$ even for bipartite graphs. Finally, we show that the MSCDS is APX-complete for graphs with $\Delta(G)=4$.

Target set selection with maximum activation time

A target set selection model is a graph G with a threshold function τ : V(G) → N upper-bounded by the vertex degree. For a given model, a set S0 ⊆ V(G) is a target set if V(G) can be partitioned into non-empty subsets S0, S1,....,St such that, for all i∈{1,....,t}, Si contains exactly every vertex v having at least τ(v) neighbors in S0∪⋯∪Si−1. We say that t is the activation time tτ(S0) of the target set S0. The problem of, given such a model, finding a target set of minimum size has been extensively studied in the literature. In this article, we investigate its variant, which we call TSS-time, in which the goal is to find a target set S0 that maximizes tτ(S0). That is, given a graph G, a threshold function τ in G, and an integer k, the objective of the TSS-time problem is to decide whether G contains a target set S0 such that tτ(S0)≥k. Let τ*=maxV∈V(G)τ(v). Our main result is the following dichotomy about the complexity of TSS-time when G belongs to a minor-closed graph class C: if C has bounded local treewidth, the problem is FPT parameterized by k and τ*; otherwise, it is NP-complete even for fixed k = 4 and τ* = 2. We also prove that, with τ = 2, the problem is NP-hard in bipartite graphs for fixed k = 5, and from previous results we observe that TSS-time is NP-hard in planar graphs and W[1]-hard parameterized by treewidth. Finally, we present a linear-time algorithm to find a target set S0 in a given tree maximizing tτ(S0).

New Method for Optimal Feature Set Reduction

A problem of searching a minimum-size feature set to use in distribution of multidimensional objects in classes, for instance with the help of classifying trees, is considered. It has an important value in developing high speed and accuracy classifying systems. A short comparative review of existing approaches is given. Formally, the problem is formulated as finding a minimum-size (minimum weighted sum) covering set of discriminating 0,1-matrix, which is used to represent capabilities of the features to distinguish between each pair of objects belonging to different classes. There is given a way to build a discriminating 0,1-matrix. On the basis of the common solving principle, called the group resolution principle, the following problems are formulated and solved: finding an exact minimum-size feature set; finding a feature set with minimum total weight among all the minimum-size feature sets (the feature weights may be defined by the known methods, e.g. the RELIEF method and its modifications); finding an optimal feature set with respect to fuzzy data and discriminating matrix elements belonging to diapason [0,1]; finding statistically optimal solution especially in the case of big data. Statistically optimal algorithm makes it possible to restrict computational time by a polynomial of the problem sizes and density of units in discriminating matrix and provides a probability of finding an exact solution close to 1.
 Thus, the paper suggests a common approach to finding a minimum-size feature set with peculiarities in problem formulation, which differs it from the known approaches. The paper contains a lot of illustrations for clarification aims. Some theoretical statements given in the paper are based on the previously published works.
 In the concluding part, the results of the experiments are presented, as well as the information on dimensionality reduction for the coverage problem for big datasets. Some promising directions of the outlined approach are noted, including working with incomplete and categorical data, integrating the control model into the data classification system.

Algorithmic Aspects of Some Variants of Domination in Graphs

Abstract A set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E, i.e., N[S] = V . A dominating set S is an isolate dominating set (IDS) if the induced subgraph G[S] has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set S ⊆ V is an independent set if G[S] has no edge. A set S ⊆ V is a secure dominating set of G if, for each vertex u ∈ V \ S, there exists a vertex v ∈ S such that (u, v) ∈ E and (S \ {v}) ∪ {u} is a dominating set of G. In addition, we initiate the study of a new domination parameter called, independent secure domination. A set S ⊆ V is an independent secure dominating set (InSDS) if S is an independent set and a secure dominating set of G. The minimum size of an InSDS in G is called the independent secure domination number of G and is denoted by γ is (G). Given a graph G and a positive integer k, the InSDM problem is to check whether G has an independent secure dominating set of size at most k. We prove that InSDM is NP-complete for bipartite graphs and linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. The MInSDS problem is to find an independent secure dominating set of minimum size, in the input graph. Finally, we show that the MInSDS problem is APX-hard for graphs with maximum degree 5.

On linear sets of minimum size

An Fq-linear set of rank k, k≤h, on a projective line PG(1,qh), containing at least one point of weight one, has size at least qk−1+1 (see De Beule and Van De Voorde (2019)). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between k∕2 and k−1. Our construction extends the known examples of linear sets of size qk−1+1 in PG(1,qh) constructed for k=h=4 (Bonoli and Polverino, 2005) and k=h in Lunardon and Polverino (2000). We determine the weight distribution of the constructed linear sets and describe them as the projection of a subgeometry. For small k, we investigate whether all linear sets of size qk−1+1 arise from our construction.Finally, we modify our construction to define rank k linear sets of size qk−1+qk−2+…+qk−l+1 in PG(l,qh). This leads to new infinite families of small minimal blocking sets which are not of Rédei type.

Liar’s dominating set problem on unit disk graphs

In this paper, we consider Euclidean versions of the 2-tuple dominating set problem and the liar’s dominating set problem. For a given set P={p1,p2,…,pn} of n points in R2, the objective of the Euclidean 2-tuple dominating set problem is to find a minimum size set D⊆P such that |N[pi]∩D|≥2 for each pi∈P, where N[pi]={pj∈P∣δ(pi,pj)≤1} and δ(pi,pj) is the Euclidean distance between pi and pj. The objective of the Euclidean liar’s dominating set problem is to find a set D(⊆P) of minimum size satisfying the following two conditions: (i) D is a 2-tuple dominating set of P, and (ii) for every distinct pair of points pi and pj in P, |(N[pi]∪N[pj])∩D|≥3.We first propose a simple O(nlogn) time 632-factor approximation algorithm for the Euclidean liar’s dominating set problem. Next, we propose approximation algorithms to improve the approximation factor to 732α for 3≤α≤183, and 846α for 3≤α≤282. The running time of both the algorithms is O(nα+1Δ), where Δ=max{|N[p]|:p∈P}. Finally, we propose a PTAS for the Euclidean 2-tuple dominating set problem.

Subset feedback vertex set on graphs of bounded independent set size

The (Weighted) Subset Feedback Vertex Set problem is a generalization of the classical Feedback Vertex Set problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although Subset Feedback Vertex Set and Feedback Vertex Set exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of structural graph parameters. Here we consider graphs of bounded independent set number for which it is known that Weighted Feedback Vertex Set can be solved in polynomial time. We provide a dichotomy result with respect to the size α of a maximum independent set. In particular we show that Weighted Subset Feedback Vertex Set can be solved in polynomial time for graphs with α≤3, whereas we prove that the problem remains NP-hard for graphs with α≥4. Moreover, we show that the (unweighted) Subset Feedback Vertex Set problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time nO(α). To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size. Node Multiway Cut is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals. Based on our findings for Subset Feedback Vertex Set, we settle the complexity of Node Multiway Cut as well as its variants where nodes are weighted and/or the terminals are deletable, for every value of the given independent set number.

The complexity of the defensive domination problem in special graph classes

A k-attack on a graph G is a set of k distinct vertices {a1,…,ak} which are said to be under attack. A k-attack A can be countered or defended by a subset of defender vertices X if and only if there exists an injective function f from A to X, such that either f(ai)=ai or (ai,f(ai)) is an edge of G, for all i, 1≤i≤k. Given a graph G, a subset D of V is a k-defensive dominating set of G if and only if D can counter any k-attack in G. We consider the k-defensive dominating set problem.We start a systematic study on the complexity of the k-defensive dominating set problem. When k is not fixed, we show that it is already co-NP-hard to decide whether a given set of vertices is a k-defensive dominating set or not. However, if k is fixed, then we show that the problem is NP-complete even in split graphs. Subsequently, we consider special graph classes, namely paths, cycles, co-chain graphs and threshold graphs. We show that in each of these graph classes, a k-defensive dominating set of minimum size can be found in linear time even for non-fixed k.

Minimal Input and Output Selection for Stability of Systems With Uncertainties

In networked control systems, selecting a subset of input and output nodes is a crucial step in designing a stabilizing controller. Most existing approaches to input and output selection focus on nominal systems with known parameters. For systems with uncertainties and time delays, current selection methods are based on exploiting the convexity after relaxing the original problem, which is inherently discrete, to continuous optimization forms, and hence lack optimality guarantees. This paper studies the problem of identifying the minimum-size sets of input and output nodes to guarantee stability of a linear system with uncertainties and time delays. We derive sufficient conditions to guarantee the existence of a stabilizing controller for an uncertain linear system, based on a subset of system modes lying within the controllability and observability subspaces induced by the selected inputs and outputs. We then formulate the problems of selecting minimum-size sets of input and output nodes to satisfy the derived conditions, and prove that they are equivalent to discrete optimization problems with bounded submodularity ratios. We develop polynomial-time selection algorithms with provable guarantees on the minimum number of inputs and outputs required. Our approach is applicable to various types of uncertainties, including additive uncertainty, multiplicative uncertainty, uncertain output delay, and structured uncertainty. In a numerical study, we test our approach for the wide-area damping control in power systems to ensure small signal stability. Our results are validated on the IEEE 39-bus test power system.

Scalable approximations to k-cycle transversal problems on dynamic networks

We study scalable approximation algorithms for the k-cycle transversal problem, which is to find a minimum-size set of edges that intersects all simple cycles of length k in a network. This problem is relevant to network reliability through the important metric of network clustering coefficient of order k. We formulate two algorithms to be both scalable and have good solution quality in practice: CARL and DARC. DARC is able to efficiently update its solution under dynamic node and edge insertion and removal to the network. In our experimental evaluation, we demonstrate that DARC is able to run on networks with billions of 3-cycles within 2 h and is able to dynamically update its solution in microseconds.