scholarly journals $${\mathcal {U}}$$-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Algorithmica ◽  
2021 ◽  
Author(s):  
Jan Kratochvíl ◽  
Tomáš Masařík ◽  
Jana Novotná
Keyword(s):  
Unit Length ◽  
Algorithm Design ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Unit Interval ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Intersection Graphs ◽  
Proper Subclass ◽  
Maxcut Problem

AbstractInterval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. 10.1016/j.ipl.2017.01.007). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

scholarly journals Computing the Weighted Isolated Scattering Number of Interval Graphs in Polynomial Time

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Fengwei Li ◽  
Xiaoyan Zhang ◽  
Qingfang Ye ◽  
Yuefang Sun
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Interval Graphs ◽  
Perfect Graphs ◽  
Network Vulnerability ◽  
Hamiltonian Properties

The scattering number and isolated scattering number of a graph have been introduced in relation to Hamiltonian properties and network vulnerability, and the isolated scattering number plays an important role in characterizing graphs with a fractional 1-factor. Here we investigate the computational complexity of one variant, namely, the weighted isolated scattering number. We give a polynomial time algorithm to compute this parameter of interval graphs, an important subclass of perfect graphs.

scholarly journals Connected Subgraph Defense Games

Algorithmica ◽  
2021 ◽  
Author(s):  
Eleni C. Akrida ◽  
Argyrios Deligkas ◽  
Themistoklis Melissourgos ◽  
Paul G. Spirakis
Keyword(s):  
Polynomial Time ◽  
Nash Equilibria ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Connected Subgraph ◽  
Expected Number ◽  
Worst Case ◽  
Induced Subgraph ◽  
Other Hand

AbstractWe study a security game over a network played between a defender and kattackers. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of $$\lambda $$ λ nodes to scan and clean. Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizing defense-optimal networks which allow for the best equilibrium defense ratio; this is the ratio of k over the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. Our algorithm requires exponential time in the worst case, but it is polynomial-time for $$\lambda $$ λ constantly close to 1 or n. For the special case of tree-networks, we further refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal Nash equilibrium. On the other hand, we prove that it is $${\mathtt {NP}}$$ NP -hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for the Price of Defense for any $$\lambda $$ λ ; this is the worst equilibrium defense ratio over all graphs.

scholarly journals Large Cliques in Sparse Random Intersection Graphs

10.37236/6233 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Mindaugas Bloznelis ◽  
Valentas Kurauskas
Keyword(s):  
Degree Distribution ◽  
Maximum Clique ◽  
Polynomial Time Algorithm ◽  
Clique Number ◽  
Time Algorithm ◽  
Intersection Graph ◽  
Optimal Order ◽  
Intersection Graphs ◽  
Random Intersection Graph ◽  
Finite Set

An intersection graph defines an adjacency relation between subsets $S_1,\dots, S_n$ of a finite set $W=\{w_1,\dots, w_m\}$: the subsets $S_i$ and $S_j$ are adjacent if they intersect. Assuming that the subsets are drawn independently at random according to the probability distribution $\mathbb{P}(S_i=A)=P(|A|){\binom{m}{|A|}}^{-1}$, $A\subseteq W$, where $P$ is a probability on $\{0, 1, \dots, m\}$, we obtain the random intersection graph $G=G(n,m,P)$. We establish  the asymptotic order of the clique number $\omega(G)$ of  a sparse random intersection graph as $n,m\to+\infty$. For $m = \Theta(n)$ we show that the maximum clique is of size $(1-\alpha/2)^{-\alpha/2}n^{1-\alpha/2}(\ln n)^{-\alpha/2}(1+o_P(1))$ in the case where the asymptotic degree distribution of $G$ is a power-law with exponent $\alpha \in (1,2)$. It is of size $\frac {\ln n} {\ln \ln n}(1+o_P(1))$ if the degree distribution has bounded variance, i.e., $\alpha>2$. We construct a simple polynomial-time algorithm which finds a clique of the optimal order $\omega(G) (1-o_P(1))$.

scholarly journals Minimal Obstructions for Partial Representations of Interval Graphs

10.37236/5862 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Pavel Klavik ◽  
Maria Saumell
Keyword(s):  
Linear Time ◽  
Interval Graph ◽  
Interval Graphs ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Intersection Graphs ◽  
Certifying Algorithm ◽  
Partial Representation ◽  
Partial Representation Extension ◽  
Linear Time Algorithms

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals.  We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension.

scholarly journals Destroying Bicolored $P_3$s by Deleting Few Edges

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Niels Grüttemeier ◽  
Christian Komusiewicz ◽  
Jannik Schestag ◽  
Frank Sommer
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Np Hard ◽  
Induced Subgraph ◽  
Bounded Degree ◽  
Red Edge ◽  
Bounded Degree Graphs

We introduce and study the Bicolored $P_3$ Deletion problem defined as follows. The input is a graph $G=(V,E)$ where the edge set $E$ is partitioned into a set $E_r$ of red edges and a set $E_b$ of blue edges. The question is whether we can delete at most $k$ edges such that $G$ does not contain a bicolored $P_3$ as an induced subgraph. Here, a bicolored $P_3$ is a path on three vertices with one blue and one red edge. We show that Bicolored $P_3$ Deletion is NP-hard and cannot be solved in $2^{o(|V|+|E|)}$ time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored $P_3$ Deletion is polynomial-time solvable when $G$ does not contain a bicolored $K_3$, that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case that $G$ contains no blue $P_3$, red $P_3$, blue $K_3$, and red $K_3$. Finally, we show that Bicolored $P_3$ Deletion can be solved in $ O(1.84^k\cdot |V| \cdot |E|)$ time and that it admits a kernel with $ O(k\Delta\min(k,\Delta))$ vertices, where $\Delta$ is the maximum degree of $G$. Comment: 25 pages

scholarly journals The graph isomorphism problem on geometric graphs

2014 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Ryuhei Uehara
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Polynomial Hierarchy ◽  
Intersection Graphs ◽  
Graph Isomorphism Problem ◽  
Graph Classes

Special issue PRIMA 2013 International audience The graph isomorphism (GI) problem asks whether two given graphs are isomorphic or not. The GI problem is quite basic and simple, however, it\textquoterights time complexity is a long standing open problem. The GI problem is clearly in NP, no polynomial time algorithm is known, and the GI problem is not NP-complete unless the polynomial hierarchy collapses. In this paper, we survey the computational complexity of the problem on some graph classes that have geometric characterizations. Sometimes the GI problem becomes polynomial time solvable when we add some restrictions on some graph classes. The properties of these graph classes on the boundary indicate us the essence of difficulty of the GI problem. We also show that the GI problem is as hard as the problem on general graphs even for grid unit intersection graphs on a torus, that partially solves an open problem.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA 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 $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ 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 $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

Sign in / Sign up

Export Citation Format

Share Document