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 Complexity of Strong Approximation on the Sphere

2019 ◽  
Author(s):  
Naser T Sardari
Keyword(s):  
Polynomial Time ◽  
Strong Approximation ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Ramanujan Graphs ◽  
Other Hand ◽  
Np Complete

Abstract By assuming some widely believed arithmetic conjectures, we show that the task of accepting a number that is representable as a sum of $d\geq 2$ squares subjected to given congruence conditions is NP-complete. On the other hand, we develop and implement a deterministic polynomial-time algorithm that represents a number as a sum of four squares with some restricted congruence conditions, by assuming a polynomial-time algorithm for factoring integers and Conjecture 1.1. As an application, we develop and implement a deterministic polynomial-time algorithm for navigating Lubotzky, Phillips, Sarnak (LPS) Ramanujan graphs, under the same assumptions.

THE EDIT-DISTANCE BETWEEN A REGULAR LANGUAGE AND A CONTEXT-FREE LANGUAGE

2013 ◽  
Vol 24 (07) ◽  
pp. 1067-1082 ◽  
Author(s):  
YO-SUB HAN ◽  
SANG-KI KO ◽  
KAI SALOMAA
Keyword(s):  
Regular Language ◽  
Edit Distance ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Optimal Alignment ◽  
Worst Case ◽  
Context Free Language ◽  
Context Free ◽  
Free Language

The edit-distance between two strings is the smallest number of operations required to transform one string into the other. The distance between languages L1and L2is the smallest edit-distance between string wi∈ Li, i = 1, 2. We consider the problem of computing the edit-distance of a given regular language and a given context-free language. First, we present an algorithm that finds for the languages an optimal alignment, that is, a sequence of edit operations that transforms a string in one language to a string in the other. The length of the optimal alignment, in the worst case, is exponential in the size of the given grammar and finite automaton. Then, we investigate the problem of computing only the edit-distance of the languages without explicitly producing an optimal alignment. We design a polynomial time algorithm that calculates the edit-distance based on unary homomorphisms.

scholarly journals Maximin-Aware Allocations of Indivisible Goods

2019 ◽  
Author(s):  
Hau Chan ◽  
Jing Chen ◽  
Bo Li ◽  
Xiaowei Wu
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Indivisible Goods ◽  
Know How

We study envy-free allocations of indivisible goods to agents in settings where each agent is unaware of the goods allocated to other agents. In particular, we propose the maximin aware (MMA) fairness measure, which guarantees that every agent, given the bundle allocated to her, is aware that she does not envy at least one other agent, even if she does not know how the other goods are distributed among other agents. We also introduce two of its relaxations, and discuss their egalitarian guarantee and existence. Finally, we present a polynomial-time algorithm, which computes an allocation that approximately satisfies MMA or its relaxations. Interestingly, the returned allocation is also 1/2-approximate EFX when all agents have sub- additive valuations, which improves the algorithm in [Plaut and Roughgarden, 2018].

scholarly journals Polynomial Time Instances for the IKHO Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Romeo Rizzi ◽  
Luca Nardin
Keyword(s):  
Dynamic Programming ◽  
Polynomial Time ◽  
Broad Class ◽  
Structural Parameters ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Heuristic Optimization ◽  
Negative Fact

The Interactive Knapsacks Heuristic Optimization (IKHO) problem is a particular knapsacks model in which, given an array of knapsacks, every insertion in a knapsack affects also the other knapsacks, in terms of weight and profit. The IKHO model was introduced by Isto Aho to model instances of the load clipping problem. The IKHO problem is known to be APX-hard and, motivated by this negative fact, Aho exhibited a few classes of polynomial instances for the IKHO problem. These instances were obtained by limiting the ranges of two structural parameters, c and u, which describe the extent to which an insertion in a knapsack in uences the nearby knapsacks. We identify a new and broad class of instances allowing for a polynomial time algorithm. More precisely, we show that the restriction of IKHO to instances where is bounded by a constant can be solved in polynomial time, using dynamic programming.

Efficient Processing of XML Tree Pattern Queries

2006 ◽  
Vol 10 (5) ◽  
pp. 738-743 ◽  
Author(s):  
Yangjun Chen ◽  
◽  
Dunren Che ◽  
Keyword(s):  
Dynamic Programming ◽  
Polynomial Time ◽  
Main Idea ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Tree Pattern ◽  
Efficient Processing

In this paper, we present a polynomial-time algorithm for TPQ (tree pattern queries) minimization without XML constraints involved. The main idea of the algorithm is a dynamic programming strategy to find all the matching subtrees within a TPQ. A matching subtree implies a redundancy and should be removed in such a way that the semantics of the original TPQ is not damaged. Our algorithm consists of two parts: one for subtree recognization and the other for subtree deletion. Both of them needs only O(<I>n</I>2) time, where <I>n</I> is the number of nodes in a TPQ.

A POLYNOMIAL-TIME ALGORITHM FOR COMPUTING THE RESILIENCE OF ARRANGEMENTS OF RAY SENSORS

2014 ◽  
Vol 24 (03) ◽  
pp. 225-236 ◽  
Author(s):  
DAVID KIRKPATRICK ◽  
BOTING YANG ◽  
SANDRA ZILLES
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Np Hard ◽  
Sensor Coverage ◽  
Entire Plane ◽  
Line Segments ◽  
Minimum Number

Given an arrangement A of n sensors and two points s and t in the plane, the barrier resilience of A with respect to s and t is the minimum number of sensors whose removal permits a path from s to t such that the path does not intersect the coverage region of any sensor in A. When the surveillance domain is the entire plane and sensor coverage regions are unit line segments, even with restricted orientations, the problem of determining the barrier resilience is known to be NP-hard. On the other hand, if sensor coverage regions are arbitrary lines, the problem has a trivial linear time solution. In this paper, we study the case where each sensor coverage region is an arbitrary ray, and give an O(n2m) time algorithm for computing the barrier resilience when there are m ⩾ 1 sensor intersections.

FROM C-CONTINUATIONS TO NEW QUADRATIC ALGORITHMS FOR AUTOMATON SYNTHESIS

2001 ◽  
Vol 11 (06) ◽  
pp. 707-735 ◽  
Author(s):  
J.-M. CHAMPARNAUD ◽  
D. ZIADI
Keyword(s):  
Time Complexity ◽  
Regular Expression ◽  
Time Algorithm ◽  
The Other ◽  
Worst Case ◽  
Partial Derivatives ◽  
Space And Time ◽  
Other Hand ◽  
Deterministic Automata ◽  
The Way

Two classical non-deterministic automata recognize the language denoted by a regular expression: the position automaton which deduces from the position sets defined by Glushkov and McNaughton–Yamada, and the equation automaton which can be computed via Mirkin's prebases or Antimirov's partial derivatives. Let |E| be the size of the expression and ‖E‖ be its alphabetic width, i.e. the number of symbol occurrences. The number of states in the equation automaton is less than or equal to the number of states in the position automaton, which is equal to ‖E‖+1. On the other hand, the worst-case time complexity of Antimirov algorithm is O(‖E‖3· |E|2), while it is only O(‖E‖·|E|) for the most efficient implementations yielding the position automaton (Brüggemann–Klein, Chang and Paige, Champarnaud et al.). We present an O(|E|2) space and time algorithm to compute the equation automaton. It is based on the notion of canonical derivative which makes it possible to efficiently handle sets of word derivatives. By the way, canonical derivatives also lead to a new O(|E|2) space and time algorithm to construct the position automaton.

scholarly journals ADVICE FOR SEMIFEASIBLE SETS AND THE COMPLEXITY-THEORETIC COST(LESSNESS) OF ALGEBRAIC PROPERTIES

2005 ◽  
Vol 16 (05) ◽  
pp. 913-928 ◽  
Author(s):  
PIOTR FALISZEWSKI ◽  
LANE A. HEMASPAANDRA
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Worst Case ◽  
Advice Complexity ◽  
The Core ◽  
Case Complexity ◽  
Worst Case Complexity ◽  
Algebraic Properties

Informally put, the semifeasible sets are the class of sets having a polynomial-time algorithm that, given as input any two strings of which at least one belongs to the set, will choose one that does belong to the set. We provide a tutorial overview of the advice complexity of the semifeasible sets. No previous familiarity with either the semifeasible sets or advice complexity will be assumed, and when we include proofs we will try to make the material as accessible as possible via providing intuitive, informal presentations. Karp and Lipton introduced advice complexity about a quarter of a century ago.18 Advice complexity asks, for a given power of interpreter, how many bits of "help" suffice to accept a given set. Thus, this is a notion that contains aspects both of descriptional/informational complexity and of computational complexity. We will see that for some powers of interpreter the (worst-case) complexity of the semifeasible sets is known right down to the bit (and beyond), but that for the most central power of interpreter—deterministic polynomial time—the complexity is currently known only to be at least linear and at most quadratic. While overviewing the advice complexity of the semifeasible sets, we will stress also the issue of whether the functions at the core of semifeasibility—so-called selector functions—can without cost be chosen to possess such algebraic properties as commutativity and associativity. We will see that this is relevant, in ways both potential and actual, to the study of the advice complexity of the semifeasible sets.

scholarly journals POLYNOMIAL-TIME COMPLEXITY FOR INSTANCES OF THE ENDOMORPHISM PROBLEM IN FREE GROUPS

2007 ◽  
Vol 17 (02) ◽  
pp. 289-328 ◽  
Author(s):  
LAURA CIOBANU
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Efficient Algorithm ◽  
Time Complexity ◽  
Free Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Polynomial Time Complexity

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism ϕ of F sending W to U. This work analyzes an approach due to Edmunds and improved by Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two-generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side.

scholarly journals Polynomial-Time Algorithm for Learning Optimal BFS-Consistent Dynamic Bayesian Networks

Entropy ◽  
2018 ◽  
Vol 20 (4) ◽  
pp. 274 ◽  
Author(s):  
◽  
Keyword(s):  
Bayesian Networks ◽  
Polynomial Time ◽  
Learning Algorithm ◽  
Search Space ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Dynamic Bayesian Networks ◽  
Worst Case ◽  
Current Time ◽  
Transition Networks

Dynamic Bayesian networks (DBN) are powerful probabilistic representations that model stochastic processes. They consist of a prior network, representing the distribution over the initial variables, and a set of transition networks, representing the transition distribution between variables over time. It was shown that learning complex transition networks, considering both intra- and inter-slice connections, is NP-hard. Therefore, the community has searched for the largest subclass of DBNs for which there is an efficient learning algorithm. We introduce a new polynomial-time algorithm for learning optimal DBNs consistent with a breadth-first search (BFS) order, named bcDBN. The proposed algorithm considers the set of networks such that each transition network has a bounded in-degree, allowing for p edges from past time slices (inter-slice connections) and k edges from the current time slice (intra-slice connections) consistent with the BFS order induced by the optimal tree-augmented network (tDBN). This approach increases exponentially, in the number of variables, the search space of the state-of-the-art tDBN algorithm. Concerning worst-case time complexity, given a Markov lag m, a set of n random variables ranging over r values, and a set of observations of N individuals over T time steps, the bcDBN algorithm is linear in N, T and m; polynomial in n and r; and exponential in p and k. We assess the bcDBN algorithm on simulated data against tDBN, revealing that it performs well throughout different experiments.

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