Polynomial-time Constant-factor Approximation Algorithm Research Articles

Minimizing the Maximum Length of Flight Paths for UAVs Providing Location Service to Ground Targets

When a UAV broadcasts its location periodically during flight, a target will observe the strongest signal when the UAV is at the closest location; thus, the target can infer its location based on the observed signals and the UAV’s trajectory. We consider how to design trajectories for UAVs such that all targets can locate themselves, and the maximum length of trajectories is minimized. To make the problem tractable, we impose constraints on the set of possible trajectories such that they only contain the edges of small squares. We propose an integer-linear-programming-based solution, which runs in exponential time, and two polynomial-time constant-factor approximation algorithms. We implement a prototype system to verify the feasibility of the localization method. Simulations show that our approach outperforms the existing ones in terms of flight length and localization error.

Matching-based capture strategies for 3D heterogeneous multiplayer reach-avoid differential games

This paper studies a 3D multiplayer reach-avoid differential game with a goal region and a play region. Multiple pursuers defend the goal region by consecutively capturing multiple evaders in the play region. The players have heterogeneous moving speeds and the pursuers have heterogeneous capture radii. Since this game is hard to analyze directly, we decompose the whole game as many subgames which involve multiple pursuers and only one evader. Then, these subgames are used as a building block for the pursuer–evader matching. First, for multiple pursuers and one evader, we introduce an evasion space (ES) method characterized by a potential function to construct a guaranteed pursuer winning strategy. Then, based on this strategy, we develop conditions to determine whether a pursuit team can guard the goal region against one evader. It is shown that in 3D, if a pursuit team is able to defend the goal region against an evader, then at most three pursuers in the team are necessarily needed. We also compute the value function of the Hamilton–Jacobi–Isaacs (HJI) equation for a special subgame of degree. To capture the maximum number of evaders in the open-loop sense, we formulate a maximum bipartite matching problem with conflict graph (MBMC). We show that the MBMC is NP-hard and design a polynomial-time constant-factor approximation algorithm to solve it. Finally, we propose a receding horizon strategy for the pursuit team where in each horizon an MBMC is solved and the strategies of the pursuers are given. We also extend our results to the case of a bounded convex play region where the evaders escape through an exit. Two numerical examples are provided to demonstrate the results.

Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

We consider a well-studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and a fixed parameter $$d\ge 1$$ , in the maximum diameter-bounded subgraph problem (MaxDBS for short) the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For $$d=1$$ , this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor $$n^{1-\epsilon }$$ , for any $$\epsilon >0$$ . Moreover, it is known that, for any $$d\ge 2$$ , it is NP-hard to approximate MaxDBS within a factor $$n^{1/2-\epsilon }$$ , for any $$\epsilon >0$$ . In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems, and several geometric properties of unit disk graphs.

Maximum parsimony distance on phylogenetic trees: A linear kernel and constant factor approximation algorithm

Maximum parsimony distance is a measure used to quantify the dissimilarity of two unrooted phylogenetic trees. It is NP-hard to compute, and very few positive algorithmic results are known due to its complex combinatorial structure. Here we address this shortcoming by showing that the problem is fixed parameter tractable. We do this by establishing a linear kernel i.e., that after applying certain reduction rules the resulting instance has size that is bounded by a linear function of the distance. As powerful corollaries to this result we prove that the problem permits a polynomial-time constant-factor approximation algorithm; that the treewidth of a natural auxiliary graph structure encountered in phylogenetics is bounded by a function of the distance; and that the distance is within a constant factor of the size of a maximum agreement forest of the two trees, a well studied object in phylogenetics.

On the Complexity and Approximability of Optimal Sensor Selection and Attack for Kalman Filtering

Given a linear dynamical system affected by stochastic noise, we consider the problem of selecting an optimal set of sensors (at design-time) to minimize the trace of the steady state a priori or a posteriori error covariance of the Kalman filter, subject to certain selection budget constraints. We show the fundamental result that there is no polynomial-time constant-factor approximation algorithm for this problem. This contrasts with other classes of sensor selection problems studied in the literature, which typically pursue constant-factor approximations by leveraging greedy algorithms and submodularity (or supermodularity) of the cost function. Here, we provide a specific example showing that greedy algorithms can perform arbitrarily poorly for the problem of design-time sensor selection for Kalman filtering. We then study the problem of attacking (i.e., removing) a set of installed sensors, under predefined attack budget constraints, to maximize the trace of the steady state a priori or a posteriori error covariance of the Kalman filter. Again, we show that there is no polynomial-time constant-factor approximation algorithm for this problem, and show specifically that greedy algorithms can perform arbitrarily poorly.

Approximability of the Problem of Finding a Vector Subset with the Longest Sum

We answer the question of existence of polynomial-time constant-factor approximation algorithms for the space of nonfixed dimension. We prove that, in Euclidean space the problem is solvable in polynomial time with accuracy $$\sqrt a $$ , where α = 2/π, and if P ≠ NP then there are no polynomial algorithms with better accuracy. It is shown that, in the case of the lp spaces, the problem is APX-complete if p ∈ [1, 2] and not approximable with constant accuracy if P ≠ NP and p ∈ (2,∞).

Improving the Betweenness Centrality of a Node by Adding Links

Betweenness is a well-known centrality measure that ranks the nodes according to their participation in the shortest paths of a network. In several scenarios, having a high betweenness can have a positive impact on the node itself. Hence, in this article, we consider the problem of determining how much a vertex can increase its centrality by creating a limited amount of new edges incident to it. In particular, we study the problem of maximizing the betweenness score of a given node—Maximum Betweenness Improvement (MBI)—and that of maximizing the ranking of a given node—Maximum Ranking Improvement (MRI). We show that MBI cannot be approximated in polynomial-time within a factor (1−1/2e) and that MRI does not admit any polynomial-time constant factor approximation algorithm, both unless P = NP . We then propose a simple greedy approximation algorithm for MBI with an almost tight approximation ratio and we test its performance on several real-world networks. We experimentally show that our algorithm highly increases both the betweenness score and the ranking of a given node and that it outperforms several competitive baselines. To speed up the computation of our greedy algorithm, we also propose a new dynamic algorithm for updating the betweenness of one node after an edge insertion, which might be of independent interest. Using the dynamic algorithm, we are now able to compute an approximation of MBI on networks with up to 10 5 edges in most cases in a matter of seconds or a few minutes.

Energy aware routing with link disjoint backup paths

Energy aware routing has attracted a wide spread attention in the last decade. However, relatively concentrated traffic after applying energy aware strategies makes the network be vulnerable to link failure and sudden traffic bursts. It is a challenge to take enhancing network reliability into account while keeping low energy consumption. Routing each source-to-terminal request via two disjoint paths can be a trade-off. In this paper, we propose a problem called Energy Aware Routing with Link Disjoint Backup Paths (EAR-LDBP) to minimize router power consumption when backup sharing is not allowed during off-peak hours. Each request is routed via an active path. A link disjoint backup path is computed for each request and can be used to route traffic in case of single link failure on the active path. A 0–1 integer linear programming model that can guarantee the network resilience to single link failure is formulated to minimize the power of used links and nodes while putting idle ones to sleep under constraints of link utilization and a link disjoint backup path for each request. We analyze the complexity of EAR-LDBP and show its Np-hardness. We prove that there is no polynomial-time constant-factor approximation algorithm even for two requests. Four different versions of EAR-LDBP (link/node-disjoint, link cost only/link and node cost) are shown to be equivalent and polynomial-time reducible to each other. Then two algorithms are proposed to solve this problem: 2 Stage Greedy Algorithm (2SG) and Greedy Two Link Disjoint Paths Algorithm (G2DP) with an iterative deletion strategy to improve the solution. For comparison purpose, we also use the state of the art reliable energy aware routing algorithm 2DP-NF in unsplittable way. Simulation results on synthetic networks and real network of Internet2 show the effectiveness of our algorithms. Particularly, our algorithms can save more energy (up to 39%) while using less computation time as compared to 2DP-NF (less than 1% at best).

Maximum Lifetime Scheduling for Target Coverage and Data Collection in Wireless Sensor Networks

Target coverage and data collection are two fundamental problems for wireless sensor networks (WSNs). Target coverage is needed to select sensors in a given area that can monitor a set of interesting points. Data collection is needed to transmit the sensed data from sensors to a sink. Since, in many applications, sensors are battery powered, it is expected that a WSN can work untended for a long period. This paper addresses the scheduling problems for both target coverage and data collection in WSNs with the objective of maximizing network lifetime. First, a polynomial-time approximation scheme is developed for the case where the density of target points is bounded, and then, a polynomial-time constant-factor approximation algorithm is developed for the general case. It is also proved that it is NP-hard to find a maximum lifetime scheduling of target cover and data collection for a WSN, even if all the sensors have the same sensing radius and the same transmission radius. Further, the practical efficiency of our algorithms is analyzed through simulation. These extensive simulation results show better performances of our algorithms compared with other research findings.

Chasing a Fast Robber on Planar Graphs and Random Graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, that is, can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let denote the number of cops needed to capture the robber in a graph G in this variant, and let denote the treewidth of G. We show that if G is planar then , and there is a polynomial-time constant-factor approximation algorithm for computing . We also determine, up to constant factors, the value of of the Erdős–Rényi random graph for all admissible values of p, and show that when the average degree is ω(1), is typically asymptotic to the domination number.

A Constant-Factor Approximation Algorithm for Unsplittable Flow on Paths

In the unsplittable flow problem on a path, we are given a capacitated path $P$ and $n$ tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks such that, for each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not exceed the capacity of $e$. This is a well-studied problem that has been described under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack, and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of $O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any $\epsilon>0$. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves to optimality a special case of the problem of finding a maximum weight independent set of rectangles. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a $(2+\epsilon)$-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either 1, 2, or 3.

On a Linear Program for Minimum-Weight Triangulation

Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial-time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: For an LP formulation due to Dantzig, Hoffman, and Hu [Math. Programming, 31 (1985), pp. 1--14], (i) the integrality gap is constant, and (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP.

Efficient rounding for the noncommutative Grothendieck inequality

$ \newcommand{\cclass}[1]{{\textsf{#1}}} $The classical Grothendieck inequality has applications to the design of approximation algorithms for $\cclass{NP}$-hard optimization problems. We show that an algorithmic interpretation may also be given for a noncommutative generalization of the Grothendieck inequality due to Pisier and Haagerup. Our main result, an efficient rounding procedure for this inequality, leads to a polynomial-time constant-factor approximation algorithm for an optimization problem which generalizes the Cut Norm problem of Frieze and Kannan, and is shown here to have additional applications to robust principal component analysis and the orthogonal Procrustes problem.

Multirobot Forest Coverage for Weighted and Unweighted Terrain

One of the main applications of mobile robots is coverage: visiting each location in known terrain. Coverage is crucial for lawn mowing, cleaning, harvesting, search-and-rescue, intrusion detection, and mine clearing. Naturally, coverage can be sped up with multiple robots. However, we show that solving several versions of multirobot coverage problems with minimal cover times is NP-hard, which provides motivation for designing polynomial-time constant-factor approximation algorithms. We then describe multirobot forest coverage (MFC), a new polynomial-time multirobot coverage algorithm based on an algorithm by Even et al. [Min-max tree covers of graphs. Oper. Res. Lett., vol. 32, pp. 309-315, 2004] for finding a tree cover with trees of balanced weights. Our theoretical results show that the cover times of MFC in weighted and unweighted terrain are at most about a factor of 16 larger than minimal. Our simulation results show that the cover times of MFC are close to minimal in all tested scenarios and smaller than the cover times of an alternative multirobot coverage algorithm.

Hardness and approximation of minimum distortion embeddings

We show that the problem of computing a minimum distortion embedding of a given graph into a path is NP-hard when the input graph is bipartite, cobipartite, or split. This problem is hard to approximate within a constant factor on arbitrary graphs. We give polynomial-time constant-factor approximation algorithms for split, cocomparability, interval and cographs.

Approximation Algorithms for Minimizing Edge Crossings in Radial Drawings

We study a crossing minimization problem of drawing a bipartite graph with a radial drawing of two orbits. Radial drawings are one of well-known drawing conventions in social network analysis and visualization, in particular, displaying centrality indices of actors (Wasserman and Faust, Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge, 1994). The main problem in this paper is called the one-sided radial crossing minimization, if the positions of vertices in the outer orbit are fixed. The problem is known to be NP-hard (Bachmaier, IEEE Trans. Vis. Comput. Graph. 13, 583–594, 2007), and a number of heuristics are available (Bachmaier, IEEE Trans. Vis. Comput. Graph. 13, 583–594, 2007). However, there is no approximation algorithm for the crossing minimization problem in radial drawings. We present the first polynomial time constant-factor approximation algorithm for the one-sided radial crossing minimization problem.

TSP with neighborhoods of varying size

In TSP with neighborhoods (TSPN) we are given a collection S of regions in the plane, called neighborhoods, and we seek the shortest tour that visits all neighborhoods. Until now constant-factor approximation algorithms have been known only for cases where the neighborhoods are of approximately the same size. In this paper we present the first polynomial-time constant-factor approximation algorithm for disjoint convex fat neighborhoods of arbitrary size. We also show that in the general case, where the neighborhoods can overlap and are not required to be convex or fat, TSPN is APX-hard and cannot be approximated within a factor of 391 / 390 in polynomial time, unless P = NP .

Comparing Top k Lists

Motivated by several applications, we introduce various distance measures between "top k lists." Some of these distance measures are metrics, while others are not. For each of these latter distance measures, we show that they are "almost" a metric in the following two seemingly unrelated aspects: (i) they satisfy a relaxed version of the polygonal (hence, triangle) inequality, and (ii) there is a metric with positive constant multiples that bound our measure above and below. This is not a coincidence---we show that these two notions of almost being a metric are the same. Based on the second notion, we define two distance measures to be equivalent if they are bounded above and below by constant multiples of each other. We thereby identify a large and robust equivalence class of distance measures. Besides the applications to the task of identifying good notions of (dis)similarity between two top k lists, our results imply polynomial-time constant-factor approximation algorithms for the rank aggregation problem with respect to a large class of distance measures. (A correction for this article has been appended to the pdf file.)