Polynomial-time Approximation Algorithm Research Articles

Approximation Algorithms for Stochastic Combinatorial Optimization Problems

Stochastic optimization has established itself as a major method to handle uncertainty in various optimization problems by modeling the uncertainty by a probability distribution over possible realizations. Traditionally, the main focus in stochastic optimization has been various stochastic mathematical programming (such as linear programming, convex programming). In recent years, there has been a surge of interest in stochastic combinatorial optimization problems from the theoretical computer science community. In this article, we survey some of the recent results on various stochastic versions of classical combinatorial optimization problems. Since most problems in this domain are NP-hard (or #P-hard, or even PSPACE-hard), we focus on the results which provide polynomial time approximation algorithms with provable approximation guarantees. Our discussions are centered around a few representative problems, such as stochastic knapsack, stochastic matching, multi-armed bandit etc. We use these examples to introduce several popular stochastic models, such as the fixed-set model, 2-stage stochastic optimization model, stochastic adaptive probing model etc, as well as some useful techniques for designing approximation algorithms for stochastic combinatorial optimization problems, including the linear programming relaxation approach, boosted sampling, content resolution schemes, Poisson approximation etc. We also provide some open research questions along the way. Our purpose is to provide readers a quick glimpse to the models, problems, and techniques in this area, and hopefully inspire new contributions .

Scheduling two parallel machines with machine-dependent availabilities

A two-parallel-machine scheduling problem with machine-dependent availabilities where one machine is subject to tool changes and the other is subject to periodic maintenance is considered. The objective is to determine the start time of each tool change activity and schedule all the jobs to the two machines such that the makespan is minimized. Due to the NP-hardness of the non-approximability of the problem, there does not exist any polynomial time approximation algorithm with a worst-case ratio less than 2 for the problem unless P=NP. To solve small sized and moderate sized instances of the problem, a mixed 0–1 programming model is proposed. To solve large sized instances of the problem, nine heuristic algorithms that employ two classical dispatching rules, two assignment mechanisms and a post-optimization procedure are proposed. To evaluate the performance of the algorithms, some machine setting featured lower bounds are provided. Computational experiment shows that the average-case relative error ratio of the heuristic algorithm that based on the classical LPT rule, two assignment procedures and a post-optimization procedure is less than 8%, which implies that it is promising for problem and suitable for real-world application.

On Distributed Virtual Network Embedding With Guarantees

To provide wide-area network services, resources from different infrastructure providers are needed. Leveraging the consensus-based resource allocation literature, we propose a general distributed auction mechanism for the (NP-hard) virtual network (VNET) embedding problem. Under reasonable assumptions on the bidding scheme, the proposed mechanism is proven to converge, and it is shown that the solutions guarantee a worst-case efficiency of (1 - (1/e)) relative to the optimal node embedding, or VNET embedding if virtual links are mapped to exactly one physical link. This bound is optimal, that is, no better polynomial-time approximation algorithm exists, unless P = NP. Using extensive simulations, we confirm superior convergence properties and resource utilization when compared to existing distributed VNET embedding solutions, and we show how by appropriate policy design, our mechanism can be instantiated to accommodate the embedding goals of different service and infrastructure providers, resulting in an attractive and flexible resource allocation solution.

Polynomial-time approximation algorithms for weighted LCS problem

We consider a variant of the well-known Longest Common Subsequence (LCS) problem for weighted sequences. A weighted sequence determines the probability for each symbol to occur at a given position of the sequence (such sequences are also called Position Weighted Matrices, PWM). Two possible such versions of the problem were proposed by (Amir et al., 2009 and 2010), they are called LCWS and LCWS2 (Longest Common Weighted Subsequence 1 and 2). We solve an open problem, stated in the paper by Amir et al., of the tractability of a log-probability version of LCWS2 problem for bounded alphabets, showing that it is NP-hard already for an alphabet of size 2. We also improve the (1/|Σ|)-approximation algorithm given by Amir et al. (where Σ is the alphabet): we show a polynomial-time approximation scheme (PTAS) for the LCWS2 problem using O(n5) space. We also give a simpler (1/2)-approximation algorithm for the same problem using only O(n2) space.

Data reductions and combinatorial bounds for improved approximation algorithms

Kernelization algorithms in the context of Parameterized Complexity are often based on a combination of data reduction rules and combinatorial insights. We will expose in this paper a similar strategy for obtaining polynomial-time approximation algorithms. Our method features the use of approximation-preserving reductions, akin to the notion of parameterized reductions. We exemplify this method to obtain the currently best approximation algorithms for Harmless Set, Differential and Multiple Nonblocker, all of them can be considered in the context of securing networks or information propagation.

Firefighting as a Strategic Game

The Firefighter Problem was proposed in 1995 as a deterministic discrete-time model for the spread and containment of a fire. The problem is defined on an undirected finite graph G = (V, E), where fire breaks out initially at f nodes. In each subsequent time-step, two actions occur: a certain number b of firefighters are placed on nonburning nodes, permanently protecting them from the fire, then the fire spreads to all nondefended neighbors of the nodes on fire. Because the graph is finite, at some point each node is either on fire or saved, and thus the fire cannot spread further. One of the objectives for the problem is to place the firefighters in such a way that the number of saved nodes is maximized. The applications of the Firefighter Problem reach from real fires to the spreading of diseases and the containment of floods. Furthermore, it can be used to model the spread of computer viruses or viral marketing in communication networks. Most research on the problem considers the case in which the fire starts in a single place (i.e., f = 1), and in which the budget of available firefighters per time-step is one (i.e., b = 1). So does the work in this study. This configuration already leads to hard problems and, even in this case, the problem is known to be NP-hard. In this work, we study the problem from a game-theoretical perspective. We introduce a strategic game model for the Firefighter Problem to tackle its complexity from a different angle. We refer to it as the Firefighter Game. Such a game-based context seems very appropriate when applied to large networks where entities may act and make decisions based on their own interests, without global coordination. At every time-step of the game, a player decides whether to place a new firefighter in a nonburning node of the graph. If so, he must decide where to place it. By placing it, the player is indirectly deciding which nodes to protect at that time-step. We define different utility functions in order to model selfish and nonselfish scenarios, which lead to equivalent games. We show that the Price of Anarchy (PoA) is linear for a particular family of graphs, but it is at most two for trees. We also analyze the quality of the equilibria when coalitions among players are allowed. It turns out that it is possible to compute an equilibrium in polynomial time, even for constant-size coalitions. This yields to a polynomial time approximation algorithm for the problem and its approximation ratio equals the PoA of the corresponding game. We show that for some specific topologies, the PoA is constant when constant-size coalitions are considered.

Cubical coloring — fractional covering by cuts and semidefinite programming

We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

Improved Approximation Algorithms for Projection Games

The projection games (aka Label-Cover) problem is of great importance to the field of approximation algorithms, since most of the NP-hardness of approximation results we know today are reductions from Label-Cover. In this paper we design several approximation algorithms for projection games: 1. A polynomial-time approximation algorithm that improves on the previous best approximation by Charikar, Hajiaghayi and Karloff. 2. A sub-exponential time algorithm with much tighter approximation for the case of smooth projection games. 3. A polynomial-time approximation scheme (PTAS) for projection games on planar graphs and a tight running time lower bound for such approximation schemes.

Approximating Maximum Agreement Forest on Multiple Binary Trees

Given a collection of phylogenetic trees on the same leaf label-set, the Maximum Agreement Forest problem (Maf) asks for a largest common subforest of these trees. The Maf problem on two binary phylogenetic trees has been studied extensively. In this paper, we are focused on the Maf problem on multiple (i.e., two or more) binary phylogenetic trees and present two polynomial-time approximation algorithms, one for the Maf problem on multiple rooted trees, and the other for the Maf problem on multiple unrooted trees. The ratio of our algorithm for the Maf problem on multiple rooted trees is 3, which is an improvement over the previous best ratio 8 for the problem. Our approximation algorithm of ratio 4 for the Maf problem on multiple unrooted trees is the first constant ratio approximation algorithm for the problem.

Matching colored points with rectangles

Let S be a point set in the plane such that each of its elements is colored either red or blue. A matching of S with rectangles is any set of pairwise-disjoint axis-aligned closed rectangles such that each rectangle contains exactly two points of S. Such a matching is monochromatic if every rectangle contains points of the same color, and is bichromatic if every rectangle contains points of different colors. We study the following two problems: (1) Find a maximum monochromatic matching of S with rectangles. (2) Find a maximum bichromatic matching of S with rectangles. For each problem we provide a polynomial-time approximation algorithm that constructs a matching with at least 1 / 4 of the number of rectangles of an optimal matching. We show that the first problem is $$\mathsf {NP}$$ -hard even if either the matching rectangles are restricted to axis-aligned segments or S is in general position, that is, no two points of S share the same x or y coordinate. We further show that the second problem is also $$\mathsf {NP}$$ -hard, even if S is in general position. These $$\mathsf {NP}$$ -hardness results follow by showing that deciding the existence of a matching that covers all points is $$\mathsf {NP}$$ -complete in each case. Additionally, we prove that it is $$\mathsf {NP}$$ -complete to decide the existence of a matching with rectangles that cover all points in the case where all the points have the same color, solving an open problem of Bereg et al. (Comput Geom 42(2):93–108, 2009).

An improved approximation algorithm for the partial-terminal Steiner tree problem with edge cost 1 or 2

Given a complete graph G=(V,E) with a metric cost function c:E→R+ and two vertex subsets R⊂V and R′⊆R, a partial-terminal Steiner tree is a Steiner tree which contains all the vertices in R such that all the vertices in R′ are leaves. The partial-terminal Steiner tree problem (PTSTP) is to find a partial-terminal Steiner tree with the minimum cost. The problem has been shown to be NP-hard and MAX SNP-hard, even when the edge costs are restricted in {1,2}, namely, the (1,2)-partial-terminal Steiner tree problem (PTSTP(1,2)). In this paper, we consider PTSTP(1,2). The previous best-known approximation ratio of PTSTP(1,2) was at most 2514. In this paper, we propose a polynomial-time approximation algorithm that improves the approximation ratio from 2514 to 53.

A faster 1.375-approximation algorithm for sorting by transpositions.

Sorting by Transpositions is an NP-hard problem for which several polynomial-time approximation algorithms have been developed. Hartman and Shamir (2006) developed a 1.5-approximation [Formula: see text] algorithm, whose running time was improved to O(nlogn) by Feng and Zhu (2007) with a data structure they defined, the permutation tree. Elias and Hartman (2006) developed a 1.375-approximation O(n(2)) algorithm, and Firoz et al. (2011) claimed an improvement to the running time, from O(n(2)) to O(nlogn), by using the permutation tree. We provide counter-examples to the correctness of Firoz et al.'s strategy, showing that it is not possible to reach a component by sufficient extensions using the method proposed by them. In addition, we propose a 1.375-approximation algorithm, modifying Elias and Hartman's approach with the use of permutation trees and achieving O(nlogn) time.

Adaptive informative path planning in metric spaces

In contrast to classic geometric motion planning, informative path planning (IPP) seeks a path for a robot to sense the world and gain information. In adaptive IPP, the robot chooses the next sensing location conditioned on all information acquired so far, and the robot’s goal is to minimize the travel cost required for identifying a true hypothesis. Adaptive IPP is NP-hard, because the robot must trade-off information gain and travel cost optimally. In this paper we present Recursive Adaptive Identification (RAId), a new polynomial-time approximation algorithm for adaptive IPP. We prove a polylogarithmic approximation bound when the robot travels in a metric space. Furthermore, our experiments suggest that RAId is practical and provides good approximate solutions for two distinct robot planning tasks. Although RAId is designed primarily for noiseless observations, a simple extension allows it to handle some tasks with noisy observations.

Control traffic balancing in software defined networks

To promise on-line and adaptive traffic engineering in software defined networks (SDNs), the control messages, e.g., the first packet of every new flow and network traffic statistics, should be forwarded from software defined switches to the controller(s) in a fast and robust manner. As many signaling events and control plane operations are required in SDNs, they could easily generate a significant amount of control traffic that must be addressed together with the data traffic. However, the usage of in-band control channel imposes a great challenge into timely and reliable transmissions of control traffic, while out-band control is usually cost-prohibitive. To counter this, in this paper, the control traffic balancing problem is first formulated as a nonlinear optimization framework with an objective to find the optimal control traffic forwarding paths for each switch in such a way the average control traffic delay in the whole network is minimized. This problem is extremely critical in SDNs because the timely delivery of control traffic initiated by Openflow switches directly impacts the effectiveness of the routing strategies. Specifically, the fundamental mathematical structures of the formulated nonlinear problem and solution set are provided and accordingly, an efficient algorithm, called polynomial-time approximation algorithm (PTAA), is proposed to yield the fast convergence to a near optimal solution by employing the alternating direction method of multipliers (ADMM). Furthermore, the optimal controller placement problem in in-band mode is examined, which aims to find the optimal switch location where the controller can be collocated by minimizing the control message delay. While it is not widely researched except quantitative or heuristic results, a simple and efficient algorithm is proposed to guarantee the optimum placement with regards of traffic statistics. Simulation results confirm that the proposed PTAA achieves considerable delay reduction, greatly facilitating controller’s traffic engineering in large-scale SDNs.

Approximability results for the converse connected p-centre problem

In this paper, we investigate a combinatorial optimization problem, called the converse connected p-centre problem which is the converse problem of the connected p-centre problem. This problem is a variant of the p-centre problem. Given an undirected graph with a nonnegative edge length function ℓ, a vertex set , and an integer p, , let denote the shortest distance from v to C of G for each vertex v in , and the eccentricity of C denote . The connected p-centre problem is to find a vertex set P in V, , such that the eccentricity of P is minimized but the induced subgraph of P must be connected. Given an undirected graph and an integer , the converse connected p-centre problem is to find a vertex set P in V with minimum cardinality such that the induced subgraph of P must be connected and the eccentricity . One of the applications of the converse connected p-centre problem has the facility location with load balancing and backup constraints. The connected p-centre problem had been shown to be NP-hard. However, it is still unclear whether there exists a polynomial time approximation algorithm for the converse connected p-centre problem. In this paper, we design the first approximation algorithm for the converse connected p-centre problem with approximation ratio of , . We also discuss the approximation complexity for the converse connected p-centre problem. We show that there is no polynomial time approximation algorithm achieving an approximation ratio of , , for the converse connected p-centre problem unless.

Computing the Fréchet distance between folded polygons

Computing the Fréchet distance for surfaces is a surprisingly hard problem and the only known polynomial-time algorithm is limited to computing it between flat surfaces. We study the problem of computing the Fréchet distance for a class of non-flat surfaces called folded polygons. We present a fixed-parameter tractable algorithm for this problem. Next, we present a polynomial-time approximation algorithm. Finally, we present a restricted class of folded polygons for which we can compute the Fréchet distance for L∞ in polynomial time.

The Thinnest Path Problem

We formulate and study the thinnest path problem for secure communication in wireless ad hoc networks. The objective is to find a path from a source to its destination that results in the minimum number of nodes overhearing the message by a judicious choice of relaying nodes and their corresponding transmission powers. We adopt a directed hypergraph model of the problem and establish the NP-completeness of the problem in 2-D networks. We then develop two polynomial-time approximation algorithms that offer √(n/2) and n/2√(n-1) approximation ratios for general directed hypergraphs (which can model nonisotropic signal propagation in space) and constant approximation ratios for ring hypergraphs (which result from isotropic signal propagation). We also consider the thinnest path problem in 1-D networks and 1-D networks embedded in a 2-D field of eavesdroppers with arbitrary unknown locations (the so-called 1.5-D networks). We propose a linear-complexity algorithm based on nested backward induction that obtains the optimal solution for both 1-D and 1.5-D networks. This algorithm does not require the knowledge of eavesdropper locations and achieves the best performance offered by any algorithm that assumes complete location information of the eavesdroppers.

Bounding Interference in Wireless Ad Hoc Networks With Nodes in Random Position

Given a set of positions for wireless nodes, the interference minimization problem is to assign a transmission radius (i.e., a power level) to each node such that the resulting communication graph is connected while minimizing the maximum (respectively, average) interference. We consider the model introduced by von Rickenbach (2005), in which each wireless node is represented by a point in Euclidean space on which is centered a transmission range represented by a ball, and edges in the corresponding graph are symmetric. The problem is NP-complete in two or more dimensions (Buchin 2008), and no polynomial-time approximation algorithm is known. We show how to solve the problem efficiently in settings typical for wireless ad hoc networks. If nodes are represented by a set P of n points selected uniformly and independently at random over a d-dimensional rectangular region, then the topology given by the closure of the Euclidean minimum spanning tree of P has O(log n) maximum interference with high probability and O(1) expected interference. We extend the first bound to a general class of communication graphs over a broad set of probability distributions. We present a local algorithm that constructs a graph from this class; this is the first local algorithm to provide an upper bound on expected maximum interference. Finally, we disprove a conjecture of Devroye and Morin (2012) relating the maximum interference of the Euclidean minimum spanning tree to the optimal maximum interference attainable.

Directed Steiner trees with diffusion costs

Given a directed arc-weighted graph G with n nodes, a root r and k terminals, the directed steiner tree problem (DST) consists in finding a minimum-weight tree rooted at r and spanning all the terminals. If this problem has several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which some nodes, called non diffusing nodes, are unable to duplicate packets. We define a more general problem, namely the directed steiner tree with a limited number of diffusing nodes (DSTLD), that enables us to model multicast in a network containing at most d diffusing nodes. We show that DSTLD is XP with respect to d, and use this result to build a $$\left\lceil \frac{k-1}{d} \right\rceil $$k-1d-approximation algorithm for DST that is XP in d. We deduce from that result a strong inapproximability property. In particular, we prove that, under the assumption that NP$$\not \subseteq $$źZTIME$$[n^{\log ^{O(1)}n}]$$[nlogO(1)n], there is no polynomial-time approximation algorithm for DSTLD with ratio $$\varOmega \left( \frac{k}{d}\right) $$Ωkd. We finally give an evaluation of performances of an exact algorithm dedicated to the case $$d \le 3$$d≤3.

A complexity and approximation framework for the maximization scaffolding problem

We explore in this paper some complexity issues inspired by the contig scaffolding problem in bioinformatics. We focus on the following problem: given an undirected graph with no loop, and a perfect matching on this graph, find a set of cycles and paths covering every vertex of the graph, with edges alternatively in the matching and outside the matching, and satisfying a given constraint on the numbers of cycles and paths. We show that this problem is NP-complete, even in planar bipartite graphs. Moreover, we show that there is no subexponential-time algorithm for several related problems unless the Exponential-Time Hypothesis fails. Lastly, we also design two polynomial-time approximation algorithms for complete graphs.