Experimental Evaluation of Approximation and Heuristic Algorithms for Maximum Distance-Bounded Subgraph Problems

In this paper we study the (in)approximability of two distance-based relaxed variants of the maximum clique problem (Max Clique), named Max d-Clique and Max d-Club: A d-clique in a graph $$G = (V, E)$$ is a subset $$S\subseteq V$$ of vertices such that for every pair of vertices $$u, v\in S$$ , the distance between u and v is at most d in G. A d-club in a graph $$G = (V, E)$$ is a subset $$S'\subseteq V$$ of vertices that induces a subgraph of G of diameter at most d. Given a graph G with n vertices, the goal of Max d-Clique (Max d-Club, resp.) is to find a d-clique (d-club, resp.) of maximum cardinality in G. Since Max 1-Clique and Max 1-Club are identical to Max Clique, the inapproximabilty for Max Clique shown by Zuckerman in 2007 is transferred to them. Namely, Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of $$n^{1-\varepsilon }$$ for any $$\varepsilon > 0$$ unless $$\mathcal{P} = \mathcal{NP}$$ . Also, in 2002, Marin $$\breve{\mathrm{c}}$$ ek and Mohar showed that it is $$\mathcal{NP}$$ -hard to approximate Max d-Club to within a factor of $$n^{1/3-\varepsilon }$$ for any $$\varepsilon >0$$ and any fixed $$d\ge 2$$ . In this paper, we strengthen the hardness result; we prove that, for any $$\varepsilon > 0$$ and any fixed $$d\ge 2$$ , it is $$\mathcal{NP}$$ -hard to approximate Max d-Club to within a factor of $$n^{1/2-\varepsilon }$$ . Then, we design a polynomial-time algorithm which achieves an optimal approximation ratio of $$O(n^{1/2})$$ for any integer $$d\ge 2$$ . By using the similar ideas, we show the $$O(n^{1/2})$$ -approximation algorithm for Max d-Clique for any $$d\ge 2$$ . This is the best possible in polynomial time unless $$\mathcal{P} = \mathcal{NP}$$ , as we can prove the $$\varOmega (n^{1/2-\varepsilon })$$ -inapproximability.

In this paper we consider the problem of constructing numerical algorithms for approximating of convex compact bodies in d-dimensional Euclidean space by polytopes with any given accuracy. It is well known that optimal with respect to the order algorithms produces polytopes for which the accuracy in Hausdorff metric is inversely proportional to the number of vertices (faces) in the degree of 2/(d-1). Numerical approximation algorithms can be adaptive (active) when the vertices or faces are constructed successively, depending on the information obtained in the process of approximation, and non-adaptive (passive) when parameters of algorithms are defined on the basis of a priory information available. Approximation algorithms differ in the use of operations applied to the approximated body. Most common are indicator, support and distance (Minkowski) functions calculations. Some optimal active algorithms for arbitrary bodies approximation are known using support or distance function calculation operation. Optimal passive algorithms for smooth bodies approximation are known using support function calculation operation and extremal curvature information. It is known that there are no optimal non-adaptive algorithms for arbitrary bodies approximation using support function calculation operation. We consider optimal non-adaptive algorithms for arbitrary bodies approximation using projection function calculation operation.

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed.

Many approximation algorithms have been presented in the last decades for ${\cal NP}$-hard search problems. The focus of this paper is on cryptographic applications, where it is desirable to design algorithms which do not leak unnecessary information. Specifically, we are interested in private approximation algorithms—efficient approximation algorithms whose output does not leak information not implied by the optimal solutions to the search problems. Privacy requirements add constraints on the approximation algorithms; in particular, known approximation algorithms usually leak a lot of information. For functions, Feigenbaum et al. [ACM Trans. Algorithms, 2 (2006), pp. 435–472] presented a natural requirement that a private algorithm should not leak information not implied by the original function. Generalizing this requirement to relations is not straightforward as an input may have many different outputs. We present a new definition that captures a minimal privacy requirement from such algorithms; applied to an input instance, it should not leak any information that is not implied by its collection of exact solutions. We argue that our privacy requirement is natural and quite minimal. We show that, even under this minimal definition of privacy, for well-studied problems such as vertex cover and max exact 3SAT, private approximation algorithms are unlikely to exist even for poor approximation ratios. Similarly to Halevi et al. [in Proceedings of the 33rd ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 550–559], we define a relaxed notion of approximation algorithms that leak (a little) information, and demonstrate the applicability of this notion by showing near optimal approximation algorithms for max exact 3SAT that leak a little information.

We consider a game of persuasion with evidence between a sender and a receiver. The sender has private information. By presenting evidence on the information, the sender wishes to persuade the receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender’s utility depends solely on whether or not the receiver takes the action. The receiver’s utility depends on both the action as well as the sender’s private information. We study three natural variations. First, we consider sequential equilibria of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and then the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and then the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, as well as polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm for a natural problem in algorithmic economics.

The purpose of reoptimization using approximation methods—application of knowledge about the solution of the initial instance I, provided to achieve a better quality of approximation (approximation ratio) of an algorithm for determining optimal or close to it solutions of some “minor” changes of instance I. To solve the problem Ins-Max-EkCSP-P (reoptimization of Max-EkCSP-P with the addition of one constraint) with approximation resistant predicate P exists a polynomial threshold (optimal) -approximation algorithm, where the threshold “random” approximation ratio of P). When the unique games conjecture (UGC) is hold there exists a polynomial threshold (optimal) -approximation algorithm (where and the integrality gap of semidefinite relaxation of Max-EkCSP-P problem Z) to solve the problem Ins-Max-EkCSP-P.

In this paper we gather several improvements in the field of exact and approximate exponential-time algorithms for the Bandwidth problem. For graphs with treewidth t we present a O(n O(t) 2n) exact algorithm. Moreover for the same class of graphs we introduce a subexponential constant-approximation scheme – for any α> 0 there exists a (1 + α)-approximation algorithm running in \(O(\exp(c(t + \sqrt{n/\alpha})\log n))\) time where c is a universal constant. These results seem interesting since Unger has proved that Bandwidth does not belong to APX even when the input graph is a tree (assuming P ≠ NP). So somewhat surprisingly, despite Unger’s result it turns out that not only a subexponential constant approximation is possible but also a subexponential approximation scheme exists. Furthermore, for any positive integer r, we present a (4r − 1)-approximation algorithm that solves Bandwidth for an arbitrary input graph in \(O^*(2^{n\over r})\) time and polynomial space. Finally we improve the currently best known exact algorithm for arbitrary graphs with a O(4.473n) time and space algorithm.In the algorithms for the small treewidth we develop a technique based on the Fast Fourier Transform, parallel to the Fast Subset Convolution techniques introduced by Björklund et al. This technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in O *(2n) time and space, what matches the best known results.KeywordsFast Fourier TransformApproximation AlgorithmChromatic NumberExact AlgorithmInput GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

• NP-Completeness and APX-Hardness • We prove that the problems MinFixMask OPT and MinFixMask BOUND are NP-complete. • We establish that FixMasks + is NP-complete and APX-hard. • We show NP-completeness of DFixMasks + . • Exact, FPT, and Approximation Algorithms for DFixMasks + • We design a fixed-parameter tractable (FPT) algorithm running in O *(2 M ) time, parameterized by the number of shield configurations M . • We present a polynomial-time 1 M -approximation algorithm. • We propose an exponential k M -approximation method (for a treatment plan length k ). • We provide a tight ( 1 − 1 e ) -approximation algorithm by exploiting monotonicity and submodularity of the objective function. • Quasi-Polynomial Algorithms and Logarithmic Approximations • We develop quasi-polynomial-time algorithms for MinFixMask OPT , MinFixMask OPT + , MinFixMask BOUND , and MinFixMask BOUND + , assuming that the maximum prescribed dose d ^ max is polynomially bounded. • Under the same assumption, we achieve polynomial-time approximation algorithms with a logarithmic factor log ( d ^ max ) for MinFixMask OPT and MinFixMask OPT + . • Polynomial-Time Solvable Special Cases • We identify polynomial-time solvable cases: FixMask and FixMasks + when M = 1 . • We show that the problems MinFixMask BOUND and MinFixMask BOUND + can be solved in polynomial time when T max 1, N ;. In this paper, we study a combinatorial problem which arises in the development of innovative treatment strategies and equipment using tunable shields in internal radiotherapy. From an algorithmic point of view, the problem is related to circular integer word decomposition into circular binary words under constraints. We consider several variants of the problem, depending on constraints and parameters and present exact, approximation, fixed parameter tractable algorithms and NP-hardness and APX-hardness results.

In the maximum edge-disjoint paths problem, we are given a graph and a collection of pairs of vertices, and the objective is to find the maximum number of pairs that can be routed by edge-disjoint paths. An r-approximation algorithm for this problem is a polynomial time algorithm that finds at least OPT / r edge-disjoint paths, where OPT is the maximum possible. Currently, an O(n1/2)-approximation algorithm is best known for this problem even if a congestion of two is allowed, i.e., each edge is allowed to be used in at most two of the paths.In this paper, we give a randomized O(n3/7 • poly (log n))-approximation algorithm with congestion two. This is the first result that breaks the O(n1/2)-approximation algorithm. In particular, we prove the following. 1. If we have a (randomized) polynomial time algorithm for finding Ω(OPT1/p) edge-disjoint paths for some p>1, then, for some α >0, we can give a randomized O(n1/2-α)-approximation algorithm for the maximum edge-disjoint paths problem by using Rao-Zhou's algorithm. 2. Based on the well-linked set of Chekuri, Khanna, and Shepherd, we show that there is a randomized algorithm for finding Ω(OPT1/4) edge-disjoint paths connecting given terminal pairs with congestion two.Our framework for this algorithm is more general. Indeed, the above two ingredients also work for the maximum edge-disjoint paths problem (with congestion one) if the following conjecture is true. Conjecture: There is a (randomized) polynomial time algorithm for finding Ω(OPT1/p/β(n)) edge-disjoint paths connecting given terminal pairs, where β is a poly-logarithmic function.

We present a dynamic voltage scaling (DVS) technique that minimizes system-wide energy consumption for both periodic and sporadic tasks. It is known that a system consists of processors and a number of other components. Energy-aware processors can be run in different speed levels; components like memory and I/O subsystems and network interface cards can be in a standby state when they are active but idle. Processor energy optimization solutions are not necessarily efficient from the perspective of systems. Current system-wide energy optimization studies are often limited to periodic tasks with heuristics in getting approximated solutions. In this paper, we develop an exact dynamic programming algorithm for periodic tasks on processors with practical discrete speed levels. The algorithm determines the lower bound of energy expenditure in pseudo-polynomial time. An approximation algorithm is proposed to provide performance guarantee with a given bound in polynomial running time. Because of their time efficiency, both the optimization and approximation algorithms can be adapted for online scheduling of sporadic tasks with irregular task releases. We prove that system-wide energy optimization for sporadic tasks is NP-hard in the strong sense. We develop (pseudo-) polynomial-time solutions by exploiting its inherent properties.

Given a set ofn points on the plane, the rectilinearm-center problem is to findn rectilinear squares covering all thesen points such that the maximum side length of these squares is minimized. In this paper we prove that there is no polynomial-time algorithm with an error ratio ɛ < 2 for the rectilinearm-center problem unless NP = P. A polynomial-time approximation algorithm with an error ratio of 2 is also proposed.

A fundamental problem in computer and communication networks is the wavelength assignment (WA) problem: given a set of routing paths on a network, assign wavelengths (channels) to the paths such that the paths with the same wavelength are edge-disjoint. The optimization problem here is to minimize the number of wavelengths. A popular network topology is a tree of rings. It is known NP-hard to find the minimum number of wavelengths for the WA problem on a tree of rings. Let L be the maximum number of paths on any edge in the network. Then L is a lower bound on the number of wavelengths for the WA problem. We give a polynomial time algorithm which uses at most 3L wavelengths for the WA problem on a tree of rings with node degree at most eight. This improves the previous result of 4L. We also show that some instances of the WA problem require at least 3L wavelengths on a tree of rings, implying that the 3L upper bound is optimal for the worst case instances. In addition, we prove that our algorithm has approximation ratios 2 and 2.5 for a tree of rings with node degrees at most four and six, respectively.

An approximate algorithm for computing multidimensional convex hulls

On the optimality of exact and approximation algorithms for scheduling problems

Optimal recovery of multivariate contractions