On the hardness of approximating Max-Satisfy

Max-Satisfy is the problem of finding an assignment that satisfies the maximum number of equations in a system of linear equations over Q. We prove that unless NP⊂BPP Max-Satisfy cannot be efficiently approximated within an approximation ratio of 1/n1−?, if we consider systems of n linear equations with at most n variables and ?>0 is an arbitrarily small constant. Previously, it was known that the problem is NP-hard to approximate within a ratio of 1/nα, but 0<α<1 was some specific constant that could not be taken to be arbitrarily close to 1.     

A note on compressed sensing and the complexity of matrix multiplication

We consider the conjectured O(N2+?) time complexity of multiplying any two N×N matrices A and B. Our main result is a deterministic Compressed Sensing (CS) algorithm that both rapidly and accurately computes A⋅B provided that the resulting matrix product is sparse/compressible. As a consequence of our main result we increase the class of matrices A, for any given N×N matrix B, which allows the exact computation of A⋅B to be carried out using the conjectured O(N2+?) operations. Additionally, in the process of developing our matrix multiplication procedure, we present a modified version of Indyk's recently proposed extractor-based CS algorithm [P. Indyk, Explicit constructions for compressed sensing of sparse signals, in: SODA, 2008] which is resilient to noise.     

The hardness of approximation: Gap location

We refine the complexity analysis of approximation problems by relating it to a new parameter calledgap location. Many of the results obtained so far for approximations yield satisfactory analysis with respect to this refined parameter, but some known results (e.g.,max-k-colorability, max 3-dimensional matching andmax not-all-equal 3sat) fall short of doing so. As a second contribution, our work fills the gap in these cases by presenting new reductions.Next, we present definitions and hardness results of new approximation versions of some NP-complete optimization problems. The problems we treat arevertex cover (for which we define a different optimization problem from the one treated in Papadimitriou & Yannakakis 1991),k-edge coloring, andset splitting.     

Parameterized approximation of dominating set problems

A problem open for many years is whether there is an FPT algorithm that given a graph G and parameter k, either: (1) determines that G has no k-Dominating Set, or (2) produces a dominating set of size at most g(k), where g(k) is some fixed function of k. Such an outcome is termed an FPT approximation algorithm. We describe some results that begin to provide some answers. We show that there is no such FPT algorithm for g(k) of the form k+c (where c is a fixed constant, termed an additive FPT approximation), unless FPT=W[2]. We answer the analogous problem completely for the related Independent Dominating Set (IDS) problem, showing that IDS does not admit an FPT approximation algorithm, for any g(k), unless FPT=W[2].     

A note on the approximation of mean-payoff games

We consider the problem of designing approximation schemes for the values of mean-payoff games. It was recently shown that (1) mean-payoff with rational weights scaled on [−1,1]$[-1,1]$ admit additive fully-polynomial approximation schemes, and (2) mean-payoff games with positive weights admit relative fully-polynomial approximation schemes. We show that the problem of designing additive/relative approximation schemes for general mean-payoff games (i.e. with no constraint on their edge-weights) is P-time equivalent to determining their exact solution.     

Approximating the Maximum Agreement Forest on k trees

The Maximum Agreement Forest problem (MAF) asks for the largest common subforest of a set of binary trees. This problem is known to be MAXSNP-complete for instances consisting of 2 trees. We show that it remains MAXSNP-complete for k?2 trees.     

A note on approximation measures for multi-valued dependencies in relational databases

We consider the problem of defining a normalized approximation measure for multi-valued dependencies in relational database theory. An approximation measure is a function mapping relation instances to real numbers. The number to which an instance is mapped, intuitively, describes the strength of the dependency in that instance. A normalized approximation measure for functional dependencies has been proposed previously: the minimum number of tuples that need be removed for the functional dependency to hold divided by the total number of tuples. This leads naturally to a normalized measure for multi-valued dependencies: the minimum number of tuples that need be removed for the multi-valued dependency to hold divided by the total number of tuples.The measure for functional dependencies can be computed efficiently, O(|r|log(|r|)) where |r| is the relation instance. However, we show that an efficient algorithm for computing the analogous measure for multi-valued dependencies is not likely to exist. A polynomial time algorithm for computing the measure would lead to a polynomial time algorithm for an NP-complete problem (proven by a reduction from the maximum edge biclique problem in graph theory). Hence, we argue that it is not a good measure. We propose an alternate measure based on the lossless join characterization of multi-valued dependencies. This measure is efficiently computable, O(|r|²).     

A note on improving the performance of approximation algorithms for radiation therapy

The segment minimization problem consists of representing an integer matrix as the sum of the fewest number of integer matrices each of which have the property that the non-zeroes in each row are consecutive. This has direct applications to an effective form of cancer treatment. Using several insights, we extend previous results to obtain constant-factor improvements in the approximation guarantees. We show that these improvements yield better performance by providing an experimental evaluation of all known approximation algorithms using both synthetic and real-world clinical data. Our algorithms are superior for 76% of instances and we argue for their utility alongside the heuristic approaches used in practice.     

A note on maximum independent sets in rectangle intersection graphs

Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld and Suri [Comput. Geom. Theory Appl. 11 (1998) 209-218] gave a (1+1/k)-factor algorithm with an O(nlogn+n^2k−1) time bound for any integer constant k?1; we describe a similar algorithm running in only O(nlogn+nΔ^k−1) time, where Δ?n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan and Ramaswami [J. Algorithms 41 (2001) 443-470] gave a ⌈log_kn⌉-factor algorithm with an O(n^k+1) time bound for any integer constant k?2; we describe similar algorithms running in O(nlogn+nΔ^k−2) and n^O(k/logk) time.     

A fast deterministic smallest enclosing disk approximation algorithm

We describe a simple and fast -time algorithm for finding a (1+?)-approximation of the smallest enclosing disk of a planar set of n points or disks. Experimental results of a readily available implementation are presented.     

Hierarchical classification of images by sparse approximation

Using image hierarchies for visual categorization has been shown to have a number of important benefits. Doing so enables a significant gain in efficiency (e.g., logarithmic with the number of categories [16,12]) or the construction of a more meaningful distance metric for image classification [17]. A critical question, however, still remains controversial: would structuring data in a hierarchical sense also help classification accuracy? In this paper we address this question and show that the hierarchical structure of a database can be indeed successfully used to enhance classification accuracy using a sparse approximation framework. We propose a new formulation for sparse approximation where the goal is to discover the sparsest path within the hierarchical data structure that best represents the query object. Extensive quantitative and qualitative experimental evaluation on a number of branches of the Imagenet database [7] as well as on the Caltech-256 [12] demonstrate our theoretical claims and show that our approach produces better hierarchical categorization results than competing techniques.     

Finding optimal paths in MREP routing

Maximum Residual Energy Path (MREP) routing has been shown an effective routing scheme for energy conservation in battery powered wireless networks. Past studies on MREP routing are based on the assumption that the transmitting node consumes power, but the receiving node does not. This assumption is false if acknowledgment is required as occurs, for example, in some Bluetooth applications.If the receiving node does not consume power then the MREP routing problem for a single message is easily solvable in polynomial time using a simple Dijkstra-like algorithm. We further show in that when the receiving node does consume power the problem becomes NP-complete and is even impossible to approximate with an exponential approximation factor in polynomial time unless P=NP.     

Surface approximation via sparse representation and parameterization optimization

Surface approximation with smooth functions suffers the problems of choosing the basis functions and representing non-smooth features. In this work, we introduce a sparse representation for surfaces with a set of redundant basis functions, which efficiently overcomes the overfitting artifacts. Moreover, we propose an approach of parameterization transformation, which makes the possibility to represent non-smooth features by the composition of a smooth function and a non-smooth domain optimization. We couple the sparse representation and the parameterization transformation in a global optimization to respect sharp features with smooth polynomial basis functions. Our approach is capable for approximating a wide range of surfaces with different level of sharp features. Experimental results have shown the feasibility and applicability of our proposed method in various applications.     

A note on the inapproximability of correlation clustering

We consider inapproximability of the correlation clustering problem defined as follows: Given a graph G=(V,E) where each edge is labeled either “+” (similar) or “−” (dissimilar), correlation clustering seeks to partition the vertices into clusters so that the number of pairs correctly (resp., incorrectly) classified with respect to the labels is maximized (resp., minimized). The two complementary problems are called MaxAgree and MinDisagree, respectively, and have been studied on complete graphs, where every edge is labeled, and general graphs, where some edge might not have been labeled. Natural edge-weighted versions of both problems have been studied as well. Let S-MaxAgree denote the weighted problem where all weights are taken from set S, we show that S-MaxAgree with weights bounded by O(|V|^1/2−δ) essentially belongs to the same hardness class in the following sense: if there is a polynomial time algorithm that approximates S-MaxAgree within a factor of λ=O(log|V|) with high probability, then for any choice of S^′, S^′-MaxAgree can be approximated in polynomial time within a factor of (λ+?), where ?>0 can be arbitrarily small, with high probability. A similar statement also holds for S-MinDisagree. This result implies it is hard (assuming NP≠RP) to approximate unweighted MaxAgree within a factor of 80/79−?, improving upon a previous known factor of 116/115−? by Charikar et al. [M. Charikar, V. Guruswami, A. Wirth, Clustering with qualitative information, Journal of Computer and System Sciences 71 (2005) 360-383].¹     

On the hardness of approximating label-cover

The Label-Cover problem, defined by S. Arora, L. Babai, J. Stern, Z. Sweedyk [Proceedings of 34th IEEE Symposium on Foundations of Computer Science, 1993, pp. 724-733], serves as a starting point for numerous hardness of approximation reductions. It is one of six ‘canonical’ approximation problems in the survey of Arora and Lund [Hardness of Approximations, in: Approximation Algorithms for NP-Hard Problems, PWS Publishing Company, 1996, Chapter 10]. In this paper we present a direct combinatorial reduction from low error-probability PCP [Proceedings of 31st ACM Symposium on Theory of Computing, 1999, pp. 29-40] to Label-Cover showing it NP-hard to approximate to within 2^{(logn)1−o(1)}. This improves upon the best previous hardness of approximation results known for this problem.We also consider the Minimum-Monotone-Satisfying-Assignment (MMSA) problem of finding a satisfying assignment to a monotone formula with the least number of 1's, introduced by M. Alekhnovich, S. Buss, S. Moran, T. Pitassi [Minimum propositional proof length is NP-hard to linearly approximate, 1998]. We define a hierarchy of approximation problems obtained by restricting the number of alternations of the monotone formula. This hierarchy turns out to be equivalent to an AND/OR scheduling hierarchy suggested by M.H. Goldwasser, R. Motwani [Lecture Notes in Comput. Sci., Vol. 1272, Springer-Verlag, 1997, pp. 307-320]. We show some hardness results for certain levels in this hierarchy, and place Label-Cover between levels 3 and 4. This partially answers an open problem from M.H. Goldwasser, R. Motwani regarding the precise complexity of each level in the hierarchy, and the place of Label-Cover in it.     

Approximating the Minimal Sensor Selection for Supervisory Control

This paper discusses the problem of selecting a set of sensors of minimum cost that can be used for the synthesis of a supervisory controller. It is shown how this sensor selection problem is related to a type of directed graph st-cut problem that has not been previously discussed in the literature. Approximation algorithms to solve the sensor selection problem can be used to solve the graph cutting problem and vice-versa. Polynomial time algorithms to find good approximate solutions to either problem most likely do not exist (under certain complexity assumptions), but a time efficient approximation algorithm is shown that solves a special case of these problems. It is also shown how to convert the sensor selection problem into an integer programming problem.