Some Relations between Approximation Problems and PCPs over the Real Numbers

In [10] it was recently shown that that is the existence of transparent long proofs for was established. The latter denotes the class of real number decision problems verifiable in polynomial time as introduced by Blum et al. [6]. The present paper is devoted to the question what impact a potential full real number theorem would have on approximation issues in the BSS model of computation. We study two natural optimization problems in the BSS model. The first, denoted by MAX-QPS, is related to polynomial systems; the other, MAX-q-CAP, deals with algebraic circuits. Our main results combine the PCP framework over with approximation issues for these two problems. We also give a negative approximation result for a variant of the MAX-QPS problem.     

Real Hypercomputation and Continuity

By the sometimes so-called Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of hypercomputation allow for the effective evaluation of also discontinuous . More precisely the present work considers the following three super-Turing notions of real function computability: - relativized computation; specifically given oracle access to the Halting Problem or its jump ; - encoding input and/or output y = f(x) in weaker ways also related to the Arithmetic Hierarchy; - nondeterministic computation. It turns out that any computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation provides the required power to evaluate for instance the discontinuous Heaviside function.     

The Complexity of Finding Top-Toda-Equivalence-Class Members

We identify two properties that for P-selective sets are effectively computable. Namely, we show that, for any P-selective set, finding a string that is in a given length's top Toda equivalence class (very informally put, a string from that the set's P-selector function declares to be most likely to belong to the set) is computable, and we show that each P-selective set contains a weakly- -rankable subset.     

Efficient Algorithms for k Maximum Sums

We study the problem of computing the k maximum sum subsequences. Given a sequence of real numbers and an integer parameter k, the problem involves finding the k largest values of for The problem for fixed k = 1, also known as the maximum sum subsequence problem, has received much attention in the literature and is linear-time solvable. Recently, Bae and Takaoka presented a -time algorithm for the k maximum sum subsequences problem. In this paper we design an efficient algorithm that solves the above problem in time in the worst case. Our algorithm is optimal for and improves over the previously best known result for any value of the user-defined parameter k < 1. Moreover, our results are also extended to the multi-dimensional versions of the k maximum sum subsequences problem; resulting in fast algorithms as well.     

The Expected Size of the Rule k Dominating Set

Dai, Li, and Wu proposed Rule k, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. In this paper we consider the "average-case" performance of two closely related versions of Rule k for the model of random unit disk graphs constructed from n random points in an square. We show that if and then for both versions of Rule k, the expected size of the Rule k dominating set is as It follows that, for in a suitable range, the expected size of the Rule k dominating sets are within a constant factor of the optimum.     

Random Geometric Graph Diameter in the Unit Ball

The unit ball random geometric graph has as its vertices n points distributed independently and uniformly in the unit ball in , with two vertices adjacent if and only if their ℓ_p-distance is at most λ. Like its cousin the Erdos-Renyi random graph, G has a connectivity threshold: an asymptotic value for λ in terms of n, above which G is connected and below which G is disconnected. In the connected zone we determine upper and lower bounds for the graph diameter of G. Specifically, almost always, , where is the ℓ_p-diameter of the unit ball B. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.     

The Hardness of Approximating Spanner Problems

This paper examines a number of variants of the sparse k-spanner problem and presents hardness results concerning their approximability. Previously, it was known that most k-spanner problems are weakly inapproximable (namely, they are NP-hard to approximate with ratio O(log n), for every k ≥ 2) and that the unit-length k-spanner problem for constant stretch requirement k ≥ 5 is strongly inapproximable (namely, it is NP-hard to approximate with ratio ). The results of this paper significantly expand the ranges of hardness for k-spanner problems. In general, strong hardness is shown for a number of k-spanner problems, for certain ranges of the stretch requirement k depending on the particular variant at hand. The problems studied differ by the types of edge weights and lengths used, and they include directed, augmentation and client-server variants. The paper also considers k-spanner problems in which the stretch requirement k is relaxed (e.g., . For these cases, no inapproximability results were known (even for a constant approximation ratio) for any spanner problem. Moreover, some versions of the k-spanner problem are known to enjoy the ratio-degradation property; namely, their complexity decreases exponentially with the inverse of the stretch requirement. So far, no hardness result existed precluding any k-spanner problem from enjoying this property. This paper establishes strong inapproximability results for the case of relaxed stretch requirement (up to , for any ), for a large variety of k-spanner problems. It is also shown that these problems do not enjoy the ratio-degradation property.     

Exact Algorithms for Graph Homomorphisms

Graph homomorphism, also called H-coloring, is a natural generalization of graph coloring: There is a homomorphism from a graph G to a complete graph on k vertices if and only if G is k-colorable. During recent years the topic of exact (exponential-time) algorithms for NP-hard problems in general, and for graph coloring in particular, has led to extensive research. Consequently, it is natural to ask how the techniques developed for exact graph coloring algorithms can be extended to graph homomorphisms. By the celebrated result of Hell and Nesetril, for each fixed simple graph H, deciding whether a given simple graph G has a homomorphism to H is polynomial-time solvable if H is a bipartite graph, and NP-complete otherwise. The case where H is the cycle of length 5, is the first NP-hard case different from graph coloring. We show that for an odd integer , whether an input graph G with n vertices is homomorphic to the cycle of length k, can be decided in time . We extend the results obtained for cycles, which are graphs of treewidth two, to graphs of bounded treewidth as follows: if H is of treewidth at most t, then whether input graph G with n vertices is homomorphic to H can be decided in time .     

Morpion Solitaire

We study a popular pencil-and-paper game called morpion solitaire. We present upper and lower bounds for the maximum score attainable for many versions of the game. We also show that, in its most general form, the game is NP-hard and the high score is inapproximable within for any unless P = NP.     

Algorithms to Compute Minimum Cycle Basis in Directed Graphs

We consider the problem of computing a minimum cycle basis in a directed graph G with m arcs and n vertices. The arcs of G have non-negative weights assigned to them. In this problem a {-1,0,1} incidence vector is associated with each cycle and the vector space over generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of weights of the cycles is minimum is called a minimum cycle basis of G. This paper presents an algorithm, which is the first polynomial-time algorithm for computing a minimum cycle basis in G. We then improve it to an algorithm. The problem of computing a minimum cycle basis in an undirected graph has been well studied. In this problem a {0,1} incidence vector is associated with each cycle and the vector space over generated by these vectors is the cycle space of the graph. There are directed graphs in which the minimum cycle basis has lower weight than any cycle basis of the underlying undirected graph. Hence algorithms for computing a minimum cycle basis in an undirected graph cannot be used as black boxes to solve the problem in directed graphs.     

Polynomial-Time Algorithms for the Ordered Maximum Agreement Subtree Problem

For a set of rooted, unordered, distinctly leaf-labeled trees, the NP-hard maximum agreement subtree problem (MAST) asks for a tree contained (up to isomorphism or homeomorphism) in all of the input trees with as many labeled leaves as possible. We study the ordered variants of MAST where the trees are uniformly or non-uniformly ordered. We provide the first known polynomial-time algorithms for the uniformly and non-uniformly ordered homeomorphic variants as well as the uniformly and non-uniformly ordered isomorphic variants of MAST. Our algorithms run in time , , , and , respectively, where n is the number of leaf labels and k is the number of input trees.     

The Effect of Faults on Network Expansion

We study the problem of how resilient networks are to node faults. Specifically, we investigate the question of how many faults a network can sustain and still contain a large (i.e., linear-sized) connected component with approximately the same expansion as the original fault-free network. We use a pruning technique that culls away those parts of the faulty network that have poor expansion. The faults may occur at random or be caused by an adversary. Our techniques apply in either case. In the adversarial setting we prove that for every network with expansion a large connected component with basically the same expansion as the original network exists for up to a constant times faults. We show this result is tight in the sense that every graph G of size n and uniform expansion can be broken into components of size o(n) with faults. Unlike the adversarial case, the expansion of a graph gives a very weak bound on its resilience to random faults. While it is the case, as before, that there are networks of uniform expansion that are not resilient against a fault probability of a constant times it is also observed that there are networks of uniform expansion that are resilient against a constant fault probability. Thus, we introduce a different parameter, called the span of a graph, which gives us a more precise handle on the maximum fault probability. We use the span to show the first known results for the effect of random faults on the expansion of d-dimensional meshes.     

Approximation Algorithms for the Unsplittable Flow Problem

We present approximation algorithms for the unsplittable flow problem (UFP) in undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the non-uniform capacity case in which the edge capacities can vary arbitrarily over the graph. Our results are: We obtain an approximation ratio for UFP, where n is the number of vertices, is the maximum degree, and is the expansion of the graph. Furthermore, if we specialize to the case where all edges have the same capacity, our algorithm gives an approximation. For certain strong constant-degree expanders considered by we obtain an approximation for the uniform capacity case. For UFP on the line and the ring, we give the first constant-factor approximation algorithms. All of the above results improve if the maximum demand is bounded away from the minimum capacity. The above results either improve upon or are incomparable with previously known results for these problems. The main technique used for these results is randomized rounding followed by greedy alteration, and is inspired by the use of this idea in recent work.     

Constant-time distributed dominating set approximation

Finding a small dominating set is one of the most fundamental problems of classical graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary, possibly constant parameter k and maximum node degree , our algorithm computes a dominating set of expected size in rounds. Each node has to send messages of size . This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of rounds.Received: 9 September 2003, Accepted: 2 September 2004, Published online: 13 January 2005The work presented in this paper was supported (in part) by the National Competence Center in Research on Mobile Information and Communication Systems (NCCR-MICS), a center supported by the Swiss National Science Foundation under grant number 5005-67322.     

Convex Drawings of 3-Connected Plane Graphs

We use Schnyder woods of 3-connected planar graphs to produce convex straight-line drawings on a grid of size The parameter depends on the Schnyder wood used for the drawing. This parameter is in the range The algorithm is a refinement of the face-counting algorithm; thus, in particular, the size of the grid is at most The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular Schnyder's and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3-connected planar graphs. The algorithm takes linear time. The drawing algorithm has been implemented and tested. The expected grid size for the drawing of a random triangulation is close to For a random 3-connected plane graph, tests show that the expected size of the drawing is      

Finding Pathway Structures in Protein Interaction Networks

The increased availability of data describing biological interactions provides important clues on how complex chains of genes and proteins interact with each other. Most previous approaches either restrict their attention to analyzing simple substructures such as paths or trees in these graphs, or use heuristics that do not provide performance guarantees when general substructures are analyzed. We investigate a formulation to model pathway structures directly and give a probabilistic algorithm to find an optimal path structure in time and space, where n and m are respectively the number of vertices and the number of edges in the given network, k is the number of vertices in the path structure, and t is the maximum number of vertices (i.e., "width") at each level of the structure. Even for the case t = 1 which corresponds to finding simple paths of length k, our time complexity is a significant improvement over previous probabilistic approaches. To allow for the analysis of multiple pathway structures, we further consider a variant of the algorithm that provides probabilistic guarantees for the top suboptimal path structures with a slight increase in time and space. We show that our algorithm can identify pathway structures with high sensitivity by applying it to protein interaction networks in the DIP database.     

Quantum Complexity of Testing Group Commutativity

We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in . The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of , we give a reduction from a special case of Element Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model.     

Cardinal interpolation with periodic polysplines on strips

Abstract  We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we prove the existence of a unique function in , polyharmonic of order p on each strip , , and periodic in its last n variables, whose restriction to the parallel hyperplanes , , coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in . The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal -splines. Keywords: cardinal -splines, polyharmonic functions, multivariable interpolation Mathematics Subject Classification (2000): 41A05, 41A15, 41A63     

A Note on MODp - MODm Circuits

We give a new proof of recent results of Grolmusz and Tardos on the computing power of constant-depth circuits consisting of a single layer of gates followed by a fixed number of layers of -gates, where p is prime.