A simple factor-3 approximation for labeling points with circles

In this paper we present a simple factor-(3+ε), 0<ε<1, approximation algorithm, which runs in O(nlogn+n(1/ε)^O(1/ε2)log(D₃/εD₂)) time, for the problem of labeling a set P of n distinct points with uniform circles. (D₂ is the closest pair of P and D₃ is the minimum diameter of all subsets of P with size three.) This problem is known to be NP-hard. Our bound improves the previous factor of 3.6+ε.     

On the complexity of finding balanced oneway cuts

A bisection of an n-vertex graph is a partition of its vertices into two sets S and T, each of size n/2. The bisection cost is the number of edges connecting the two sets. In directed graphs, the cost is the number of arcs going from S to T. Finding a minimum cost bisection is NP-hard for both undirected and directed graphs. For the undirected case, an approximation of ratio O(log²n) is known. We show that directed minimum bisection is not approximable at all. More specifically, we show that it is NP-hard to tell whether there exists a directed bisection of cost 0, which we call oneway bisection. In addition, we study the complexity of the problem when some slackness in the size of S is allowed, namely, (1/2−ε)n?|S|?(1/2+ε)n. We show that the problem is solvable in polynomial time when , and provide evidence that the problem is not solvable in polynomial time when ε=o(1/(logn)⁴).     

Spanning trees with minimum weighted degrees

Given a metric graph G, we are concerned with finding a spanning tree of G where the maximum weighted degree of its vertices is minimum. In a metric graph (or its spanning tree), the weighted degree of a vertex is defined as the sum of the weights of its incident edges. In this paper, we propose a 4.5-approximation algorithm for this problem. We also prove it is NP-hard to approximate this problem within a 2−ε factor.     

Pathwidth of cubic graphs and exact algorithms

We prove that for any ?>0 there exists an integer n? such that the pathwidth of every cubic (or 3-regular) graph on n>n? vertices is at most (1/6+?)n. Based on this bound we improve the worst case time analysis for a number of exact exponential algorithms on graphs of maximum vertex degree three.     

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.     

The Complexity of the Empire Colouring Problem

We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in Q. J. Pure Appl. Math. 24:332–338, 1890) for maps containing empires formed by exactly r>1 countries each. We prove that the problem can be solved in polynomial time using s colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than s/r. However, if s≥3, the problem is NP-hard even if the graph is a for forests of paths of arbitrary lengths (for any r≥2, provided $s < 2r - \sqrt{2r + \frac{1}{4}}+ \frac{3}{2}$ ). Furthermore we obtain a complete characterization of the problem’s complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any r≥2, if 3≤s≤2r?1 (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if s<7 for r=2, and s<6r?3, for r≥3. The result for planar graphs also proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than 6r?3 colours graphs of thickness r≥3.     

A Simple Parallel Algorithm for the Single-Source Shortest Path Problem on Planar Digraphs

We present a simple parallel algorithm for the single-source shortest path problem in planar digraphs with nonnegative real edge weights. The algorithm runs on the EREW PRAM model of parallel computation in O((n^2ε+n^1−ε) log n) time, performing O(n^1+ε log n) work for any 0<ε<1/2. The strength of the algorithm is its simplicity, making it easy to implement and presumable quite efficient in practice. The algorithm improves upon the work of all previous parallel algorithms. Our algorithm is based on a region decomposition of the input graph and uses a well-known parallel implementation of Dijkstra's algorithm. The logarithmic factor in both the work and the time can be eliminated by plugging in a less practical, sequential planar shortest path algorithm together with an improved parallel implementation of Dijkstra's algorithm.     

Finding a dominating set on bipartite graphs

Finding a dominating set of minimum cardinality is an NP-hard graph problem, even when the graph is bipartite. In this paper we are interested in solving the problem on graphs having a large independent set. Given a graph G with an independent set of size z, we show that the problem can be solved in time O^∗(²n−z), where n is the number of vertices of G. As a consequence, our algorithm is able to solve the dominating set problem on bipartite graphs in time O^∗(²n/2). Another implication is an algorithm for general graphs whose running time is O(n^1.7088).     

An improved FPTAS for Restricted Shortest Path

Given a graph with a cost and a delay on each edge, Restricted Shortest Path (RSP) aims to find a min-cost s-t path subject to an end-to-end delay constraint. The problem is NP-hard. In this note we present an FPTAS with an improved running time of O(mn/ε) for acyclic graphs, where m and n denote the number of edges and nodes in the graph. Our algorithm uses a scaling and rounding technique similar to that of Hassin [Math. Oper. Res. 17 (1) (1992) 36-42]. The novelty of our algorithm lies in its “adaptivity”. During each iteration of our algorithm the approximation parameters are fine-tuned according to the quality of the current solution so that the running time is kept low while progress is guaranteed at each iteration. Our result improves those of Hassin [Math. Oper. Res. 17 (1) (1992) 36-42], Phillips [Proc. 25th Annual ACM Symposium on the Theory of Computing, 1993, pp. 776-785], and Raz and Lorenz [Technical Report, 1999].     

Inapproximability of the Tutte polynomial of a planar graph

The Tutte polynomial of a graph G is a two-variable polynomial T(G; x, y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: given as input a planar graph G, determine T(G; x, y). Vertigan completely mapped the complexity of exactly computing the Tutte polynomial of a planar graph. He showed that the problem can be solved in polynomial time if (x, y) is on the hyperbola H _q given by (x ? 1)(y ? 1) = q for q = 1 or q = 2 or if (x, y) is one of the two special points (x, y) = (?1, ?1) or (x, y) = (1, 1). Otherwise, the problem is #P-hard. In this paper, we consider the problem of approximating T(G; x, y), in the usual sense of “fully polynomial randomized approximation scheme” or FPRAS. Roughly speaking, an FPRAS is required to produce, in polynomial time and with high probability, an answer that has small relative error. Assuming that NP is different from RP, we show that there is no FPRAS for the Tutte polynomial in a large portion of the (x, y) plane. In particular, there is no FPRAS if x > 1, y < ?1 or if y > 1, x < ?1 or if x < 0, y < 0 and q > 5. Also, there is no FPRAS if x < 1, y < 1 and q = 3. For q > 5, our result is intriguing because it shows that there is no FPRAS at (x, y) =?(1 ? q/(1 + ε), ?ε) for any positive ε but it leaves open the limit point ε =?0, which corresponds to approximately counting q-colorings of a planar graph.     

A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths that either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) it computes a clique partition having size that is guaranteed to be within (1+ε) of the optimum size if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDG’s is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an $O(\frac{\log^{*} n}{{\varepsilon}^{O(1)}})$ time distributed PTAS. We consider a weighted version of the clique partition problem on vertex-weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS break down. Yet, surprisingly, it admits a (2+ε)-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the unweighted case for UDGs expressed in standard form.     

Approximation Schemes for Packing Splittable Items with Cardinality Constraints

We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of n items must be packed into as few bins as possible. Items may be split, but each bin may contain at most?k (parts of) items, where k is some given parameter. Complicating the problem further is the fact that items may be larger than?1, which is the size of a bin. The problem is known to be strongly NP-hard for any fixed value of?k. We essentially close this problem by providing an efficient polynomial-time approximation scheme (EPTAS) for most of its versions. Namely, we present an efficient polynomial time approximation scheme for k=o(n). A?PTAS for k=Θ(n) does not exist unless P = NP. Additionally, we present dual approximation schemes for k=2 and for constant values of?k. Thus we show that for any ε>0, it is possible to pack the items into the optimal number of bins in polynomial time, if the algorithm may use bins of size 1+ε.     

The super connectivity of augmented cubes

The augmented cube AQn, proposed by Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a (2n−1)-regular (2n−1)-connected graph (n≠3). This paper determines that the super connectivity of AQn is 4n−8 for n?6 and the super edge-connectivity is 4n−4 for n?5. That is, for n?6 (respectively, n?5), at least 4n−8 vertices (respectively, 4n−4 edges) of AQn are removed to get a disconnected graph that contains no isolated vertices. When the augmented cube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.     

Circulant block-factorization preconditioning of anisotropic elliptic problems

The recently introduced circulant block-factorization preconditioners are studied in this paper. The general approach is first formulated for the case of block-tridiagonal sparse matrices. Then an estimate of the condition number of the preconditioned matrix for a model anisotropic Dirichlet boundary value problem is derived in the formκ<√2ε(n+1)+2, whereN=n ² is the size of the discrete problem, andε stands for the ratio of the anisotropy. Various numerical tests demonstrating the behavior of the circulant block-factorization preconditioners for anisotropic problems are presented.     

Faster Algorithms for All-Pairs Small Stretch Distances in Weighted Graphs

Let G=(V,E) be a weighted undirected graph, with non-negative edge weights. We consider the problem of efficiently computing approximate distances between all pairs of vertices in?G. While many efficient algorithms are known for this problem in unweighted graphs, not many results are known for this problem in weighted graphs. Zwick?(J. Assoc. Comput. Mach. 49:289–317, 2002) showed that for any fixed ε>0, stretch 1+ε distances (a path in G between u,v∈V is said to be of stretch t if its length is at most t times the distance between u and v in G) between all pairs of vertices in a weighted directed graph on n vertices can be computed in $\tilde{O}(n^{\omega})$ time, where ω<2.376 is the exponent of matrix multiplication and n is the number of vertices. It is known that finding distances of stretch less than 2 between all pairs of vertices in G is at least as hard as Boolean matrix multiplication of two n×n matrices. Here we show that all pairs stretch 2+ε distances for any fixed ε>0 in G can be computed in expected time O(n ^9/4). This algorithm uses a fast rectangular matrix multiplication subroutine. We also present a combinatorial algorithm (that is, it does not use fast matrix multiplication) with expected running time O(n ^9/4) for computing all-pairs stretch 5/2 distances in?G. This combinatorial algorithm will serve as a key step in our all-pairs stretch 2+ε distances algorithm.     

A new approximation algorithm for labeling points with circle pairs

We study the NP-hard problem of labeling points with maximum-radius circle pairs: given n point sites in the plane, find a placement for 2n interior-disjoint uniform circles, such that each site touches two circles and the circle radius is maximized. We present a new approximation algorithm for this problem that runs in time and O(n) space and achieves an approximation factor of (≈1.486+ε), which improves the previous best bound of 1.491+ε.     

On Chromatic Sums and Distributed Resource Allocation

This paper studies an optimization problem that arises in the context of distributed resource allocation: Given a conflict graph that represents the competition of processors over resources, we seek an allocation under which no two jobs with conflicting requirements are executed simultaneously. Our objective is to minimize theaverage response timeof the system. In alternative formulation this is known as theMinimum Color Sum (MCS)problem (E. Kubicka and A. J. Schwenk, 1989. An introduction to chromatic sums,in“Proceedings of the ACM Computer Science Conference,” pp. 39–45.). We show that the algorithm based on finding iteratively a maximum independent set (MaxIS) is a 4-approximation to the MCS. This bound is tight to within a factor of 2. We give improved ratios for the classes of bipartite, bounded-degree, and line graphs. The bound generalizes to a 4ρ-approximation of MCS for classes of graphs for which the maximum independent set problem can be approximated within a factor ofρ. On the other hand, we show that ann^1−ε-approximation is NP-hard, for someε>0. For some instances of the resource allocation problem, such as theDining Philosophers, an efficient solution requiresedgecoloring of the conflict graph. We introduce theMinimum Edge Color Sum (MECS)problem which is shown to be NP-hard. We show that a 2-approximation to MECS(G) can be obtained distributively usingcompactcoloring withinO(log² n) communication rounds     

Regular connected bipancyclic spanning subgraphs of hypercubes

An n-dimensional hypercube Q_n is a Hamiltonian graph; in other words Q_n (n≥2) contains a spanning subgraph which is 2-regular and 2-connected. In this paper, we explore yet another strong property of hypercubes. We prove that for any integer k with 3≤k≤n, Q_n (n≥3) contains a spanning subgraph which is k-regular, k-connected and bipancyclic. We also obtain the result that every mesh P_m×P_n (m,n≥2) is bipancyclic, which is used to prove the property above.     

Random Walks,Bisections and Gossiping in Circulant Graphs

Circulant graphs are regular graphs based on Cayley graphs defined on the Abelian group \(\mathbb{Z}_{n}\) . They are popular network topologies that arise in distributed computing. Using number theoretical tools, we first prove two main results for random directed k-regular circulant graphs with n vertices, when n is sufficiently large and k is fixed. First, for any fixed ε>0, n=p prime and L≥p ^1/k (logp)^1+1/k+ε , walks of length at most L terminate at every vertex with asymptotically the same probability. Second, for any n, there is a polynomial time algorithm that for almost all undirected 2r-regular circulant graphs finds a vertex bisector and an edge bisector, both of size less than n ^1?1/r+o(1). We then prove that the latter result also holds for all (rather than for almost all) 2r-regular circulant graphs with n=p, prime, vertices, while, in general, it does not hold for composite n. Using the bisection results, we provide lower bounds on the number of rounds required by any gossiping algorithms for any n. We introduce generic distributed algorithms to solve the gossip problem in any circulant graphs. We illustrate the efficiency of these algorithms by giving nearly matching upper bounds of the number of rounds required by these algorithms in the vertex-disjoint and the edge-disjoint paths communication models in particular circulant graphs.     

Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs

A bipartite graph G is bipancyclic if G has a cycle of length l for every even 4?l?|V(G)|. For a bipancyclic graph G and any edge e, G is edge-bipancyclic if e lies on a cycle of any even length l of G. In this paper, we show that the bubble-sort graph Bn is bipancyclic for n?4 and also show that it is edge-bipancyclic for n?5. Assume that F is a subset of E(Bn). We prove that Bn−F is bipancyclic, when n?4 and |F|?n−3. Since Bn is a (n−1)-regular graph, this result is optimal in the worst case.