Inapproximability of the Tutte polynomial

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: take as input a graph G, and output a value which is a good approximation to T(G;x,y). Jaeger et al. have completely mapped the complexity of exactly computing the Tutte polynomial. They have shown that this is #P-hard, except along the hyperbola (x-1)(y-1)=1 and at four special points. We are interested in determining for which points (x,y) there is a fully polynomial randomised approximation scheme (FPRAS) for T(G;x,y). Under the assumption RP≠NP, we prove that there is no FPRAS at (x,y) if (x,y) is in one of the half-planes x<-1 or y<-1 (excluding the easy-to-compute cases mentioned above). Two exceptions to this result are the half-line x<-1,y=1 (which is still open) and the portion of the hyperbola (x-1)(y-1)=2 corresponding to y<-1 which we show to be equivalent in difficulty to approximately counting perfect matchings. We give further intractability results for (x,y) in the vicinity of the origin. A corollary of our results is that, under the assumption RP≠NP, there is no FPRAS at the point (x,y)=(0,1-λ) when λ>2 is a positive integer. Thus, there is no FPRAS for counting nowhere-zero λ flows for λ>2. This is an interesting consequence of our work since the corresponding decision problem is in P for example for λ=6. Although our main concern is to distinguish regions of the Tutte plane that admit an FPRAS from those that do not, we also note that the latter regions exhibit different levels of intractability. At certain points (x,y), for example the integer points on the x-axis, or any point in the positive quadrant, there is a randomised approximation scheme for T(G;x,y) that runs in polynomial time using an oracle for an NP predicate. On the other hand, we identify a region of points (x,y) at which even approximating T(G;x,y) is as hard as #P.     

Finding the closed partition of a planar graph

In this paper we consider the problem of finding aclosed partition in a directed graph. This problem has applications in concurrent probabilistic program verification. The best sequential algorithm known for this problem runs inO(mn) time wherem is the number of directed edges andn is the number of vertices in the given digraph. In this paper we present a linear-time sequential algorithm to solve the closed partition problem for planar digraphs that arecompact. We then build on this algorithm to obtain an O(n^1.5)-time sequential algorithm to solve the closed partition problem for a general planar digraph.This work was supported in part by NSF Grant CCR 89-10707.     

Algorithm 24 algorithm for deriving the chromatic polynomial of a graph

An algorithm for deriving the chromatic polynomial of a graph in coefficient form is described. A FORTRAN implementation is given, and some computational experience is summarized.     

A planar graph drawing program

This paper describes a program, written in FORTRAN, for a 64K I.C.L. 4130 which will accept data describing the ordering of nodes and branches within the regions of a planar graph, and generate a two-dimensional representation, without crossovers, from these data. The program was written to provide one of the basic ‘tools’ required in the development of a printed wiring board layout program for the electronics industry The program generates display code which is sent to a satellite PDP-7 computer driving a 340 display. On completion of the automatic drawing phase an interactive phase is entered in which the user can ‘tidy’ and label the drawing, by means of keyboard and light-pen commands Brief notes are included on the hardware, the data structures package (MINIJASP, derived from ASP) and the graphics package, employed.     

On the feedback vertex set problem for a planar graph

An algorithm solving the feedback-vertex-set problem for planar digraphs is described. In particular, planar graphs with a certain additional condition are considered as they arise from solving systems of linear equations obtained from convection-dominated flow problems. The proposed algorithm requires a computational work linear in the size of the graph. Furthermore, a side-product is a decomposition of the graph into subsets that are of interest for block-Gauß-Seidel smoothers in multi-grid methods.     

The relative neighbourhood graph of a finite planar set

The relative neighbourhood graph (RNG) of a set of n points on the plane is defined. The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree (MST) and the Delaunay (Voronoi) triangulation (DT). It is shown that the RNG is a superset of the MST and a subset of the DT. Two algorithms for obtaining the RNG of n points on the plane are presented. One algorithm runs in 0(n²) time and the other runs in 0(n³) time but works also for the d-dimensional case. Finally, several open problems concerning the RNG in several areas such as geometric complexity, computational perception, and geometric probability, are outlined.     

Speeding up computation of the reliability polynomial coefficients for a random graph

We consider the problem of computing the coefficients of the reliability polynomial (RP) for a random graph with reliable vertices and unreliable edges. To speed up the computation, we use the meaning of RP coefficients in one of its representations and prove vector relations over vectors of binomial coefficients.     

Optimal parallel algorithm for findingst-ambitus of a planar biconnected graph

A cycleC passing through two specific verticess andt of a biconnected graph is said to be anst-ambitus if its bridges do not interlace in some special way. We present algorithms forst-ambitus for planar biconnected graphs, which are much simpler than the one known for general graphs [MT]. Our algorithm runs inO(n) time on a sequential machine and (logn) parallel time usingO(n/logn) processors on an EREW PRAM.     

NP-completeness of maximum-cut and cycle-covering problems for a planar graph

We consider problems of finding the maximum cut and a cycle covering for a planar graph with edge weights of arbitrary sign. Methods that find the maximum cut in graphs with only positive edge weights are shown to be inapplicable in this case. NP-completeness of the problem is proved.Translated from Kibernetika, No. 1, pp. 13–16, January–February, 1991.     

Some properties of the planar Euclidean relative neighbourhood graph

Some simple properties of the planar Euclidean relative neighbourhood graph are derived. Interesting properties are already known for a similar structure - the Gabriel graph - and a comparison of analogous properties is given.     

Predicting nearly as well as the best pruning of a planar decision graph

We design efficient on-line algorithms that predict nearly as well as the best pruning of a planar decision graph. We assume that the graph has no cycles. As in the previous work on decision trees, we implicitly maintain one weight for each of the prunings (exponentially many). The method works for a large class of algorithms that update its weights multiplicatively. It can also be used to design algorithms that predict nearly as well as the best convex combination of prunings.     

A linear algorithm to find a rectangular dual of a planar triangulated graph

We develop anO(n) algorithm to construct a rectangular dual of ann-vertex planar triangulated graph.     

A linear algorithm to find a rectangular dual of a planar triangulated graph

We develop anO(n) algorithm to construct a rectangular dual of ann-vertex planar triangulated graph.This research was supported in part by the National Science Foundation under Grant MCS-83-05567.     

Exponential Inapproximability of Selecting a Maximum Volume Sub-matrix

Given a matrix A∈? ^m×n (n vectors in m dimensions), and a positive integer k<n, we consider the problem of selecting k column vectors from A such that the volume of the parallelepiped they define is maximum over all possible choices. We prove that there exists δ<1 and c>0 such that this problem is not approximable within 2^?ck for k=δn, unless P=NP.     

A sufficient condition for a planar graph to be 3-choosable

It is shown that a planar graph without cycles of length 4, 6, 8, or 9 is 3-choosable.     

Inapproximability of the kidney exchange problem

To overcome the shortage of kidneys available for transplantation, several countries have started various programmes of kidney exchanges between incompatible patient-donor pairs. This situation can be modeled as a cooperative game in which the players are the patient-donor pairs and their preferences are derived in the first step from the suitability of the donated kidney and in the second step from the length of the obtained cycle of exchanges. Although the core of such a cooperative game is always nonempty and one core solution can be found by the famous Top Trading Cycles algorithm, many questions concerning the structure of the core are NP-hard. In this paper we show that the problem of finding a core permutation that maximizes the number of patients who obtain a suitable kidney is not approximable within n1−ε for any ε>0 unless P=NP.     

On simultaneous straight-line grid embedding of a planar graph and its dual

Simultaneous representations of planar graphs and their duals normally require that the dual vertices to be placed inside their corresponding primal faces, and the edges of the dual graph to cross only their corresponding primal edges. Erten and Kobourov [C. Erten, S.G. Kobourov, Simultaneous embedding of a planar graph and its dual on the grid, Theory Computer Systems 38 (2005) 313-327] provided a linear time algorithm on simultaneous straight-line grid embedding of a 3-connected planar graph and its dual such that all the vertices are placed on grid points and each edge is drawn as one straight-line segment except for one which is drawn using two segments. Their drawing size is (2n−2)×(2n−2), where n is the total number of vertices in the graph and its dual. They raised an open question on whether there is a large class of planar graphs that allows this simultaneous straight-line grid embedding on a smaller grid. We answer this open question by giving a linear time simultaneous straight-line grid embedding algorithm for a 3-connected planar graph and its dual on a grid of size (n−1)×n.     

MTPSO algorithm for solving planar graph coloring problem

In this paper, we proposed a modified turbulent particle swarm optimization (named MTPSO) model for solving planar graph coloring problem based on particle swarm optimization. The proposed model is consisting of the walking one strategy, assessment strategy and turbulent strategy. The proposed MTPSO model can solve the planar graph coloring problem using four-colors more efficiently and accurately. Compared to the results shown in Cui et al. (2008), not only the experimental results of the proposed model can get smaller average iterations but can get higher correction coloring rate when the number of nodes is greater than 30.     

Inapproximability of maximal strip recovery

In comparative genomics, the first step of sequence analysis is usually to decompose two or more genomes into syntenic blocks that are segments of homologous chromosomes. For the reliable recovery of syntenic blocks, noise and ambiguities in the genomic maps need to be removed first. Maximal Strip Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff for reliably recovering syntenic blocks from genomic maps in the midst of noise and ambiguities. Given d genomic maps as sequences of gene markers, the objective of MSR-d is to find d subsequences, one subsequence of each genomic map, such that the total length of syntenic blocks in these subsequences is maximized. For any constant d≥2, a polynomial-time 2d-approximation for MSR-d was previously known. In this paper, we show that for any d≥2, MSR-d is APX-hard, even for the most basic version of the problem in which all gene markers are distinct and appear in positive orientation in each genomic map. Moreover, we provide the first explicit lower bounds on approximating MSR-d for all d≥2. In particular, we show that MSR-d is NP-hard to approximate within Ω(d/logd). From the other direction, we show that the previous 2d-approximation for MSR-d can be optimized into a polynomial-time algorithm even if d is not a constant but is part of the input. We then extend our inapproximability results to several related problems including CMSR-d, δ-gap-MSR-d, and δ-gap-CMSR-d.