Parametric problems on graphs of bounded tree-width

Self-avoiding walks interacting with a surface are a useful model of polymer adsorption and one can choose to weight the walk according to the number of vertices or the number of edges in the surface. We consider a combined model of self-avoiding walks on the d-dimensional hypercubic lattice with weights associated with the numbers of vertices and edges in the surface and investigate its phase diagram. We also derive inequalities relating the critical points in the edge and vertex weighting schemes.

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomialZ(G;x,y,z). This polynomial was studied with respect to its approximability by Goldberg, Jerrum and Paterson.Z(G;x,y,z) generalizes a bivariate polynomialZ(G;t,y), which was studied in by Andrén and Markström.We consider the complexity ofZ(Gt,y) andZ(G;x,y,z) in comparison to that of the Tutte polynomial, which is well known to be closely related to the Potts model in the absence of an external field. We show thatZ(G;x,y,z) is #P-hard to evaluate at all points in3, except those in an exceptional set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by Dell, Husfeldt and Wahlén in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETHa dichotomy theorem stating that evaluations ofZ(G;t,y) either take exponential time in the number of vertices ofGto compute, or can be done in polynomial time. Finally, we give an algorithm for computingZ(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.

Multi-way graph partition by stochastic probe

We consider optimization problems on weighted trees, including weighted versions of the minimum vertex cover and the minimum dominating set problems, where vertex and edge weights vary as linear functions of time. For each of the problems studied here, we derive bounds on the number of breakpoints of the piecewise linear functions describing the cost of the optimum solution and we present efficient methods for constructing these functions. We also consider the problem of determining optimum steady-state solutions and finding the time t∞ at which the steady state is reached.

Given a set W of k sequences (strings) and a tree structure T with k leaves, each of which is labeled with a unique sequence in W, a tree alignment is to label a sequence to each internal node of T. The weight of an edge of the tree alignment is the distance, such as Hamming distance, Levenshtein (edit) distance or reversal distance, between the two sequences labeled to the two ends of the edge. The bottleneck tree alignment problem is to find a tree alignment such that the weight of the largest edge is minimized. A lifted tree alignment is a tree alignment, where each internal node v is labeled one of the sequences that was labeled to the children of v. The bottleneck lifted tree alignment problem is to find a lifted tree alignment such that the weight of the largest edge is minimized. In this paper, we show that the bottleneck tree alignment problem is NP-complete even when the tree structure is the binary tree and the weight function is metric. For special cases, we present an exact algorithm to solve the bottleneck lifted tree alignment problem in polynomial time. If the weight function is ultrametric, we show that any lifted tree alignment is an optimal bottleneck tree alignment.KeywordsEdit distancebottleneck tree alignmentmetricultrametricNP-complete

Fully Dynamic Algorithms for Maintaining Shortest Paths Trees

Randomized online graph coloring

A linear time algorithm for the maximum capacity path problem

Let $G$ be a graph embedded in an orientable surface $\Sigma$, possibly with edge weights, and denote by ${\rm len}(\gamma)$ the length (the number of edges or the sum of the edge weights) of a cycle $\gamma$ in $G$. The stretch of a graph embedded on a surface is the minimum of ${\rm len}(\alpha)\cdot {\rm len}(\beta)$ over all pairs of cycles $\alpha$ and $\beta$ that cross exactly once. We provide two algorithms to compute the stretch of an embedded graph, each based on a different principle. The first algorithm is based on surgery and computes the stretch in time $O(g^4 n \log n)$ with high probability, or in time $O(g^4 n \log^2 n)$ in the worst case, where $g$ is the genus of the surface $\Sigma$ and $n$ is the number of vertices in $G$. The second algorithm is based on using a short homology basis and computes the stretch in time $O(n^2\log n + n^2g + ng^3)$.

This paper deals with the labeling of vertices and edges of a graph. Let G be a graph with vertex set V and edge set E, where |V| be the number of vertices and |E| edges of G. The two bijection methods for which we have designed algorithms are as follows. Initially A bijection λ 1:V∪E → {1, 2…|V| + |E|} is called a Vertex-Magic Total Labeling (VMTL) if there is a vertex magic constant vk such that the weight of vertex m is, λ 1(m) + Σ n εA (m) λ 1(mn) = vk, ∀ m εv Where A(m) is the set of vertices adjacent to x. In the similar fashion the bijection λ 2:V∪E → {1, 2…|V| + |E|} is called Edge-Magic Total Labeling (EMTL) if there is a edge magic constant ek such that the weight of an edge es(mn), λ 2(m) + λ 2(n) + λ 2 e(mn) = ek, ∀ e εE. A resultant Graph which consists of both VMTL and EMTL are said to be Total Magic Labeling (TML) for different vertex magic constant and edge magic constant values. Baker and Sawada proposed algorithms to find VMTLs on cycles and wheels. In this paper we enhanced these algorithms and also we proposed new algorithms to generate EMTLs and TMLs of cycles and wheels. We used the concept variations and sum set sequences to produce VMTLs and EMTLs on cycles and wheels. Also we designed modules to identify TML’s.KeywordsMagic labelingMagic labeling algorithmsMagic constantCyclesWheelsVertex Magic Total LabelingEdge Magic Total LabelingTotal Magic Labeling etc

In this paper, we define a distributionally robust k-sum optimization problem as the problem of finding a solution that minimizes the worst-case expected sum of up to the k largest costs of the elements in the solution. The costs are random with a joint probability distribution that is not completely specified but rather assumed to be known to lie in a set of probability distributions. For k=1, this reduces to a distributionally robust bottleneck optimization problem while for k=n, this reduces to distributionally robust minimum sum optimization problem. Our main result is that for a Fréchet class of discrete marginal distributions with finite support, the distributionally robust k-sum combinatorial optimization problem is solvable in polynomial time if the deterministic minimum sum problem is solvable in polynomial time. This extends the result of Punnen and Aneja [On k-sum optimization, Oper. Res. Lett. 18(5)(1996), pp. 233–236] from the deterministic to the robust case. We show that this choice of the set of distributions helps preserve the submodularity of the k-sum objective function which is an useful structural property for optimization problems.

Let G=(V,E) be an n-vertex connected graph with positive edge weights. A subgraph G′ = (V,E′) is a t-spanner of G if for all u, v e V,the weighted distance between u and v in G′ is at most t times the weighted distance between u and v in G. We consider the problem of constructing sparse spanners, and the weight, defined as the sum of the edge weights in the spanner. In this paper, we concentrate on constructing spanners of small weight.For an arbitrary positive edge-weighted graph G, for any t> 1, and any e>0, we show that a t-spanner of G with weight O(n(2+e)/(t-1>) •wt(MST) can be constructed in polynomial time. We also show that (log2n)-spanners of weight O(log n)•wt(MST) can be constructed.We then consider spanners for complete graphs induced by a set of points in d-dimensional real normed space. The weight of an edge xy is the norm of the xy vector. We show that for these graphs, t-spanners with total weight O(log n)•wt(MST) can be constructed in polynomial time.

We study a problem of energy-efficiently connecting a symmetric wireless communication network: given an n-vertex graph with edge weights, find a connected spanning subgraph of minimum cost, where the cost is determined by each vertex paying the heaviest edge incident to it in the subgraph. The problem is known to be NP-hard. Strengthening this hardness result, we show that even o(log n)-approximating the difference d between the optimal solution cost and a natural lower bound is NP-hard. Moreover, we show that under the exponential time hypothesis, there are no exact algorithms running in 2o(n) time or in [Formula: see text] time for any computable function f. We also show that the special case of connecting c network components with minimum additional cost generally cannot be polynomial-time reduced to instances of size cO(1) unless the polynomial-time hierarchy collapses. On the positive side, we provide an algorithm that reconnects O(log n)-connected components with minimum additional cost in polynomial time. These algorithms are motivated by application scenarios of monitoring areas or where an existing sensor network may fall apart into several connected components because of sensor faults. In experiments, the algorithm outperforms CPLEX with known integer linear programming (ILP) formulations when n is sufficiently large compared with c. Summary of Contribution: Wireless sensor networks are used to monitor air pollution, water pollution, and machine health; in forest fire and landslide detection; and in natural disaster prevention. Sensors in wireless sensor networks are often battery-powered and disposable, so one may be interested in lowering the energy consumption of the sensors in order to achieve a long lifetime of the network. We study the min-power symmetric connectivity problem, which models the task of assigning transmission powers to sensors so as to achieve a connected communication network with minimum total power consumption. The problem is NP-hard. We provide perhaps the first parameterized complexity study of optimal and approximate solutions for the problem. Our algorithms work in polynomial time in the scenario where one has to reconnect a sensor network with n sensors and O(log n)-connected components by means of a minimum transmission power increase or if one can find transmission power lower bounds that already yield a network with O(log n)-connected components. In experiments, we show that, in this scenario, our algorithms outperform previously known exact algorithms based on ILP formulations.

Location of an undesirable facility in a polygonal region with forbidden zones

An approximation algorithm for K-best enumeration of minimal connected edge dominating sets with cardinality constraints