Optimization Problems in Multiple Subtree Graphs

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multiple-interval graphs by considering three classical problems: MINIMUM VERTEX COVER, MINIMUM DOMINATING SET, and MAXIMUM CLIQUE. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multiple-interval graph. For MINIMUM VERTEX COVER, we give a (2--1/t)-approximation algorithm which equals the best known ratio for 2t--1 bounded degree graphs. Since these graphs are known to be included in multiple-interval graphs with t intervals associated to each vertex, this ratio is in some sense tight. Following this, we give a t2-approximation algorithm for MINIMUM DOMINATING SET which adapts well to more general variants of the problem. We then proceed to prove that MAXIMUM CLIQUE is NP-complete for the case of t = 3, and provide a (t2--t+1)/2-approximation algorithm for the problem, using recent bounds proven for the so-called transversal number of t-interval families.

Borodin, Nielsen, and Rackoff ([5]) gave a model of greedy-like algorithms for scheduling problems and [1] extended their work to facility location and set cover problems. We generalize their notion to include other optimization problems, and apply the generalized framework to graph problems. Our goal is to define an abstract model that captures the intrinsic power and limitations of greedy algorithms for various graph optimization problems. We prove bounds on the approximation ratio achievable by such algorithms for basic graph problems such as shortest path, vertex cover, and others. Shortest path is an example of a problem where no algorithm in the FIXED priority model can achieve any approximation ratio (even one dependent on the graph size), but for which the well-known Dijkstra's algorithm shows that an ADAPTIVE priority algorithm can be optimal. We also prove that the approximation ratio for vertex cover achievable by ADAPTIVE priority algorithms is exactly 2. Here, a new lower bound matches the known upper bounds ([8]).

This paper studies the semidefinite programming SDP problem, i.e., the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First the classical cone duality is reviewed as it is specialized to SDP is reviewed. Next an interior point algorithm is presented that converges to the optimal solution in polynomial time. The approach is a direct extension of Ye’s projective method for linear programming. It is also argued that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity carrying over in a similar fashion. Finally, the significance of these results is studied in a variety of combinatorial optimization problems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graph...

It is known that the problem of computing the adjacency dimension of a graph is NP-hard. This suggests finding the adjacency dimension for special classes of graphs or obtaining good bounds on this invariant. In this work we obtain general bounds on the adjacency dimension of a graph G in terms of known parameters of G . We discuss the tightness of these bounds and, for some particular classes of graphs, we obtain closed formulae. In particular, we show the close relationships that exist between the adjacency dimension and other parameters, like the domination number, the location-domination number, the 2 -domination number, the independent 2 -domination number, the vertex cover number, the independence number and the super domination number.

We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, t -Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W[i]-hard for some i≥1 in general graphs. We also show that the Dominating Set problem is W[2]-hard for bipartite graphs and hence for triangle free graphs. In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, parameterized by k, is W[1]-hard even on graphs with girth at least six. Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph.

In communication networks such as social networks, wireless networks and biology networks, it is of importance to find all cliques which can help understand the network topology. The clique structures can also be utilized in facilitating message forwarding in wireless networks. For instance, using a set of cliques of maximal and disjoint, one of the nodes in each clique can be elected to forward messages, by which the duplicate message transmissions can be efficiently reduced. Further, it is well known that the maximum clique problem (MCP) is closely related to the maximum independent set and the vertex cover problems. In recent years, the fundamental problem of finding maximal cliques or maximum cliques has attracted lots of attentions. However, few of these works focus on distributed solutions in wireless networks. In this paper, we pay our attention to this missing corner of research. Specifically, we first give a distributed algorithm which can compute all maximal cliques in a wireless network represented by a graph. The algorithm takes O(n) time and uses O(mn) messages, where n is the number of nodes and m the number of edges. Then, with the proposed algorithms MCP (maximum clique problem) and UMCP (unique MCP), we show that a unique maximum clique can be selected from all maximal cliques in O(n) rounds and using O(mn) messages. To the best of our knowledge, our algorithms are the first deterministic distributed solutions for MCP in wireless networks.

We introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments. We find that for several real-world hypergraphs their two cores tend to be much smaller than those of their null models, suggesting that covering problems in those real-world hypergraphs can actually be solved in polynomial time.

We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (WUDC) problem asks for a subset of disks of minimum total weight that covers all given points. WUDC is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted WUDC problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted WUDC problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for WUDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.

We present a fixed-parameter algorithm that constructively solves the $k$-dominating set problem on any class of graphs excluding a single-crossing graph (a graph that can be drawn in the plane with at most one crossing) as a minor in $O(4^{9.55\sqrt{k}}n^{O(1)})$ time. Examples of such graph classes are the $K_{3,3}$-minor-free graphs and the $K_{5}$-minor-free graphs. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, clique-transversal set, kernels in digraphs, feedback vertex set, and a collection of vertex-removal problems. Our work generalizes and extends the recent results of exponential speedup in designing fixed-parameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.

We prove a characterization of all polynomial-time computable queries on the class of interval graphs by sentences of fixed-point logic with counting. More precisely, it is shown that on the class of unordered interval graphs, any query is polynomial-time computable if and only if it is definable in fixed-point logic with counting. This result is one of the first establishing the capturing of polynomial time on a graph class which is defined by forbidden induced subgraphs. For this, we define a canonical form of interval graphs using a type of modular decomposition, which is different from the method of tree decomposition that is used in most known capturing results for other graph classes, specifically those defined by forbidden minors. The method might also be of independent interest for its conceptual simplicity. Furthermore, it is shown that fixed-point logic with counting is not expressive enough to capture polynomial time on the classes of chordal graphs or incomparability graphs.

In this paper, some basic traffic management problems in future mobile communications networks are considered: thechannel minimization problem and themaximum channel reuse problem. These are formulated as mathematical optimization problems in a bipartite graph. We prove the intractability of these problems and discuss relations to well-known graph optimization problems as, for example, vertex coloring. Simple and fast heuristic methods for finding reasonable solutions to the problems mentioned above are presented. Results from computational work using these heuristics on simulated traffic are also given.

The weighted independent domination problem: Integer linear programming models and metaheuristic approaches

Improved Approximation Algorithms for Optimization Problems in Graphs with Superlogarithmic Treewidth

The undirected negative cost cycle detection (UNCCD) problem is concerned with checking whether an undirected, weighted graph contains a negative cost cycle. Known approaches for solving this problem involve reducing it to either the minimum weight$$b$$b-matching problem or the minimum weight$$T$$T-join problem. In this paper, we formally describe these two approaches and provide the tightest known analysis of the $$b$$b-matching approach. Our analysis establishes that the $$b$$b-matching approach runs in $$O((n+m)^{2}\cdot \log (n+m))$$O((n+m)2·log(n+m)) time on a graph with $$n$$n nodes and $$m$$m edges. We also explore the case where the edge costs are integers in the range $$\{-K,\ldots , K\}$${-K,?,K}. For $$K = O(1)$$K=O(1), we provide improved time bounds for both the above-mentioned approaches by exploiting the existence of specialized procedures for optimization problems in graphs with integral positive edge costs. We show that while the $$T$$T-join algorithm has a better time bound for general graphs, the $$b$$b-matching algorithm is more efficient for sparse graphs. Finally, we present one of the first, extensive empirical studies for the UNCCD problem. We examine both approaches on several different families of undirected graphs. Our findings provide insight into the actual efficiency of each approach and reinforce the asymptotic analysis of our algorithms.

The class of fork-free graphs is an extension of claw-free graphs and their subclass of line graphs. The first polynomial-time solution to the maximum weight independent set problem in the class of line graphs, which is equivalent to the maximum matching problem in general graphs, has been proposed by Edmonds in 1965 and then extended to the entire class of claw-free graphs by Minty in 1980. Recently, Alekseev proposed a solution for the larger class of fork-free graphs, but only for the unweighted version of the problem, i.e. finding an independent set of maximum cardinality. In the present paper, we describe the first polynomial-time algorithm to solve the problem for weighted fork-free graphs.