Graph Optimization Problem Research Articles

Semi-nice tree-decompositions: The best of branchwidth, treewidth and pathwidth with one algorithm

Branchwidth and treewidth are connectivity parameters of graphs of high importance in algorithm design. By dynamic programming along the associated branch- or tree-decomposition one can solve most graph optimization problems in time linear in the graph size and exponential in the parameter. If one of these parameters is bounded on a class of graphs, then so is the other, but they can differ by a small constant factor and this difference can be crucial for the resulting runtime. In this paper we introduce semi-nice tree-decompositions and show that they combine the best of both branchwidth and treewidth. We first give simple algorithms to transform a given tree-decomposition or branch-decomposition into a semi-nice tree-decomposition. We then give two templates for dynamic programming along a semi-nice tree-decomposition, one for optimization problems over vertex subsets and another for optimization problems over edge subsets. We show that the resulting runtime will match or beat the runtimes achieved by doing dynamic programming directly on either a branch- or tree-decomposition. For example, given a graph G on n vertices with path-, tree- and branch-decompositions of width p w , t w and b w respectively, the Minimum Dominating Set problem on G is solved in time O ( n 2 min { 1.58 p w , 2 t w , 2.38 b w } ) by a single dynamic programming algorithm along a semi-nice tree-decomposition. On the applied side the immediate gain is that for each optimization problem one can achieve the benefits of both treewidth, branchwidth and pathwidth while developing and implementing only one dynamic programming algorithm. On the theoretical side the combination of the best properties of both branchwidth and treewidth in a single decomposition is a step towards a more general framework giving the fastest possible algorithms on tree-like graphs.

Probabilistic graph-coloring in bipartite and split graphs

We revisit in this paper the stochastic model for minimum graph-coloring introduced in (Murat and Paschos in Discrete Appl. Math. 154:564–586, 2006), and study the underlying combinatorial optimization problem (called probabilistic coloring) in bipartite and split graphs. We show that the obvious 2-coloring of any connected bipartite graph achieves standard-approximation ratio 2, that when vertex-probabilities are constant probabilistic coloring is polynomial and, finally, we propose a polynomial algorithm achieving standard-approximation ratio 8/7. We also handle the case of split graphs. We show that probabilistic coloring is NP-hard, even under identical vertex-probabilities, that it is approximable by a polynomial time standard-approximation schema but existence of a fully a polynomial time standard-approximation schema is impossible, even for identical vertex-probabilities, unless P=NP. We finally study differential-approximation of probabilistic coloring in both bipartite and split graphs.

Models of Greedy Algorithms for Graph Problems

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]).

Simple and fast surrogate constraint heuristics for the maximum independent set problem

In a recent paper Glover (J. Heuristics 9:175---227, 2003) discussed a variety of surrogate constraint-based heuristics for solving optimization problems in graphs. The key ideas put forth in the paper were illustrated by giving specializations designed for certain covering and coloring problems. In particular, a family of methods designed for the maximum cardinality independent set problem was presented. In this paper we report on the efficiency and effectiveness of these methods based on considerable computational testing carried out on test problems from the literature as well as some new test problems.

WAOA 2005 Special Issue of TOCS

This special issue of Theory of Computing Systems is devoted to selected papers from the 3rd Workshop on Approximation and Online Algorithms (WAOA 2005) that took place in Palma de Mallorca, Spain in October 2005. Of 68 papers submitted to the workshop, 26 were accepted and 6 were invited to this special issue. The papers in this special issue deal with several of the most fundamental topics in approximation and online algorithms: optimization problems in graphs and networks, packing problems, and scheduling problems. Some of the papers consider new variants of classical problems. Bar-Yehuda, Feldman and Rawitz investigate the convex recoloring problem for trees, a problem arising in the context of phylogenetic tree reconstruction. They present an improved approximation algorithm, based on the local ratio technique and dynamic programming, as well as a fixed-parameter tractable algorithm. Blaser and Ram study the classical traveling salesman problem in a game-theoretic setting. They prove that it is NP-hard to determine whether the core of a metric TSP game is empty or not and provide approximately fair cost allocation schemes. Han, Iwama and Zhang consider the square packing problem, a two-dimensional geometric variant of the knapsack problem, in an online setting where previously packed squares can be removed later on. They give tight bounds on the competitive ratio for the

The Complexity of Combinatorial Optimization Problems on d‐Dimensional Boxes

The MAXIMUM INDEPENDENT SET problem in d‐box graphs, i.e., in intersection graphs of axis‐parallel rectangles in $\mathbb{R}^d$, is known to be NP‐hard for any fixed $d\geq 2$. A challenging open problem is that of how closely the solution can be approximated by a polynomial time algorithm. For the restricted case of d‐boxes with bounded aspect ratio a PTAS exists [T. Erlebach, K. Jansen, and E. Seidel, SIAM J. Comput., 34 (2005), pp. 1302–1323]. In the general case no polynomial time algorithm with approximation ratio $o(\log^{d-1} n)$ for a set of n d‐boxes is known. In this paper we prove APX‐hardness of the MAXIMUM INDEPENDENT SET problem in d‐box graphs for any fixed $d\geq 3$. We give an explicit lower bound $\frac{245}{244}$ on efficient approximability for this problem unless $\PP=\text{\rm NP}$. Additionally, we provide a generic method how to prove APX‐hardness for other graph optimization problems in d‐box graphs for any fixed $d\geq 3$.

Shortest‐path metric approximation for random subgraphs

AbstractWe consider graph optimization problems where the cost of a solution depends only on the shortest‐path metric in the graph, such as Steiner Tree or Traveling Salesman. We study a scenario where such a problem needs to be solved repeatedly on random subgraphs of a given graph G. With the goal of speeding up the repeated queries and saving space, we describe the construction of a sparse subgraph Q ⊂ G, which contains approximately an optimal solution for any such problem on a random subgraph of G, with high probability. More precisely, the subgraph Q has the property that after some vertices or edges are removed randomly, Q still contains c‐approximate shortest paths between all pairs of vertices with high probability. The number of edges in Q is O(p−cn1+2/c log n) for edge‐induced random subgraphs and O(p−2cn1+2/c log 2n) for vertex‐induced random subgraphs, where n is the number of vertices in G, p the sampling probability of edges/vertices, and c ∈ ℤ, c ≥ 3 is the desired approximation factor. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007

Genetic algorithm for the personnel assignment problem with multiple objectives

The assignment problem is a well-known graph optimization problem defined on weighted-bipartite graphs. The objective of the standard assignment problem is to maximize the summation of the weights of the matched edges of the bipartite graph. In the standard assignment problem, any node in one partition can be matched with any node in the other partition without any restriction. In this paper, variations of the standard assignment problem are defined with matching constraints by introducing structures in the partitions of the bipartite graph, and by defining constraints on these structures. According to the first constraint, the matching between the two partitions should respect the hierarchical-ordering constraints defined by forest and level graph structures produced by using the nodes of the two partitions respectively. In order to define the second constraint, the nodes of the partitions of the bipartite graph are distributed into mutually exclusive sets. The set-restriction constraint enforces the rule that in one of the partitions all the elements of each set should be matched with the elements of a set in the other partition. Even with one of these constraints the assignment problem becomes an NP-hard problem. Therefore, the extended assignment problem with both the hierarchical-ordering and set-restriction constraints becomes an NP-hard multi-objective optimization problem with three conflicting objectives; namely, minimizing the numbers of hierarchical-ordering and set-restriction violations, and maximizing the summation of the weights of the edges of the matching. Genetic algorithms are proven to be very successful for NP-hard multi-objective optimization problems. In this paper, we also propose genetic algorithm solutions for different versions of the assignment problem with multiple objectives based on hierarchical and set constraints, and we empirically show the performance of these solutions.

A near-optimal multicast scheme for mobile ad hoc networks using a hybrid genetic algorithm

Multicast routing is an effective way to communicate among multiple hosts in a network. It outperforms the basic broadcast strategy by sharing resources along general links, while sending information to a set of predefined multiple destinations concurrently. However, it is vulnerable to component failure in ad hoc network due to the lack of redundancy, multiple paths and multicast tree structure. Tree graph optimization problems (GOP) are usually difficult and time consuming NP-hard or NP-complete problems. Genetic algorithms (GA) have been proven to be an efficient technique for solving the GOP, in which well-designed chromosomes and appropriate operators are key factors that determine the performance of the GAs. Limited link, path constraints, and mobility of network hosts make the multicast routing protocol design particularly challenging in wireless ad hoc networks. Encoding trees is a critical scheme in GAs for solving these problems because each code should represent a tree. Prufer number is the most representative method of vertex encoding, which is a string of n-2 integers and can be transformed to an n-node tree. However, genetic algorithm based on Prufer encoding (GAP) does not preserve locality, while changing one element of its vector causes dramatically change in its corresponding tree topology. In this paper, we propose a novel GA based on sequence and topology encoding (GAST) for multicast protocol is introduced for multicast routing in wireless ad hoc networks and generalizes the GOP of tree-based multicast protocol as well as three associated operators. It has revealed an efficient method of the reconstruction of multicast tree topology and the experimental results demonstrated the effectiveness of GAST compare to GAP technique.

Tree decompositions of graphs: Saving memory in dynamic programming

We propose a simple and effective heuristic to save memory in dynamic programming on tree decompositions when solving graph optimization problems. The introduced “anchor technique” is based on a tree-like set covering problem. We substantiate our findings by experimental results. Our strategy has negligible computational overhead concerning running time but achieves memory savings for nice tree decompositions and path decompositions between 60% and 98%.

Length-constrained MAP decoding of variable-length encoded Markov sequences

In this paper, we consider the problem of length-constrained maximum a posteriori decoding of a Markov sequence that is variable-length encoded and transmitted over a binary symmetric channel. We convert this problem into one of a maximum-weight k-link path in a weighted directed acyclic graph. The induced graph-optimization problem can be solved by a fast parameterized search algorithm that finds either the optimal solution with high probability, or a good approximate solution otherwise. The proposed algorithm has lower complexity and superior performance than the previous approximation algorithms

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S ⊆ V , such that, for each u ∈ V , | N [ u ] ∩ S | ⩾ | N [ u ] | / 2 in a given graph G = ( V , E ) . We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs ( k ⩾ 5 ) , unless P = NP . We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2 k -regular graphs and to minimum monopoly problem for ( 2 k - 1 ) -regular graphs, where k ⩾ 3 . We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.

Approximation hardness of edge dominating set problems

We provide the first interesting explicit lower bounds on efficient approximability for two closely related optimization problems in graphs, MINIMUM EDGE DOMINATING SET and MINIMUM MAXIMAL MATCHING. We show that it is NP-hard to approximate the solution of both problems to within any constant factor smaller than $${\frac{7}{6}}$$ . The result extends with negligible loss to bounded degree graphs and to everywhere dense graphs.

An online distributed gossiping protocol for mobile networks

Gossiping in a communication network has long been studied as a combinatorial optimization problem in graphs under many different objective functions and communication models. In this paper, we propose a new online distributed gossiping protocol in the multicasting communication environment, where each node knows only its immediate neighbors. We show that the protocol can tolerate multiple node and link faults, mobility of nodes in the network (as long as the network remains connected) and it is more efficient than the a recently proposed protocol.

An efficient simulated annealing algorithm for the minimum vertex cover problem

The minimum vertex cover problem is a classic graph optimization problem. It is well known that it is an NP-complete problem. In this paper, an efficient simulated annealing algorithm is presented for the minimum vertex cover problem. In this algorithm, an acceptance function is defined for every vertex. This can help the algorithm in finding a near-optimal solution to a problem. Simulations are performed on several benchmark graphs, and the simulation results show that the proposed algorithm provides a high probability of finding optimal solutions.

Lower Bounds for On-line Graph Problems with Application to On-line Circuit and Optical Routing

We present lower bounds on the competitive ratio of randomized algorithms for a wide class of on-line graph optimization problems, and we apply such results to on-line virtual circuit and optical routing problems. Lund and Yannakakis [The approximation of maximum subgraph problems, in Proceedings of the 20th International Colloquium on Automata, Languages and Programming, 1993, pp. 40-51] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any nontrivial, hereditary property pi--e.g., independent set, planar, acyclic, bipartite. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H of G is presented on-line, vertex by vertex. The on-line algorithm must choose a subset of the vertices of H, choosing or rejecting a vertex when it is presented, whose vertex induced subgraph satisfies property pi. Furthermore, we study the on-line version of graph coloring whose off-line version has also been shown to be inapproximable [C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, in Proceedings of the 25th ACM Symposium on Theory of Computing, 1993], on-line max edge-disjoint paths, and on-line path coloring problems. Irrespective of the time complexity, we show an Omega(nepsilon) lower bound on the competitive ratio of randomized on-line algorithms for any of these problems. As a consequence, we obtain an Omega(nepsilon) lower bound on the competitive ratio of randomized on-line algorithms for virtual circuit routing on general networks, in contrast to the known results for some specific networks. Similar lower bounds are obtained for on-line optical routing as well.

The datapath merging problem in reconfigurable systems

In this paper, we investigate the data path merging problem (DPM) in reconfigurable systems. DPM is modeled as a graph optimization problem and is shown to be NP -hard. An Integer Programming (IP) formulation of the problem is presented and some valid inequalities for the convex hull of integer solutions are introduced. These inequalities form the basis of a branch-and-cut algorithm that we implemented. This algorithm was used to compute lower bounds for a set of DPM instances, allowing us to assess the performance of two heuristics proposed earlier in the literature for the problem. Moreover, the branch-and-cut algorithm also was proved to be a valuable tool to solve small-sized DPM instances to optimality.

Approximating minimum coloring and maximum independent set in dotted interval graphs

Dotted interval graphs are introduced by Aumann et al. as a generalization of interval graphs. We study two optimization problems in dotted interval graphs that find application in high-throughput genotyping. We present improved approximations for minimum coloring and the first approximation for maximum independent set in dotted interval graphs.

On the typical case complexity of graph optimization

In combinatorial optimization problems that exhibit phase transition it is a frequently observed phenomenon that the algorithmically hard instances are concentrated around the phase transition region. The location, the size and sometimes the mere existence of this critical region, however, may depend on several factors: on the choice of an “order parameter”, on the solving algorithm or on the probabilistic model. We investigate a large class of graph optimization problems and show that this concentration of hardness is in fact a more general phenomenon, if we focus on the complexity of finding or approximating the optimal value (such as the size of a maximum clique), rather than finding a witness (an actual maximum clique). Specifically, we prove that for a general class of graph optimization problems there is always a critical region of input instances in which the hardness is sharply concentrated in the following sense: (1) if the inputs that fall in the critical region are excluded, then the remaining task cannot be NP-hard, unless unlikely complexity collapses happen; (2) the critical region is a small, vanishing subset of all inputs. Thus, in this sense, the hardness of the overall task is necessarily caused by a small, exponentially vanishing critical region of the possible inputs. This concentration of hardness is invariant in the sense that it does not depend on the choice of any order parameter, or on a specific solving algorithm or on the choice of a particular probabilistic model within the considered broad family. Since a random input, drawn by any probability distribution in the family, falls almost surely outside the critical region, therefore, it is justified in a rigorous sense that the typical case complexity of these problems is easier than their worst case complexity and this phenomenon remains invariant for a broad class of models.

Self-Stabilizing Global Optimization Algorithms for Large Network Graphs

The paradigm of self-stabilization provides a mechanism to design efficient localized distributed algorithms that are proving to be essential for modern day large networks of sensors. We provide self-stabilizing algorithms (in the shared-variable ID-based model) for three graph optimization problems: a minimal total dominating set (where every node must be adjacent to a node in the set) and its generalizations, a maximal k-packing (a set of nodes where every pair of nodes are more than distance k apart), and a maximal strong matching (a collection of totally disjoint edges).