Combinatorial Approximation Algorithm Research Articles

A Combinatorial Approximation Algorithm for the Vector Scheduling with Submodular Penalties on Parallel Machines

In this paper, we focus on solving the vector scheduling problem with submodular penalties on parallel machines. We are given n jobs and m parallel machines, where each job is associated with a d -dimensional vector. Each job can either be rejected, incurring a rejection penalty, or accepted and processed on one of the m parallel machines. The objective is to minimize the sum of the maximum load overall dimensions of the total vector for all accepted jobs, along with the total penalty for rejected jobs. The penalty is determined by a submodular function. Our main work is to design a 2 − 1 / m min r , d -approximation algorithm to solve this problem. Here, r denotes the maximum ratio of the maximum load to the minimum load on the d -dimensional vectors among all jobs.

Approximating sparse quadratic programs

Given a matrix A∈Rn×n, we consider the problem of maximizing xTAx subject to the constraint x∈{−1,1}n. This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an Ω(1/lg⁡n) approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an Ω(1) approximation when A corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is O˜(n1.5⋅min⁡{N,n1.5}), where N is the number of nonzero entries in A and O˜ ignores polylogarithmic factors.In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where A is sparse (i.e., has O(n) nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that:•MaxQP admits a (1/2Δ)-approximation in O(nlg⁡n) time, where Δ=O(1) is the maximum degree of the corresponding graph.•Unit MaxQP, where A∈{−1,0,1}n×n, admits a (1/2d)-approximation in O(n) time when the corresponding graph is d-degenerate, and a (1/3δ)-approximation in O(n1.5) time when the corresponding graph has δn edges for δ=O(1).•MaxQP admits a (1−ε)-approximation in O(n) time when the corresponding graph and each of its minors have bounded local treewidth.•Unit MaxQP admits a (1−ε)-approximation in O(n2) time for H-minor free graphs.

A combinatorial approximation algorithm for k-level facility location problem with submodular penalties

A combinatorial approximation algorithm for k-level facility location problem with submodular penalties

Combinatorial approximation algorithms for the maximum bounded connected bipartition problem

Combinatorial approximation algorithms for the maximum bounded connected bipartition problem

Approximating Sparse Quadratic Programs

Given a matrix $A \in \mathbb{R}^{n\times n}$, we consider the problem of maximizing $x^TAx$ subject to the constraint $x \in \{-1,1\}^n$. This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an $\Omega(1/\lg n)$ approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an $\Omega(1)$ approximation when $A$ corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is $\tilde{O}(n^{1.5}\cdot \min\{N,n^{1.5}\})$, where $N$ is the number of nonzero entries in $A$ and $\tilde{O}$ ignores polylogarithmic factors. In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where $A$ is sparse (i.e., has $O(n)$ nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that: - MaxQP admits a $(1/2\Delta)$-approximation in $O(n \lg n)$ time, where $\Delta$ is the maximum degree of the corresponding graph. - UnitMaxQP, where $A \in \{-1,0,1\}^{n\times n}$, admits a $(1/2d)$-approximation in $O(n)$ time when the corresponding graph is $d$-degenerate, and a $(1/3\delta)$-approximation in $O(n^{1.5})$ time when the corresponding graph has $\delta n$ edges. - MaxQP admits a $(1-\varepsilon)$-approximation in $O(n)$ time when the corresponding graph and each of its minors have bounded local treewidth. - UnitMaxQP admits a $(1-\varepsilon)$-approximation in $O(n^2)$ time when the corresponding graph is $H$-minor free.

DEFECT: discover and eradicate fool around node in emergency network using combinatorial techniques

Mining the frequent pattern deals with the finding patterns in large set of data, subsequences and substructures that occur in a database frequently. Likewise, We can use Frequent pattern mining for MANET nodes in order to identify the paths which are participated in frequent data transaction among the various Mobile adhoc network nodes. The network data stream is a long and continuous sequence of data sets transmitted over the network. The OCA (Online Combinatorial Approximation) algorithm is used in the data stream for mining online data. The processing time of OCA was much less and accuracy of its approximate result was quite high like other traditional mining methods. The Data Path Combinatorial Approximation (DPCA) algorithm deals with a frequent pathset mining over the MANET data flow. The pathset is generation of path from the set of paths on any node which are provided paths to various other nodes participating in the data transmission. The mining algorithm is based on Approximate Inclusion–Exclusion technique. Without continual path scanning, approximate counts are calculated for the pathsets. Skip and complete technique and group count technique were combined together and integrated into the DPCA algorithm to improve the MANET performance in terms of identifying fool around (misbehaving) nodes.

Combinatorial approximation algorithms for the submodular multicut problem in trees with submodular penalties

In this paper, we introduce the submodular multicut problem in trees with submodular penalties, which generalizes the prize-collecting multicut problem in trees and the submodular vertex cover with submodular penalties. We present a combinatorial approximation algorithm, based on the primal-dual algorithm for the submodular set cover problem. In addition, we present a combinatorial 3-approximation algorithm for a special case where the edge cost is a modular function, based on the primal-dual scheme for the multicut problem in trees.

Purely combinatorial approximation algorithms for maximum [formula omitted]-vertex cover in bipartite graphs

We study the polynomial time approximation of the NP-hard maxk-vertex cover problem in bipartite graphs and propose purely combinatorial approximation algorithms. The main result of the paper is a simple combinatorial algorithm and a computer-assisted analysis of its approximation guarantee giving strong evidence that the worst approximation ratio achieved is bounded below by 0.821. We also study two simpler strategies with provable approximation ratios of 23 and 3447≈0.72 respectively that already beat the only such known algorithm, namely the greedy approach which guarantees ratio (1−1e)≈0.632. Our principal motivation is to bring a satisfactory answer in the following question: to what extent combinatorial methods for maxk-vertex cover compete with linear programming ones?

Finding connected k-subgraphs with high density

Given an edge-weighted connected graph G on n vertices and a positive integer k≤n, a subgraph of G on k vertices is called a k-subgraph in G. We design combinatorial approximation algorithms for finding a connected k-subgraph in G such that its weighted density is at least a factor Ω(max⁡{1/k,k2/n2}) of the maximum weighted density among all k-subgraph in G (which are not necessarily connected), where max⁡{1/k,k2/n2}≥n−2/3 implies an O(n2/3)-approximation ratio. We obtain improved O(n2/5)-approximation for unit weights. These particularly provide the first non-trivial approximations for the heaviest/densest connected k-subgraph problem on general graphs. We also give O(nlog⁡n)-approximation for the problem on general weighted interval graphs.

Simpler and Better Approximation Algorithms for the Unweighted Minimum Label s-t Cut Problem

Given a graph $$G=(V,E)$$ with a label set $$L = \{\ell _1, \ell _2, \ldots , \ell _q \}$$ , in which each edge has a label from L, and a source $$s \in V$$ together with a sink $$t \in V$$ , the Minimum Label s-t Cut problem asks to pick a set $$L' \subseteq L$$ of labels with minimized cardinality, such that the removal of all edges with labels in $$L'$$ from G disconnects s and t. Let $$n = |V|$$ and $$m = |E|$$ . The previous best known approximation ratio for this problem in literature is $$O(m^{1/2})$$ . We present two simple and purely combinatorial approximation algorithms for the problem with ratios $$O(n^{2/3}/\text {OPT}^{1/3})$$ and $$O(m^{1/2} / \text {OPT}^{1/2})$$ , where $$\text {OPT}$$ is the optimal value of the input instance. The former result gives the first approximation ratio which is sublinear in n for the problem, and in particular applies to the instances with dense graphs (e.g., $$m = \varTheta (n^2)$$ ). The latter result improves the previous ratio $$O(m^{1/2})$$ , as we can always assume that $$\text {OPT}$$ is a super-constant.

Combinatorial approximation algorithms for buy-at-bulk connected facility location problems

We consider generalizations of the connected facility location problem, where clients connect to open facilities via access trees that are shared by multiple clients. For each edge of these trees, a combination of several cable types can be chosen, according to the demand carried by the edge. The task is to choose open facilities, a Steiner tree among them as a core network, and access trees with appropriate cable types on their edges to connect clients to the open facilities in such a way that the sum of the facility opening costs and cable costs for the core and access networks is minimal. Problems of this type arise widely in the planning of access and distribution networks in telecommunications, logistics, or energy networks.In this paper, we introduce and analyze two fundamental versions of these problems. In the Buy-at-Bulk version of the problem, each access cable type has a fixed setup cost and a fixed capacity, whereas in the Deep-Discount problem version, each cable type has unlimited capacity but a traffic-dependent variable cost in addition to its fixed setup cost. We derive the first constant-factor approximation algorithms for these problems, using different algorithmic and analytical techniques.

A Combinatorial, Primal-Dual Approach to Semidefinite Programs

Semidefinite programs (SDPs) have been used in many recent approximation algorithms. We develop a general primal-dual approach to solve SDPs using a generalization of the well-known multiplicative weights update rule to symmetric matrices. For a number of problems, such as S parsest C ut and B alanced S eparator in undirected and directed weighted graphs, M in U n C ut and M in 2CNF D eletion , this yields combinatorial approximation algorithms that are significantly more efficient than interior point methods. The design of our primal-dual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones. Our ideas have proved useful in quantum computing, especially the recent result of Jain et al. [2011] that QIP = PSPACE.

Partial degree bounded edge packing problem for graphs and $$k$$ k -uniform hypergraphs

Given a graph $$G=(V,E)$$G=(V,E) and a non-negative integer $$c_u$$cu for each $$u\in V$$uźV, partial degree bounded edge packing problem is to find a subgraph $$G^{\prime }=(V,E^{\prime })$$Gź=(V,Eź) with maximum $$|E^{\prime }|$$|Eź| such that for each edge $$(u,v)\in E^{\prime }$$(u,v)źEź, either $$deg_{G^{\prime }}(u)\le c_u$$degGź(u)≤cu or $$deg_{G^{\prime }}(v)\le c_v$$degGź(v)≤cv. The problem has been shown to be NP-hard even for uniform degree constraint (i.e., all $$c_u$$cu being equal). In this work we study the general degree constraint case (arbitrary degree constraint for each vertex) and present two combinatorial approximation algorithms with approximation factors $$4$$4 and $$2$$2. Then we give a $$\log _2 n$$log2n approximation algorithm for edge-weighted version of the problem and an efficient exact algorithm for edge-weighted trees with time complexity $$O(n\log n)$$O(nlogn). We also consider a generalization of this problem to $$k$$k-uniform hypergraphs and present a constant factor approximation algorithm based on linear programming using Lagrangian relaxation.

Combinatorial approximation algorithms for the robust facility location problem with penalties

In this paper, we consider the robust facility location problem with penalties, aiming to serve only a specified fraction of the clients. We formulate this problem as an integer linear program to identify which clients must be served. Based on the corresponding LP relaxation and dual program, we propose a primal---dual (combinatorial) 3-approximation algorithm. Combining the greedy augmentation procedure, we further improve the above approximation ratio to 2.

Parameterized and approximation algorithms for finding two disjoint matchings

We first present a randomized fixed-parameter algorithm for the NP-hard problem of deciding if there are two matchings M1 and M2 in a given graph G such that |M1|+|M2| is a given number k. The algorithm runs in O(2kk(m+n)) expected time and can be derandomized to run in O(22k+12log2⁡(2k)kn(m+n)) time, where n (respectively, m) is the number of vertices (respectively, edges) in G. We then extend the algorithm to the weighted version of the problem. We further present a combinatorial approximation algorithm for the NP-hard problem of finding two disjoint matchings in a given edge-weighted graph G so that their total weight is maximized. The algorithm achieves an approximation ratio close to 0.76 and runs in O(m+n3α(n)) time, where α is the inverse Ackermann function.

New combinatorial direction stochastic approximation algorithms

The stochastic approximation problem is to find some roots or minimizers of a nonlinear function whose expression is unknown and whose evaluations are contaminated with noise. In order to accelerate the classical RM algorithm, this paper proposes a new three-term combinatorial direction stochastic approximation algorithm and its general framework which employ a weighted combination of the current noisy gradient and several previous noisy gradients as the iterative direction. Both the almost sure convergence and the asymptotic rate of convergence of the new algorithms are established. Numerical experiments show that the new algorithm outperforms the RM algorithm and another existing combined direction algorithm.

Decentralized utilitarian mechanisms for scheduling games

Game Theory and Mechanism Design are by now standard tools for studying and designing massive decentralized systems. Unfortunately, designing mechanisms that induce socially efficient outcomes often requires full information and prohibitively large computational resources. In this work we study simple mechanisms that require only local information. Specifically, in the setting of a classic scheduling problem, we demonstrate local mechanisms that induce outcomes with social cost close to that of the socially optimal solution. Somewhat counter-intuitively, we find that mechanisms yielding Pareto dominated outcomes may in fact enhance the overall performance of the system, and we provide a justification of these results by interpreting these inefficiencies as externalities being internalized. We also show how to employ randomization to obtain yet further improvements. Lastly, we use the game-theoretic insights gained to obtain a new combinatorial approximation algorithm for the underlying optimization problem.

A primal-dual approximation algorithm for the Asymmetric Prize-Collecting TSP

We present a primal-dual Â?log(n)Â?-approximation algorithm for the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. The previous algorithm for the problem (V.H. Nguyen and T.T Nguyen in Int. J. Math. Oper. Res. 4(3):294---301, 2012) which is not combinatorial, is based on the Held-Karp relaxation and heuristic methods such as the Frieze et al.'s heuristic (Frieze et al. in Networks 12:23---39, 1982) or the recent Asadpour et al.'s heuristic for the ATSP (Asadpour et al. in 21st ACM-SIAM symposium on discrete algorithms, 2010). Depending on which of the two heuristics is used, it gives respectively 1+Â?log(n)Â? and $3+ 8\frac{\log(n)}{\log(\log(n))}$ as an approximation ratio. Our algorithm achieves an approximation ratio of Â?log(n)Â? which is weaker than $3+ 8\frac{\log(n)}{\log(\log(n))}$ but represents the first combinatorial approximation algorithm for the Asymmetric Prize-Collecting TSP.

Efficient approximation algorithms for shortest cycles in undirected graphs

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. Given an adjacency-list representation of an undirected graph G with n vertices and unknown girth k, our algorithm returns with high probability a cycle of length at most 2 k for even k and 2 k + 2 for odd k, in time O ( n 3 2 log n ) . Thus, in general, it yields a 2 2 3 approximation. For a weighted, undirected graph, with non-negative edge weights in the range { 1 , 2 , … , M } , we present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle that runs in time O ( n 2 log n ( log n + log M ) ) .

Bandwidth Guaranteed Routing With Fast Restoration Against Link and Node Failures

An important feature of MPLS networks is local restoration where detour paths are set-up a priori. The detour is such that failed links or nodes can be bypassed locally from the first node that is upstream from the failures. This local bypass activation from the first detection point for failures permits much faster recovery than end-to-end path based mechanisms that require failure information to propagate to the network edges. However, local restoration of bandwidth guaranteed connections can be expensive in the additional network capacity needed. Hence, it is important to minimize and share restoration capacity. The problem of routing with local restoration requirements has been studied previously in a dynamic on-line setting. However, there are no satisfactory algorithms for the problem of pre-provisioning fast restorable connections when the aggregate traffic demands are known (as would be the case when a set of routers are to be interconnected over an optical network or for pre-provisioned ATM over MPLS overlays). The contribution of this paper is a fast combinatorial approximation algorithm for maximizing throughput when the routed traffic is required to be locally restorable. To the best of our knowledge, this is the first combinatorial algorithm for the problem with a performance guarantee. Our algorithm is a fully polynomial time approximation scheme (FPTAS), i.e., for any given epsiv > 0, it guarantees (1+epsiv)-factor closeness to the optimal solution, and runs in time polynomial in the network size and [ 1/( epsiv)]. We compare the throughput of locally restorable routing with that of unprotected routing and 1+1-dedicated path protection on actual US/European ISP topologies taken from the Rocketfuel project .