Approximation Algorithms for the Partition Vertex Cover Problem

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

We investigate the following above-guarantee parameterization of the classical Vertex Cover problem: Given a graph $G$ and $k\in\mathbb{N}$ as input, does $G$ have a vertex cover of size at most $(2LP-MM)+k$? Here $MM$ is the size of a maximum matching of $G$, $LP$ is the value of an optimum solution to the relaxed (standard) LP for Vertex Cover on $G$, and $k$ is the parameter. Since $(2LP-MM)\geq{LP}\geq{MM}$, this is a stricter parameterization than those---namely, above-$MM$, and above-$LP$---which have been studied so far. We prove that Vertex Cover is fixed-parameter tractable for this stricter parameter $k$: We derive an algorithm which solves Vertex Cover in time $O^{*}(3^{k})$, pushing the envelope further on the parameterized tractability of Vertex Cover.

We discuss the problem of sequencing precedence-constrained jobs on a single machine to minimize the average weighted completion time. This problem has attracted much attention in the mathematical programming community since Sidney’s pioneering work in 1975. We look at the problem from a polyhedral perspective and uncover a relation between Sidney’s decomposition theorem and different linear programming relaxations. More specifically, we present a generalization of Sidney’s result, which particularly allows us to reason that virtually all known 2-approximation algorithms comply with his decomposition. Moreover, we establish a connection between the single-machine scheduling problem and the vertex cover problem. Indeed, in the special case of series-parallel precedence constraints, we prove that the sequencing problem can be seen as a special case of vertex cover. We also argue that this result is true for general precedence constraints if one can show that a certain integer program represents a valid formulation of the sequencing problem. Finally, we provide a characterization of the active inequalities of a linear programming relaxation in completion time variables.KeywordsSchedule ProblemCompletion TimeSingle MachinePrecedence ConstraintVertex CoverThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The generalized vertex cover problem and some variations

In the minimum entropy set cover problem, one is given a collection of k sets which collectively cover an n-element ground set. A feasible solution of the problem is a partition of the ground set into parts such that each part is included in some of the k given sets. Such a partition defines a probability distribution, obtained by dividing each part size by n. The goal is to find a feasible solution minimizing the (binary) entropy of the corresponding distribution. Halperin and Karp have recently proved that the greedy algorithm always returns a solution whose cost is at most the optimum plus a constant. We improve their result by showing that the greedy algorithm approximates the minimum entropy set cover problem within an additive error of 1 nat =log 2 e bits $\simeq$1.4427 bits. Moreover, inspired by recent work by Feige, Lovasz and Tetali on the minimum sum set cover problem, we prove that no polynomial-time algorithm can achieve a better constant, unless P = NP. We also discuss some consequences for the related minimum entropy coloring problem.

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of the complexity of infinite-domain VCSPs with piecewise linear homogeneous cost functions. Such VCSPs can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by reducing the problem to a finite-domain VCSP which can be solved using the basic linear program relaxation. It follows that VCSPs for submodular PLH cost functions can be solved in polynomial time; in fact, we show that submodular PLH functions form a maximally tractable class of PLH cost functions.

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.

We consider single processor preemptive scheduling with job-dependent setup times. In this model, a job-dependent setup time is incurred when a job is started for the first time, and each time it is restarted after preemption. This model is a common generalization of preemptive scheduling, and actually of non-preemptive scheduling as well. The objective is to minimize the sum of any general non-negative, non-decreasing cost functions of the completion times of the jobs -- this generalizes objectives of minimizing weighted flow time, flow-time squared, tardiness or the number of tardy jobs among many others. Our main result is a randomized polynomial time O(1)-speed O(1)-approximation algorithm for this problem. Without speedup, no polynomial time finite multiplicative approximation is possible unless P=NP. We extend the approach of Bansal et al. (FOCS 2007) of rounding a linear programming relaxation which accounts for costs incurred due to the non-preemptive nature of the schedule. A key new idea used in the rounding is that a point in the intersection polytope of two matroids can be decomposed as a convex combination of incidence vectors of sets that are independent in both matroids. In fact, we use this for the intersection of a partition matroid and a laminar matroid, in which case the decomposition can be found efficiently using network flows. Our approach gives a randomized polynomial time offline O(1)-speed O(1)-approximation algorithm for the broadcast scheduling problem with general cost functions as well.

We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed.

A fixed-parameter algorithm for the vertex cover [formula omitted] problem

The minimum disc cover set can be used to construct the dominating set on the fly for energy-efficient communications in mobile ad hoc networks. The approach used to compute the minimum disc cover set proposed in previous study has been considered too expensive. In this paper, we show that the disc cover set problem is in fact a special case of the general α-hull problem. In addition, we prove that the disc cover set problem is not any easier than the α-hull problem by linearly reducing the element uniqueness problem to the disc cover set problem. In addition to the known α-hull approach, we provide an optimal divide-and-conquer algorithm that constructs the minimum disc cover set for arbitrary cases, including the degenerate ones where the traditional α-hull algorithm incapable of handling.

Consider a good (such as a hotel room) which, if not sold on time, is worth nothing to the seller. For a customer who is considering a choice of such goods, their prices may change dramatically by the time the customer needs to use the good; thus a customer who is aware of this fact might choose to gamble, delaying buying until the last moment in the hopes of better prices. While this gamble can yield large savings, it also carries much risk. However, a coordinator can offer customers a compromise between these extremes and benefits in aggregate. Here we explore how a coordinator might profit from forecasts of such future price fluctuations. Our results can be used in a general setting where customers buy products or services in advance and where market prices may significantly change in the future. We model this as a two-stage optimization problem, where the coordinator first agrees to serve some buyers, and then later executes all agreements once the final values have been revealed. Agreements with buyers consist of a set of acceptable options and a price where the details of agreements are proposed by the buyer. We investigate both the profit maximization and loss minimization problems in this setting. For the profit maximization problem, we show that the profit objective function is a non-negative submodular function, and thus we can approximate its optimal solution within an approximation factor of 0:5 in polynomial time. For the loss minimization problem, we first leverage a sampling technique to formulate our problem as an integer program. We show that there is no polynomial algorithm to solve this problem optimally, unless P = NP. In addition, we show that the corresponding integer program has a high integrality gap and it cannot lead us to an approximation algorithm via a linear-programming relaxation. Nevertheless, we propose a bicriteria-style approximation that gives a constant-factor approximation to the minimal loss by allowing a fraction of our options to overlap. Importantly, however, we show that our algorithm provides a strong, uniform bound on the amount the overlap per options. We propose our algorithm by rounding the optimal solution of the relaxed linear program via a novel dependent-rounding method.

The tile assembly model is a novel biological computing model where information is encoded in DNA tiles. It is an efficient way to solve NP-complete problems due to its scalability and parallelism. In this paper, we apply the tile assembly model to solve the minimum and exact set cover problems, which are well-known NP-complete problems. To solve the minimum set cover problem, we design a MinSetCover system composed of three parts, i.e., the seed configuration subsystem, the nondeterministic choice subsystem, and the detection subsystem. Moreover, we improve the MinSetCover system and propose a MinExactSetCover system for solving the problem of exact cover by 3-sets. Finally we analyze the computation complexity and perform a simulation experiment to verify the effectiveness and correctness of the proposed systems.

Polar graphs and maximal independent sets

The notion of augmenting graphs generalizes Berge's idea of augmenting chains, which was used by Edmonds in his celebrated solution of the maximum matching problem. This problem is a special case of the more general maximum independent set (MIS) problem. Recently, the augmenting graph approach has been successfully applied to solve MIS in various other special cases. However, our knowledge of augmenting graphs is still very limited and we do not even know what the minimal infinite classes of augmenting graphs are. In the present paper, we find an answer to this question and apply it to extend the area of polynomial-time solvability of the maximum independent set problem.