Using Combinatorial and LP-based Methods to Design Approximation Algorithms

Our goal in this proposal is to explore the connection between the local ratio technique and linear programming. We believe that a better understanding of this connection will strengthen and extend the local ratio technique and will enable us to apply the technique to a wide variety of problems. Specifically, we intend to improve the best performance guarantee and/or running time of approximation algorithms for fundamental combinatorial optimization problems. Lately, we have started to investigate this relationship. We showed that the primal-dual schema (for designing approximation algorithms) is equivalent to local ratio. We presented local ratio interpretations of algorithms that were previously analyzed by the primal-dual method (for solving linear programs). These algorithms were the first local ratio algorithms to use negative weights. Finally, fractional local ratio and fractional primal-dual were developed. In this proposal we consider several possible research directions. We intend to study the primal-dual method for solving linear programs and problems that can be solved using this method such as maximum weight matching and minimum cost flow. We plan to utilize the insight that was gained by the equivalence between local ratio and the primal-dual schema in order to design approximation algorithms. Specifically, our goal is to simplify and improve primal-dual algorithms by using local ratio. We are especially interested in problems related to network design and to algorithmic game theory. We would like to focus on extending the applicability of the fractional local ratio technique. And finally, our forth goal is to explore the possibility of using local ratio in the context of online algorithms. Over the years the local ratio technique has become much stronger than it was first believed to be, and this in contrast to its simplicity and elegance. The technique works well on many problems and was used in many papers in parallel with the primal-dual schema. We think that further exploring the connection between local ratio and linear programming will result in novel ways to use local ratio and in new local ratio algorithms. 1 Grant Application No. 808/08 Reuven Bar-Yehuda Using Combinatorial and LP-based Methods to Design Approximation Algorithms 1 Scientific background In this proposal we focus on the connection between the local ratio technique and linear programming (LP). Below we describe the local ratio technique and another well known LP-based approximation method for solving combinatorial optimization problems—the primal-dual schema. Primal-dual schema. A key step in designing approximation algorithms is to find a good bound on the value of the optimum, and this is where linear programming helps out. Many combinatorial optimization problems can be expressed as integer programs, and the value of an optimal solution to their LP-relaxation provides the desired bound. One way to obtain approximate solutions is to solve the LP-relaxation and then to round the solution while ensuring that the cost does not change by much. Another way is to use the dual of the LP-relaxation in the design of approximation algorithms and their analyses. A primal-dual r-approximation algorithm constructs a feasible integral primal solution and a feasible dual solution such that the value of the primal solution is no more than r times (or, in the maximization case, at least 1/r times) the value of the dual solution. This proposal focuses on classical primal-dual approximation algorithms. Specifically, the ones that fall within the, so called, primal-dual schema. The primal-dual schema can be seen as a modified version of the primal-dual method for solving linear programs that was originally proposed by Dantzig, Ford, and Fulkerson [22]. While the complementary slackness conditions are imposed in the primal-dual method, one enforces the primal conditions and relaxed the dual conditions when working with the primal-dual schema. A primal-dual algorithm typically constructs an approximate primal solution and a feasible dual solution simultaneously. The approximation ratio is derived from comparing the values of both solutions. The first algorithm to use the primal-dual schema was Bar-Yehuda and Even’s approximation algorithm for the set cover problem [9], and since then many approximations algorithms for NP-hard optimization problems have been devised using this approach, among which are algorithms for network design problems [43, 1, 27]. In fact, this line of research introduced the idea of looking at minimal solutions with respect to set inclusion. Several primal-dual approximation frameworks were proposed. Goemans and Williamson [27] gave a generic algorithm for network design problems. They based a subsequent survey of the primal-dual schema [28] on this algorithm. Another survey by Williamson [47] described the primal-dual schema and several extensions of the primal-dual approach. In [28] it was shown that the primal-dual schema can be used to explain many classical algorithms for special cases of the hitting set problem, such as shortest path, minimum spanning tree, and vertex cover. Bertsimas and Teo [16] proposed a primal-dual framework to design approximation algorithms for covering problems. As in [27, 28], this framework enforces the primal complementary slackness conditions while relaxing the dual conditions. However, in contrast to previous studies, they expressed each advancement step as the construction of a single inequality, and an increase of the corresponding dual variable (as opposed to an increase of several dual variables). The approximation ratio depends on the strength of the inequalities that are used. Local ratio technique. An advancement step of a local ratio algorithm typically consists of the construction of a new weight function, which is then subtracted from the current objective function. Each subtraction changes the optimum, but incurs a cost. The ratio between this cost and the change in the optimum is called the effectiveness of the weight function. The approximation ratio of a local ratio algorithm depends on the effectiveness of the weight functions it constructs. The local ratio approach was developed by Bar-Yehuda and Even [10] in order to approximate the set cover and vertex cover problems. They presented a local ratio analysis of their primal-dual algorithm for 2 Grant Application No. 808/08 Reuven Bar-Yehuda set cover [9], and a (2 − log logn 2 logn )-approximation algorithm for vertex cover. Bafna et al. [3] extended the local ratio lemma from [10] in order to construct a 2-approximation algorithm for the feedback vertex set problem. Their algorithm was the first local ratio algorithm that used the notion of minimal solutions. This work and the 2-approximation from [15] were essential in the design of primal-dual algorithms for feedback vertex set [21]. Following [3] Fujito [26] presented a generic local ratio algorithm for certain node deletion problems. Later, Bar-Yehuda [6] presented a unified local ratio framework for developing approximation algorithms for covering problems. This framework, which extends the one in [26], can be used to explain many known algorithms for covering problems. By the turn of the century local ratio and primal dual were being used extensively in the context of minimization algorithms. However, no application of either method approximated a maximization problem. The first study to present a local ratio and primal-dual approximation algorithm for a maximization problem was by Bar-Noy et al. [4]. They developed a local ratio approximation framework for resource allocation and scheduling problems. A primal-dual interpretation was presented in [4] as well. Bar-Noy et al.’s [4] results paved the way for other studies dealing with maximization problems. For example, Bar-Noy et al. [5] developed approximation algorithms for variants of the problem of scheduling jobs on identical machines with batching. Akcoglu et al. [2] presented approximation algorithms for several types of combinatorial auctions. Bar-Yehuda et al. [7] presented approximation algorithms for resource allocation problems in bounded degree trees. A detailed survey on the local ratio technique that includes recent developments is given in [8]. Equivalence. It has often been observed that primal-dual algorithms have local ratio interpretations, and vice versa. Bar-Yehuda and Even’s primal-dual algorithm for set cover [9] was later analyzed using local ratio [10]. Bafna et al.’s [3] local ratio 2-approximation algorithm for feedback vertex set was explained using primal-dual [21]. The approximation algorithm for network design problems by Goemans and Williamson [27] was restated using local ratio [6, 8]. Finally, Bar-Noy et al.’s [4] approximation framework for resource allocation and scheduling problems was developed using the local ratio approach, and then explained in primal-dual terms. Thus, over the years there has been a growing sense that the two seemingly distinct approaches share a common ground, but the exact nature of the connection between them remained unclear (see, e.g., [47], where this was posed as an open question). The issue was resolved in a paper by Bar-Yehuda and Rawitz [13]. In this paper two approximation frameworks were defined, one encompassing the primal-dual schema, and the other encompassing the local ratio technique, and it is shown that these two frameworks are equivalent. The equivalence between the methods is constructive, meaning that an algorithm formulated within one paradigm can be “translated” quite mechanically to the other paradigm. A corollary to this equivalence is that the integrality gap of a certain LP serves as a lower bound to the approximation ratio of a local ratio algorithm. Fractional local ratio. The latest important development in the context of local ratio is fractional local ratio [11]. A typical local ratio algorithm is recursive, and it constructs, in each recursive call, a new weight function w1. In essence, a local ratio analysis consists of comparing, at each level of the recursion, the solution found in this level to an optimal solution for the problem instance passed to this level, where the comparison is made with respect to w1. Thus, different optima are used at different recursion levels. The superposition of these “local optima” may be significantly worse than the “global optimum,” i.e., the optimum of the original problem instance. Conceivably, we could obtain a better bound if at each level of the recursion we approximated the weight of a solution that is optimal with respect to the original weight function. This is the idea behind fractional local ratio. We use a single solution x to the original problem instance as the yardstick against which all intermediate solutions (at all levels of the recursion) are compared. In fact, x is not even feasible for the original problem instance but rather for a relaxation of it. Typically, x is an optimal solution to an LP relaxation of the problem. 3 Grant Application No. 808/08 Reuven Bar-Yehuda Bar-Yehuda and Rawitz [14] showed that the fractional approach extends to primal-dual. As in fractional local ratio the first step in a fractional primal-dual r-approximation algorithm is to compute an optimal solution x to an LP relaxation denoted by P . Next, as usual in primal-dual algorithms, it produces an integral primal solution x and a dual solution y, such that r times the value of y bounds the weight of x (we use minimization terms). However, in contrast to other primal-dual algorithms, y is not a solution to the dual of P . The algorithm induces a new LP, denoted by P , that has the same objective function as P , but contains inequalities that may not be valid with respect to the original problem. Nevertheless, we ensure that x is a feasible solution of P . The dual solution y is a feasible solution of the dual of P . The primal solution x is r-approximate, since the optimum value of P ′ is not greater than the optimum value of P . 1.1 Formal description of both methods Primal-dual. The description is given in minimization terms. Similar arguments can be used in the maximization case. We assume basic knowledge of linear programming (see, e.g., [41, 34]). Consider the following linear program, min ∑n j=1wjxj s.t. ∑n j=1 aijxj ≥ bi ∀i ∈ {1, . . . ,m} xj ≥ 0 ∀j ∈ {1, . . . , n} and its dual, max ∑n i=1 biyi s.t. ∑n i=1 aijyi ≤ wj ∀j ∈ {1, . . . , n} yi ≥ 0 ∀i ∈ {1, . . . ,m} A primal-dual r-approximation algorithm produces an integral primal solution x and a dual solution y such that the weight of the primal solution is no more than r times the value of the dual solution. Namely, it produces an integral solution x and a solution y such that