A Bicriteria Approximation Algorithm for the Min-Max Rural Postmen Cover Problem

The diameter is a fundamental graph parameter and its computation is necessary in many applications. The fastest known way to compute the diameter exactly is to solve the All-Pairs Shortest Paths (APSP) problem.In the absence of fast algorithms, attempts were made to seek fast algorithms that approximate the diameter. In a seminal result Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] designed an algorithm that computes in O (n2 + m√n) time an estimate D for the diameter D in directed graphs with nonnegative edge weights, such that [EQUATION], where M is the maximum edge weight in the graph. In recent work, Roditty and Vassilevska W. [STOC 13] gave a Las Vegas algorithm that has the same approximation guarantee but improves the (expected) runtime to O (m√n). Roditty and Vassilevska W. also showed that unless the Strong Exponential Time Hypothesis fails, no O (n2-e) time algorithm for sparse unweighted undirected graphs can achieve an approximation ratio better than 3/2. Thus their algorithm is essentially tight for sparse unweighted graphs. For weighted graphs however, the approximation guarantee can be meaningless, as M can be arbitrarily large.In this paper we exhibit two algorithms that achieve a genuine 3/2-approximation for the diameter, one running in O (m3/2) time, and one running in O (mn2/3). time. Furthermore, our algorithms are deterministic, and thus we present the first deterministic (2 -- e)-approximation algorithm for the diameter that takes subquadratic time in sparse graphs.In addition, we address the question of obtaining an additive c-approximation for the diameter, i.e. an estimate D such that D -- c ≤ D ≤ D. An extremely simple O (mn1-e) time algorithm achieves an additive ne-approximation; no better results are known. We show that for any e > 0, getting an additive ne-approximation algorithm for the diameter running in O (n2-e) time for any δ > 2e would falsify the Strong Exponential Time Hypothesis. Thus the simple algorithm is probably essentially tight for sparse graphs, and moreover, obtaining a subquadratic time additive c-approximation for any constant c is unlikely.Finally, we consider the problem of computing the eccentricities of all vertices in an undirected graph, i.e. the largest distance from each vertex. Roditty and Vassilevska W. [STOC 13] show that in O (m√n) time, one can compute for each v e V in an undirected graph, an estimate e(v) for the eccentricity e (v) such that max{R, 2/3 · e(v)} ≤ e (v) ≤ min {D, 3/2 · e(v)} where R = minv e (v) is the radius of the graph. Here we improve the approximation guarantee by showing that a variant of the same algorithm can achieve estimates e' (v) with 3/5 · e (v) ≤ e' (v) ≤ e (v).

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. For an undirected graph G of unknown girth k, our algorithm returns with high probability a cycle of length at most 2k for even k and 2k + 2 for odd k, in time O(n3/2 √log n). Thus, in general, it yields a 2 2/3 approximation. We study also the problem of finding a simple cycle of minimum total weight in an undirected graph with nonnegative edge weights. We present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle in an undirected graph with nonnegative integer edge weights in the range {1, 2, ..., M}. This algorithm runs in time O(n2 log n log M).

We study two problems related to the Small Set Expansion Conjecture [14]: the Maximum weight \(m'\) -edge cover (MWEC) problem and the Fixed cost minimum edge cover (FCEC) problem. In the MWEC problem, we are given an undirected simple graph \(G=(V,E)\) with integral vertex weights. The goal is to select a set \(U\subseteq V\) of maximum weight so that the number of edges with at least one endpoint in \(U\) is at most \(m'\). Goldschmidt and Hochbaum [8] show that the problem is NP-hard and they give a \(3\)-approximation algorithm for the problem. The approximation guarantee was improved to \(2+\epsilon \), for any fixed \(\epsilon > 0\) [12]. We present an approximation algorithm that achieves a guarantee of \(2\). Interestingly, we also show that for any constant \(\epsilon > 0\), a \((2-\epsilon )\)-ratio for MWEC implies that the Small Set Expansion Conjecture [14] does not hold. Thus, assuming the Small Set Expansion Conjecture, the bound of 2 is tight. In the FCEC problem, we are given a vertex weighted graph, a bound \(k\), and our goal is to find a subset of vertices \(U\) of total weight at least \(k\) such that the number of edges with at least one edges in \(U\) is minimized. A \(2(1+\epsilon )\)-approximation for the problem follows from the work of Carnes and Shmoys [3]. We improve the approximation ratio by giving a \(2\)-approximation algorithm for the problem and show a \((2-\epsilon )\)-inapproximability under Small Set Expansion Conjecture conjecture. Only the NP-hardness result was known for this problem [8]. We show that a natural linear program for FCEC has an integrality gap of \(2-o(1)\). We also show that for any constant \(\rho >1\), an approximation guarantee of \(\rho \) for the FCEC problem implies a \(\rho (1+o(1))\) approximation for MWEC. Finally, we define the Degrees density augmentation problem which is the density version of the FCEC problem. In this problem we are given an undirected graph \(G=(V,E)\) and a set \(U\subseteq V\). The objective is to find a set \(W\) so that \((e(W)+e(U,W))/deg(W)\) is maximum. This problem admits an LP-based exact solution [4]. We give a combinatorial algorithm for this problem.KeywordsVertex WeightsApproximation GuaranteeMinimum Edge CoverNatural Integer Linear ProgramMaximum WeightThese 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.

Shortest odd paths in undirected graphs with conservative weight functions

Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this article, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen’s theorem and of Strojohann’s determinant algorithm. It allows us to give new and simple solutions to the following problems: Finding Shortest Cycles . We give a simple Õ ( Wnω ) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with nonnegative weights), this matches the time bounds obtained in 2011 by Roditty and Williams. On the other hand, no algorithm working in Õ ( Wn ω ) time was previously known for undirected graphs with negative weights. Furthermore, our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. Computing Diameter and Radius . We give a simple Õ ( Wnω ) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge, no algorithm with this running time was known for undirected graphs with negative weights. Finding Minimum-Weight Perfect Matchings . We present an Õ ( Wnω ) time algorithm for finding minimum-weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski [2009] who presented such an algorithm but only in the case of bipartite graphs. These three problems that are solved in the full generality demonstrate the utility of this framework. Hence, we believe that it can find applications for solving larger spectra of related problems. As an illustrative example, we apply it to the problem of computing a set of vertices that lie on cycles of length at most t , for some given t . We give a simple Õ ( Wnω ) time algorithm for this problem that improves over the Õ ( Wnωt ) time algorithm given by Yuster in 2011. Besides giving this flexible framework, the other main contribution of this article is the development of a novel combinatorial interpretation of the dual solution for the minimum-weight perfect matching problem. Despite the long history of the matching problem, such a combinatorial interpretation was not known previously. This result sheds a new light on the problem, as there exist many structural theorems about unweighted matchings, but almost no results that could cope with the weighted case.

Approximation and hardness results for the Max k-Uncut problem

Given an undirected graph G = (V,E) and three specified terminal nodes t1, t2, t3, a 3-cut is a subset A of E such that no two terminals are in the same component of G\A. If a non-negative edge weight ce is specified for each e ∈ E, the optimal 3-cut problem is to find a 3-cut of minimum total weight. This problem is NP-hard, and in fact, is max-SNP-hard. An approximation algorithm having performance guarantee 7/6 has recently been given by Calinescu, Karloff, and Rabani. It is based on a certain linear programming relaxation, for which it is shown that the optimal 3-cut has weight at most 7/6 times the optimal LP value. It is proved here that 7/6 can be improved to 12/11, and that this is best possible. As a consequence, we obtain an approximation algorithm for the optimal 3-cut problem having performance guarantee 12/11.

Given an undirected graph G=(V,E) and three specified terminal nodes t1,t2,t3, a 3-cut is a subset A of E such that no two terminals are in the same component of G\A. If a non-negative edge weight ce is specified for each e∈E, the optimal 3-cut problem is to find a 3-cut of minimum total weight. This problem is **-hard, and in fact, is max-** -hard. An approximation algorithm having performance guarantee ** has recently been given by Călinescu, Karloff, and Rabani. It is based on a certain linear-programming relaxation, for which it is shown that the optimal 3-cut has weight at most ** times the optimal LP value. It is proved here that ** can be improved to **, and that this is best possible. As a consequence, we obtain an approximation algorithm for the optimal 3-cut problem having performance guarantee **. In addition, we show that ** is best possible for this algorithm.

Consider a salesperson that must sell some quota of brushes in order to win a trip to Hawaii. This salesperson has a map (a weighted graph) in which each city has an attached demand specifying the number of brushes that can be sold in that city. What is the best route to take to sell the quota while traveling the least distance possible? Notice that unlike the standard traveling salesman problem, not only do we need to figure out the order in which to visit the cities, but we must decide the more fundamental question: which cities do we want to visit? In this paper we give the first approximation algorithms with poly-logarithmic performance guarantees for this problem, as well as for the slightly more general PCTSP of Balas, and a variation we call the bank-robber problem (also called the orienteering problem by Golden, Levi, and Vohra). We do this by providing an O(log^2 k) approximation to the k-MST which is defined as follows. Given an undirected graph on n nodes with non-negative edge weights and an integer k > n, find the tree of least weight that spans k vertices. (If desired, one may specify in the a root vertex that must be in the tree as well.) Our result improves on the previous best bound of O(k^0.5) of Ravi et al. and comes quite close to the bound of O(log k) of Garg and Hochbaum for the special case of points in 2-dimensional Euclidean space.

We consider a formalization of the following problem. A salesperson must sell some quota of brushes in order to win a trip to Hawaii. This salesperson has a map (a weighted graph) in which each city has an attached demand specifying the number of brushes that can be sold in that city. What is the best route to take to sell the quota while traveling the least distance possible? Notice that unlike the standard traveling salesman problem, not only do we need to figure out the order in which to visit the cities, but we must decide the more fundamental question: which cities do we want to visit? In this paper we give the first approximation algorithm having a polylogarithmic performance guarantee for this problem, as well as for the slightly more general prize-collecting traveling salesman (PCTSP) of Balas, and a variation we call the bank robber (also called the orienteering by Golden, Levi, and Vohra). We do this by providing an O(log2 k) approximation to the somewhat cleaner k-MST problem which is defined as follows. Given an undirected graph on n nodes with nonnegative edge weights and an integer $k \leq n$, find the tree of least weight that spans k vertices. (If desired, one may specify in the problem a root vertex that must be in the tree as well.) Our result improves on the previous best bound of $O(\sqrt{k})$ of Ravi et al.

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Their classic algorithm was the best known result for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. The first improvement over the Even-Shiloach algorithm was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented an approximate (1+) algorithm with constant query time and a total update time of O(n2+O(1/√logn)). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a -approximate algorithm whose total update time is near linear O(m1+ O(1/√logn)). In this paper they posed as a major open problem the question of derandomizing their result. In fact, all known improvements over the Even-Shiloach algorithm are randomized. All these algorithms maintain some truncated shortest path trees from a small subset of nodes. While in the randomized setting it is possible to “hide” these nodes from the adversary, in the deterministic setting this is impossible: the adversary can delete all edges touching these nodes, thus forcing the algorithm to choose a new set of nodes and incur a new computation of shortest paths. In this paper we present the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs. Our algorithm is (1 + є) approximate and achieves a total update time of O(n2). Our algorithm can also achieve the same bounds in the incremental setting. It is worth mentioning that for dense instances where m = Ω(n2 − 1/√log(n)), our algorithm is also faster than all existing randomized algorithms.

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first objective, find a subgraph from a given subgraph-class that minimizes the second objective subject to the budget on the first objective. We consider three different criteria—the total edge cost, the diameter, and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the previously mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same we present a “black box” parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms—the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudo-polynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudo-polynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.

We study network-design problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network. A prototypical example is the degree-constrained node-weighted Steiner tree problem: We are given an undirected graph G(V,E), with a non-negative integral function d that specifies an upper bound d(v) on the degree of each vertex v∈V in the Steiner tree to be constructed, nonnegative costs on the nodes, and a subset of k nodes called terminals. The goal is to construct a Steiner tree T containing all the terminals such that the degree of any node v in T is at most the specified upper bound d(v) and the total cost of the nodes in T is minimum. Our main result is a bicriteria approximation algorithm whose output is approximate in terms of both the degree and cost criteria—the degree of any node v∈V in the output Steiner tree is O(d(v)log k) and the cost of the tree is O(log k) times that of a minimum-cost Steiner tree that obeys the degree bound d(v) for each node v. Our result extends to the more general problem of constructing one-connected networks such as generalized Steiner forests. We also consider the special case in which the edge costs obey the triangle inequality and present simple approximation algorithms with better performance guarantees.

We study network-design problems with two different design objectives: the total cost of the edges and nodes in the network and the maximum degree of any node in the network. A prototypical example is the degree-constrained node-weighted Steiner tree problem: We are given an undirected graph G(V,E), with a non-negative integral function d that specifies an upper bound d(v) on the degree of each vertex v∈V in the Steiner tree to be constructed, nonnegative costs on the nodes, and a subset of k nodes called terminals. The goal is to construct a Steiner tree T containing all the terminals such that the degree of any node v in T is at most the specified upper bound d(v) and the total cost of the nodes in T is minimum. Our main result is a bicriteria approximation algorithm whose output is approximate in terms of both the degree and cost criteria—the degree of any node v∈V in the output Steiner tree is O(d(v)log k) and the cost of the tree is O(log k) times that of a minimum-cost Steiner tree that obeys the degree bound d(v) for each node v. Our result extends to the more general problem of constructing one-connected networks such as generalized Steiner forests. We also consider the special case in which the edge costs obey the triangle inequality and present simple approximation algorithms with better performance guarantees.

In the classical survivable network design problem (SNDP), we are given an undirected graph $G-(V,E)$ with costs on edges and a connectivity requirement $k(5,t)$ for each pair of vertices. The goal is to find a minimum-cost subgraph $H\sqsubseteq G$ such that every pair $(s,t)$ are connected by $k(s,t)$ edge or (openly) vertex disjoint paths, abbreviated as EC-SNDP and VC-SNDP, respectively. The seminal result of Jain [FOCS’98, Combinatorica’01] gives a 2-approximation algorithm for EC-SNDP, and a decade later, an $O(k^{3}\log n)-$ approximation algorithm for VC-SNDP, where k is the largest connectivity requirement, was discovered by Chuzhoy and Khanna [FOCS’09, Theory Comput’12]. While there is a rich literature on point-to-point settings of SNDP, the viable case of connectivity between subsets is still relatively poorly understood. This paper concerns the generalization of SNDP into the subset-to-subset setting, namely Group EC-SNDR We develop the framework, which yields the first non-trivial (true) approximation algorithm for Group. EC-SNDE Previously only a bicriteria approximation algorithm is known for Group EC-SNDP [Chalermsook, Grandoni, and Laekhanukit, SODA’15l, and a true approximation algorithm is known only for the single-source variant with connectivity requirement $k(S,T)\in\{0,1,2\}$ [Gupta, Krishnaswamy, and Ravi, SODA’10; Khandekar, Kortsarz, and Nutov, FSTTCS’09 and Theor Comput. Sci’12].