Polylogarithmic Approximation Research Articles

Polynomial Integrality Gap of Flow LP for Directed Steiner Tree

In the Directed Steiner Tree (DST) problem, we are given a directed graph \(G=(V,E)\) on \(n\) vertices with edge-costs \(c\in{\mathbb{R}}_{\geq 0}^{E}\) , a root vertex \(r\in V\) , and a set \(K\subseteq V\setminus\{r\}\) of \(k\) terminals. The goal is to find a minimum-cost subgraph of \(G\) that contains a path from \(r\) to every terminal \(t\in K\) . DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of \(O(k^{\epsilon})\) for any constant \(\epsilon {\gt} 0\) , and the best quasi-polynomial-time approximation ratio of \(O\left(\frac{\log^{2}k}{\log\log k}\right)\) . Towards understanding this gap, we study the integrality gap of the standard flow linear programming relaxation for the problem. We show that the linear program (LP) has an integrality gap of \(\Omega(n^{0.0418})\) . Previously, the integrality gap of the LP is only known to be \(\Omega\left(\frac{\log^{2}n}{\log\log n}\right)\) [Halperin et al., SODA’03 & SIAM J. Comput.] and \(\Omega(\sqrt{k})\) [Zosin-Khuller, SODA’02] in some instance with \(\sqrt{k}=O\left(\frac{\log n}{\log\log n}\right)\) . Our result gives the first known lower bound on the integrality gap of this standard LP that is polynomial in \(n\) , the number of vertices. Consequently, we rule out the possibility of developing a poly-logarithmic approximation algorithm for the problem based on the flow LP relaxation.

2-node-connectivity network design

We consider 2-connectivity network design problems in which we are given a graph and seek a min-size 2-connected subgraph that satisfies a prescribed property.•In the 1-Connectivity Augmentation problem the goal is to augment a connected graph by a minimum size edge subset of a specified edge set such that the augmented graph is 2-connected. We breach the natural approximation ratio 2 for this problem, and also for the more general Crossing Family Cover problem.•In the 2-Connected Dominating Set problem, we seek a minimum size 2-connected subgraph that dominates all nodes. We give the first non-trivial approximation algorithm with expected approximation ratio O(σ(n)⋅log3⁡n), where σ(n)=O(log⁡n⋅log⁡log⁡n⋅(log⁡log⁡log⁡n)3).The unifying technique of both results is a reduction to the Subset Steiner Connected Dominating Set problem. Such a reduction was known for edge-connectivity, and we extend it to 2-node connectivity problems. We show that the same method can be used to obtain easily polylogarithmic approximation ratios that are not too far from the best known ones for several other problems.

A polylogarithmic approximation algorithm for 2-edge-connected dominating set

In the Connected Dominating Set (CDS) problem we are given a graph G and seek a min-size dominating set S such that the subgraph G[S] of G induced by S is connected. In the 2-Edge-Connected Dominating Set problem G[S] should be 2-edge-connected. We give the first non-trivial approximation algorithm for this problem, with expected approximation ratio O(log2⁡n⋅log⁡log⁡n⋅(log⁡log⁡log⁡n)3). We also show that the Subset Steiner CDS problem is approximation equivalent to the Group Steiner Tree problem.

Distributed Graph Diameter Approximation

We present an algorithm for approximating the diameter of massive weighted undirected graphs on distributed platforms supporting a MapReduce-like abstraction. In order to be efficient in terms of both time and space, our algorithm is based on a decomposition strategy which partitions the graph into disjoint clusters of bounded radius. Theoretically, our algorithm uses linear space and yields a polylogarithmic approximation guarantee; most importantly, for a large family of graphs, it features a round complexity asymptotically smaller than the one exhibited by a natural approximation algorithm based on the state-of-the-art Δ-stepping SSSP algorithm, which is its only practical, linear-space competitor in the distributed setting. We complement our theoretical findings with a proof-of-concept experimental analysis on large benchmark graphs, which suggests that our algorithm may attain substantial improvements in terms of running time compared to the aforementioned competitor, while featuring, in practice, a similar approximation ratio.

Approximating Spanners and Directed Steiner Forest

It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA’16 8 J.ACM’17, Bodwin-Williams SODA’16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances, we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O ( n 3/5 + ε )-approximation for distance preservers and pairwise spanners (for arbitrary constant ε > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O (log n )-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O (1)-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O ( n 3/5 + ε )-approximation for the Directed Steiner Forest problem (for arbitrary constant ε > 0) when all edges have uniform costs, improving the previous best O ( n 2/3 + ε )-approximation due to Berman et al. [ICALP’11] (which holds for general edge costs).

Approximation algorithms for connected maximum cut and related problems

An instance of the Connected Maximum Cut problem consists of an undirected graph G=(V,E) and the goal is to find a subset of vertices S⊆V that maximizes the number of edges in the cut δ(S) such that the induced graph G[S] is connected. We present the first non-trivial Ω(1log⁡n) approximation algorithm for the Connected Maximum Cut problem in general graphs using novel techniques. We then extend our algorithm to edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in contrast to the classical Max-Cut problem that can be solved in polynomial time on planar graphs, we show that the Connected Maximum Cut problem remains NP-hard on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the Connected Maximum Cut problem on planar graphs and more generally on bounded genus graphs.

Approximation Algorithms for Optimal Decision Trees and Adaptive TSP Problems

We consider the problem of constructing optimal decision trees: given a collection of tests that can disambiguate between a set of m possible diseases, each test having a cost, and the a priori likelihood of any particular disease, what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease? This problem has been studied in several works, with O(log m)-approximations known in the special cases when either costs or probabilities are uniform. In this paper, we settle the approximability of the general problem by giving a tight O(log m)-approximation algorithm. We also consider a substantial generalization, the adaptive traveling salesman problem. Given an underlying metric space, a random subset S of vertices is drawn from a known distribution, but S is initially unknown—we get information about whether any vertex is in S only when it is visited. What is a good adaptive strategy to visit all vertices in the random subset S while minimizing the expected distance traveled? This problem has applications in routing message ferries in ad hoc networks and also models switching costs between tests in the optimal decision tree problem. We give a polylogarithmic approximation algorithm for adaptive TSP, which is nearly best possible due to a connection to the well-known group Steiner tree problem. Finally, we consider the related adaptive traveling repairman problem, where the goal is to compute an adaptive tour minimizing the expected sum of arrival times of vertices in the random subset S; we obtain a polylogarithmic approximation algorithm for this problem as well.

Computing large independent sets in a single round

Independent sets play a central role in distributed algorithmics. We examine here the minimal requirements for computing non-trivial independent sets. In particular, we focus on algorithms that operate in a single communication round. A classic result of Linial shows that a constant number of rounds does not suffice to compute a maximal independent set. We are therefore interested in the size of the solution that can be computed, especially in comparison to the optimal. Our main result is a randomized one-round algorithm that achieves poly-logarithmic approximation on graphs of polynomially bounded-independence. Specifically, we show that the algorithm achieves the Caro-Wei bound (an extension of the Turán bound for independent sets) in general graphs up to a constant factor, and that the Caro-Wei bound yields a poly-logarithmic approximation on bounded-independence graphs. The algorithm uses only a single bit message and operates in a beeping model, where a node receives only the disjunction of the bits transmitted by its neighbors. We give limitation results that show that these are the minimal requirements for obtaining non-trivial solutions. In particular, a sublinear approximation cannot be obtained in a single round on general graphs, nor when nodes cannot both transmit and receive messages. We also show that our analysis of the Caro-Wei bound on polynomially bounded-independence graphs is tight, and that the poly-logarithmic approximation factor does not extend to mathrm {O}(1)-claw free graphs.

A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2

In the Edge-Disjoint Paths with Congestion problem (EDPwC), we are given an undirected n -vertex graph G , a collection M ={ ( s 1 , t 1 ),… ,( s k , t k ) } of pairs of vertices called demand pairs, and an integer c . The goal is to connect the maximum possible number of the demand pairs by paths, so that the maximum edge congestion - the number of paths sharing any edge - is bounded by c . When the maximum allowed congestion is c = 1, this is the classical Edge-Disjoint Paths problem (EDP). The best current approximation algorithm for EDP achieves an O (√ n )-approximation by rounding the standard multi-commodity flow relaxation of the problem. This matches the Ω (√ n ) lower bound on the integrality gap of this relaxation. We show an O (poly log k )-approximation algorithm for EDPwC with congestion c = 2 by rounding the same multi-commodity flow relaxation. This gives the best possible congestion for a sub-polynomial approximation of EDPwC via this relaxation. Our results are also close to optimal in terms of the number of pairs routed, since EDPwC is known to be hard to approximate to within a factor of ~ Ω ((log n ) 1/( c +1) ) for any constant congestion c . Prior to our work, the best approximation factor for EDPwC with congestion 2 was Õ( n 3/7 ), and the best algorithm achieving a polylogarithmic approximation required congestion 14.

Minimum-Cost Network Design with (Dis)economies of Scale

Given a network, a set of demands, and a cost function $f(\cdot)$, the min-cost network design problem is to route all demands with the objective of minimizing $\sum_e f(\ell_e)$, where $\ell_e$ is the total traffic load under the routing. We focus on cost functions of the form $f(x)= \sigma + x^{\alpha}$ for $x > 0$, with $f(0) = 0$. For $\alpha \le 1$, $f(\cdot)$ is subadditive. This case corresponds to the well-studied buy-at-bulk network design problem and admits polylogarithmic approximation and hardness. In this paper, we focus on the less-studied scenario of $\alpha > 1$ with a positive start-up cost $\sigma > 0$. Now, the cost function $f(\cdot)$ is neither subadditive nor superadditive. It aims to model a range of computing and communication devices for which doubling processing speed more than doubles their power consumption. We begin by discussing why existing routing techniques such as randomized rounding and tree-metric embedding fail to generalize directly. We then present our main contribution, which is a polylogarithmic approximation algorithm. We obtain this result by first deriving a bicriteria approximation for a related capacitated min-cost flow problem that we believe is interesting in its own right. Our approach for this problem builds upon the well-linked decomposition due to Chekuri, Khanna, and Shepherd [Proceedings of ACM STOC, ACM, New York, pp. 183--192], the construction of expanders via matchings due to Khandekar, Rao, and Vazirani [J. ACM, 56 (2009), 19], and edge-disjoint routing in well-connected graphs due to Rao and Zhou [SIAM J. Comput., 39 (2010), pp. 1856--1887]. However, we also develop new techniques that allow us to keep a handle on the total cost, which was not a concern in the aforementioned literature.

Approximation Algorithms and Hardness of the k -Route Cut Problem

We study the k -route cut problem: given an undirected edge-weighted graph G = ( V , E ), a collection {( s 1 , t 1 ), ( s 2 , t 2 ), …, ( s r , t r )} of source-sink pairs, and an integer connectivity requirement k , the goal is to find a minimum-weight subset E ′ of edges to remove, such that the connectivity of every pair ( s i , t i ) falls below k . Specifically, in the edge-connectivity version, EC-kRC, the requirement is that there are at most ( k − 1) edge-disjoint paths connecting s i to t i in G ∖ E ′, while in the vertex-connectivity version, VC-kRC, the same requirement is for vertex-disjoint paths. Prior to our work, poly-logarithmic approximation algorithms have been known for the special case where k ⩽ 3, but no non-trivial approximation algorithms were known for any value k > 3, except in the single-source setting. We show an O ( k log 3/2 r )-approximation algorithm for EC-kRC with uniform edge weights, and several polylogarithmic bi-criteria approximation algorithms for EC-kRC and VC-kRC, where the connectivity requirement k is violated by a constant factor. We complement these upper bounds by proving that VC-kRC is hard to approximate to within a factor of k ϵ for some fixed ϵ > 0. We then turn to study a simpler version of VC-kRC, where only one source-sink pair is present. We give a simple bi-criteria approximation algorithm for this case, and show evidence that even this restricted version of the problem may be hard to approximate. For example, we prove that the single source-sink pair version of VC-kRC has no constant-factor approximation, assuming Feige’s Random κ-AND assumption.

Adaptive informative path planning in metric spaces

In contrast to classic geometric motion planning, informative path planning (IPP) seeks a path for a robot to sense the world and gain information. In adaptive IPP, the robot chooses the next sensing location conditioned on all information acquired so far, and the robot’s goal is to minimize the travel cost required for identifying a true hypothesis. Adaptive IPP is NP-hard, because the robot must trade-off information gain and travel cost optimally. In this paper we present Recursive Adaptive Identification (RAId), a new polynomial-time approximation algorithm for adaptive IPP. We prove a polylogarithmic approximation bound when the robot travels in a metric space. Furthermore, our experiments suggest that RAId is practical and provides good approximate solutions for two distinct robot planning tasks. Although RAId is designed primarily for noiseless observations, a simple extension allows it to handle some tasks with noisy observations.

Optimizing Energy Efficiency for Minimum Latency Broadcast in Low-Duty-Cycle Sensor Networks

Multihop broadcasting in low-duty-cycle Wireless Sensor Networks (WSNs) is a very challenging problem, since every node has its own working schedule. Existing solutions usually use unicast instead of broadcast to forward packets from a node to its neighbors according to their working schedules, which is, however, not energy efficient. In this article, we propose to exploit the broadcast nature of wireless media to further save energy for low-duty-cycle networks, by adopting a novel broadcasting communication model. The key idea is to let some early wake-up nodes postpone their wake-up slots to overhear broadcasting messages from its neighbors. This model utilizes the spatiotemporal locality of broadcast to reduce the total energy consumption, which can be essentially characterized by the total number of broadcasting message transmissions. Based on such model, we aim at minimizing the total number of broadcasting message transmissions of a broadcast for low-duty-cycle WSNs, subject to the constraint that the broadcasting latency is optimal. We prove that it is NP-hard to find the optimal solution, and design an approximation algorithm that can achieve a polylogarithmic approximation ratio. Extensive simulation results show that our algorithm outperforms the traditional solutions in terms of energy efficiency.

Optimal nearest neighbor queries in sensor networks

Given a set of m mobile objects in a sensor network, we consider the problem of finding the nearest object among them from any node in the network at any time. These mobile objects are tracked by nearby sensors called proxy nodes. This problem requires an object tracking mechanism which typically relies on two basic operations: query and update. A query is invoked by a node each time when there is a need to find the closest object from it in the network. Updates of an object's location are initiated when the object moves from one location (proxy node) to another. We present a scalable distributed algorithm for tracking these mobile objects such that both the query cost and the update cost are small. The main idea in our algorithm is to maintain a virtual tree of downward paths pointing to the objects. Our algorithm guarantees an asymptotically optimal O(1) approximation for query cost and an O(min⁡{log⁡n,log⁡D}) approximation for update cost in the constant-doubling graph model, where n and D, respectively, are the number of nodes and the diameter of the network. We also give polylogarithmic approximations for both query and update cost in the general graph model. Our algorithm requires only polylogarithmic bits of memory per node. To the best of our knowledge, this is the first algorithm that is asymptotically optimal in handling nearest neighbor queries with low update cost in a distributed setting.

Near-Optimal Location Tracking Using Sensor Networks

We consider the problem of tracking mobile objects using a sensor network. We present a distributed tracking algorithm, called Mobile Object Tracking using Sensors (MOT), that scales well with the number of sensors and also with the number of mobile objects. MOT maintains a hierarchical structure of detection lists of objects that can efficiently track mobile objects and resolve object queries at any time. MOT guarantees that the cost to update (or maintain) its data structures will be at most O (min { log n, log D } ) times the optimal update cost and query cost will be within O (1) of the optimal query cost in the constant-doubling graph model, where n and D , respectively, are the number of nodes and the diameter of the network. MOT achieves polylogarithmic approximations for both costs in the general graph model. MOT balances the object and bookkeeping information load at each node in the expense of only O (log n ) increase in the update and query costs. The experimentation evaluation in both one by one and concurrent execution situations shows that MOT performs well in practical scenarios. To the best of our knowledge, MOT is the first algorithm for this problem in a distributed setting that is traffic-oblivious, i.e. agnostic to a priori knowledge of objects movement patterns, mobility and query rate, etc., and is load balanced. All previous solutions for this problem assumed traffic-consciousness in constructing the tracking data structure.Â

The all-or-nothing flow problem in directed graphs with symmetric demand pairs

We study the approximability of the All-or-Nothing multicommodity flow problem in directed graphs with symmetric demand pairs (SymANF). The input consists of a directed graph $$G = (V, E)$$G=(V,E) and a collection of (unordered) pairs of nodes $$\mathcal {M}= \left\{ s_1t_1, s_2t_2, \ldots , s_kt_k\right\} $$M=s1t1,s2t2,?,sktk. A subset $$\mathcal {M}'$$M? of the pairs is routable if there is a feasible multicommodity flow in $$G$$G such that, for each pair $$s_it_i \in \mathcal {M}'$$siti?M?, the amount of flow from $$s_i$$si to $$t_i$$ti is at least one and the amount of flow from $$t_i$$ti to $$s_i$$si is at least one. The goal is to find a maximum cardinality subset of the given pairs that can be routed. Our main result is a poly-logarithmic approximation with constant congestion for SymANF. We obtain this result by extending the well-linked decomposition framework of Chekuri et al. (2005) to the directed graph setting with symmetric demand pairs. We point out the importance of studying routing problems in this setting and the relevance of our result to future work.

Approximation algorithms for the arc orienteering problem

In this article we present approximation algorithms for the Arc Orienteering Problem (AOP). We propose a polylogarithmic approximation algorithm in directed graphs, while in undirected graphs we give a (6+ϵ+o(1)) and a (4+ϵ)-approximation algorithm for arbitrary instances and instances of unit profit, respectively. Also, an inapproximability result for the AOP is obtained as well as approximation algorithms for the Mixed Orienteering Problem.

Network Capacity Under Traffic Symmetry: Wireline and Wireless Networks

The problem of designing near optimal strategies for multiple unicast traffic in wireline networks is wide open; however, channel symmetry or traffic symmetry can be leveraged to show that routing can achieve with a polylogarithmic approximation factor of the edge-cut bound. For the same problem, the edge-cut bound is known to only upper bound rates of routing flows and unlike the information theoretic cut-set bound, it does not upper bound (capacity-achieving) information rates with general strategies. In this paper, we demonstrate that under channel or traffic symmetry, the edge-cut bound upper-bounds general information rates, thus providing a capacity approximation result. The key technique is a combinatorial result relating edge-cut bounds to generalized network sharing bounds. Finally, we generalize the results to wireless networks via an intermediary class of combinatorial graphs known as polymatroidal networks-our main result is that a natural architecture separating the physical and networking layers is near optimal when the traffic is symmetric among source-destination pairs, even when the channel is asymmetric (due to asymmetric power constraints, or prior frequency allocation like frequency division duplexing). This result is complementary to an earlier work of two of the authors proving a similar result under channel symmetry.

Approximating minimum-cost edge-covers of crossing biset-families

Part of this paper appeared in the preliminary version [16]. An ordered pair Ŝ = (S, S +) of subsets of a groundset V is called a biset if S ⊆ S+; (V S +;V S) is the co-biset of Ŝ. Two bisets $\hat X,\hat Y$ intersect if X X ? Y ? $\not 0$ and cross if both X ∩ Y $\not 0$ and X + ∪ Y + ≠= V. The intersection and the union of two bisets $\hat X,\hat Y$ are defined by $\hat X \cap \hat Y = (X \cap Y,X^ + \cap Y^ + )$ and $\hat X \cup \hat Y = (X \cup Y,X^ + \cup Y^ + )$ . A biset-family $\mathcal{F}$ is crossing (intersecting) if $\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}$ for any $\hat X,\hat Y \in \mathcal{F}$ that cross (intersect). A directed edge covers a biset Ŝ if it goes from S to V S +. We consider the problem of covering a crossing biset-family $\mathcal{F}$ by a minimum-cost set of directed edges. While for intersecting $\mathcal{F}$ , a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing $\mathcal{F}$ is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently or is not known. Let us say that a biset-family $\mathcal{F}$ is k-regular if $\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}$ for any $\hat X,\hat Y \in \mathcal{F}$ with |V (X∪Y)≥k+1 that intersect. In this paper we obtain an O(log |V|)-approximation algorithm for arbitrary crossing $\mathcal{F}$ if in addition both $\mathcal{F}$ and the family of co-bisets of $\mathcal{F}$ are k-regular, our ratios are: $O\left( {\log \frac{{|V|}} {{|V| - k}}} \right) $ if |S + S| = k for all $\hat S \in \mathcal{F}$ , and $O\left( {\frac{{|V|}} {{|V| - k}}\log \frac{{|V|}} {{|V| - k}}} \right) $ if |S + S| = k for all $\hat S \in \mathcal{F}$ . Using these generic algorithms, we derive for some network design problems the following approximation ratios: $O\left( {\log k \cdot \log \tfrac{n} {{n - k}}} \right) $ for k-Connected Subgraph, and O(logk) $\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\} $ for Subset k-Connected Subgraph when all edges with positive cost have their endnodes in the subset.

Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

An elementary h-route flow, for an integer h≥1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h≤3: The size of a minimum h-route cut is at least f/h and at most O(log4 k⋅f) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h=3 that has an approximation ratio of O(log4 k). Previously, polylogarithmic approximation was known only for h-route cuts for h≤2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem.