Non-trivial Approximation Algorithm Research Articles

Sketching Approximability of All Finite CSPs

A constraint satisfaction problem (CSP), \(\textsf {Max-CSP}(\mathcal {F})\) , is specified by a finite set of constraints \(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\) for positive integers q and k . An instance of the problem on n variables is given by m applications of constraints from \(\mathcal {F}\) to subsequences of the n variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family \(\mathcal {F}\) and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in \(o(\sqrt {n})\) space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of q = k =2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of \(\mathcal {F}\) support a distribution on \([q]^k\) with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables q = 2, binary constraints k =2, and singleton families \(|\mathcal {F}|=1\) and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [ 47 ] and [ 41 ], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [ 56 ], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to general q -ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.

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.

Complementary influence maximization under comparative linear threshold model

The influence maximization problem asks to find a small number of early adopters of a product in a social network, such that the expected number of total adoptions is maximized over the network. The problem has been well-studied, but most of the studies focus on the case of a single product or purely competitive products. This paper proposes a new influence diffusion model for multiple complementary products, namely, the comparative linear threshold (Com-LT) model. Under the Com-LT model, we model the complementary relation by reducing the thresholds of nodes. With this model, we study two problems: SelfInfMax and CompInfMax. We prove that these two problems are both NP-Hard under the Com-LT model. For both the SelfInfMax and the CompInfMax problem, we theoretically analyze the monotonicity and submodularity, and accordingly leverage lower bound optimization to devise non-trivial effective approximation algorithms. We conduct experiments over 4 real-world datasets. The experimental results demonstrate the correctness and efficiency of the proposed algorithms.

Minimization of Energy-Efficient Outage Probability in AF-Relayed PLC

Energy-aware resource allocation to achieve desired throughput over cooperative power line communications (PLCs) has gained growing interest from past few years. In this article, we design an energy-efficient amplify-and-forward relay-assisted PLC, where outage probability based on energy-efficiency is minimized by jointly optimizing the relay location and power allocation (PA) in two scenarios: 1) absence and 2) presence of the end-to-end direct link. To reduce complexity, using a novel and nontrivial approach, we obtain a closed-form expression of the outage probabilities for both the scenarios. Further, an equivalent problem is formulated to make the optimization problem more tractable to obtain its solution. In the absence of direct link, a closed-form solution of the problem is obtained by the optimization schemes where underlying variables are optimized individually and jointly. While, in the presence of the direct link, a closed-form solution is obtained using a nontrivial tight approximation algorithm. Via numerical results, the outage expressions are validated, and various insights on optimization of PA and relay location are obtained. Finally, with respect to energy-efficiency, it is shown that the joint optimization gives an outage improvement of around 55% and 37% in the absence and presence of the direct link, respectively.

Approximating Multistage Matching Problems

In multistage perfect matching problems, we are given a sequence of graphs on the same vertex set and are asked to find a sequence of perfect matchings, corresponding to the sequence of graphs, such that consecutive matchings are as similar as possible. More precisely, we aim to maximize the intersections, or minimize the unions between consecutive matchings. We show that these problems are NP-hard even in very restricted scenarios. As our main contribution, we present the first non-trivial approximation algorithms for these problems: On the one hand, we devise a tight approximation on graph sequences of length two (2-stage graphs). On the other hand, we propose several general methods to deduce multistage approximations from blackbox approximations on 2-stage graphs.

Log-concave polynomials, I: Entropy and a deterministic approximation algorithm for counting bases of matroids

We give a deterministic polynomial-time 2O(r)-approximation algorithm for the number of bases of a given matroid of rank r and the number of common bases of any two matroids of rank r. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze, this is almost the best possible result assuming oracle access to independent sets of the matroid. There are two main ingredients in our result. For the first, we build upon recent results of Adiprasito, Huh, Katz, and Wang on combinatorial Hodge theory to show that the basis generating polynomial of any matroid is a (completely) log-concave polynomial. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is (and all of its directional derivatives along the positive orthant are) log-concave as functions over the positive orthant. For the second ingredient, we develop a general framework for approximate counting in discrete problems, based on convex optimization. The connection goes through subadditivity of the entropy. For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the log-concavity of the basis generating polynomial.

Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem . Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O (log n / log log n ) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, Mądry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem.

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.

Approximation algorithms for the maximally balanced connected graph tripartition problem

Given a vertex-weighted connected graph $$G = (V, E, w(\cdot ))$$, the maximally balanced connected graphk-partition (k-BGP) seeks to partition the vertex set V into k non-empty parts such that the subgraph induced by each part is connected and the weights of these k parts are as balanced as possible. When the concrete objective is to maximize the minimum (to minimize the maximum, respectively) weight of the k parts, the problem is denoted as max–mink-BGP (min–maxk-BGP, respectively), and has received much study since about four decades ago. On general graphs, max–mink-BGP is strongly NP-hard for every fixed $$k \ge 2$$, and remains NP-hard even for the vertex uniformly weighted case; when k is part of the input, the problem is denoted as max–min BGP, and cannot be approximated within 6/5 unless P $$=$$ NP. In this paper, we study the tripartition problems from approximation algorithms perspective and present a 3/2-approximation for min–max 3-BGP and a 5/3-approximation for max–min 3-BGP, respectively. These are the first non-trivial approximation algorithms for 3-BGP, to our best knowledge.

A QPTAS for the base of the number of crossing-free structures on a planar point set

The number of triangulations of a planar n point set S is known to be cn, where the base c lies between 2.43 and 30. Similarly, the number of crossing-free spanning trees on S is known to be dn, where the base d lies between 6.75 and 141.07. The fastest known algorithm for counting triangulations of S runs in 2(1+o(1))nlog⁡n time while that for counting crossing-free spanning trees runs in O⁎(7.125n) time. The fastest known, non-trivial approximation algorithms for the number of triangulations of S and the number of crossing-free spanning trees of S, respectively, run in time subexponential in n. We present the first non-trivial approximation algorithms for these numbers running in quasi-polynomial time. They yield the first quasi-polynomial approximation schemes for the base of the number of triangulations of S and the base of the number of crossing-free spanning trees on S, respectively.

Spider Covers for Prize-Collecting Network Activation Problem

In the network activation problem, each edge in a graph is associated with an activation function that decides whether the edge is activated from weights assigned to its end nodes. The feasible solutions of the problem are node weights such that the activated edges form graphs of required connectivity, and the objective is to find a feasible solution minimizing its total weight. In this article, we consider a prize-collecting version of the network activation problem and present the first nontrivial approximation algorithms. Our algorithms are based on a new linear programming relaxation of the problem. They round optimal solutions for the relaxation by repeatedly computing node weights activating subgraphs, called spiders, which are known to be useful for approximating the network activation problem. For the problem with node-connectivity requirements, we also present a new potential function on uncrossable biset families and use it to analyze our algorithms.

Approximate association via dissociation

A vertex set X of a graph G is an association set if each component of G−X is a clique, and a dissociation set if each of these cliques has only one or two vertices. We observe some special structures and show that if none of them exists, then the minimum association set problem can be reduced to the minimum weighted dissociation set problem. This yields the first nontrivial approximation algorithm for the association set problem, with approximation ratio 2.5. The reduction is based on a combinatorial study of modular decomposition of graphs free of these special structures. Further, a novel algorithmic use of modular decomposition enables us to implement our algorithm in O(mn+n2) time.

Blind, Greedy, and Random: Algorithms for Matching and Clustering Using Only Ordinal Information

We study the Maximum Weighted Matching problem in a partial information setting where the agents' utilities for being matched to other agents are hidden and the mechanism only has access to ordinal preference information. Our model is motivated by the fact that in many settings, agents cannot express the numerical values of their utility for different outcomes, but are still able to rank the outcomes in their order of preference. Specifically, we study problems where the ground truth exists in the form of a weighted graph, and look to design algorithms that approximate the true optimum matching using only the preference orderings for each agent (induced by the hidden weights) as input. If no restrictions are placed on the weights, then one cannot hope to do better than the simple greedy algorithm, which yields a half optimal matching. Perhaps surprisingly, we show that by imposing a little structure on the weights, we can improve upon the trivial algorithm significantly: we design a 1.6-approximation algorithm for instances where the hidden weights obey the metric inequality. Our algorithm is obtained using a simple but powerful framework that allows us to combine greedy and random techniques in unconventional ways. These results are the first non-trivial ordinal approximation algorithms for such problems, and indicate that we can design robust matchings even when we are agnostic to the precise agent utilities.

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.

English

PASTRY andOverlie routing is a very attractive scheme that allows improving certain properties of the routing (such as delay or TCP throughput) without the need to change the standards of the current underlying routing. PASTRY is an implementation of a Distributed Hash Table(DHT) algorithm for P2P routing overlay using the PASTRY implementation the silent features to achieve the reliability,fully centralized and high fault tolerant . However, deploying overlie routing requires the placement and maintenance of overlie infrastructure. This gives rise to the optimization,reliability,high fault tolerant and highly distributed problem which we overcome in this project using . PASTRY distributed algorithm and non-trivial approximation algorithm. The current BGP routing in the Internet , where from a single source to all autonomous systems (ASs), that a small number of less than 100 relay servers is sufficient to enable routing over shortest paths reducing the average path length.

Shorter tours by nicer ears: 7/5-Approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs

We prove new results for approximating the graph-TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graph-TSP itself, we improve the approximation ratio to 7=5. For a generalization, the minimum T-tour problem, we obtain the first nontrivial approximation algorithm, with ratio 3=2. This contains the s-t-path graph-TSP as a special case. Our approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4=3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.

Cost-Effective Resource Allocation of Overlay Routing Relay Nodes

Overlay routing is a very attractive scheme that allows improving certain properties of the routing (such as delay or TCP throughput) without the need to change the standards of the current underlying routing. However, deploying overlay routing requires the placement and maintenance of overlay infrastructure. This gives rise to the following optimization problem: Find a minimal set of overlay nodes such that the required routing properties are satisfied. In this paper, we rigorously study this optimization problem. We show that it is NP-hard and derive a nontrivial approximation algorithm for it, where the approximation ratio depends on specific properties of the problem at hand. We examine the practical aspects of the scheme by evaluating the gain one can get over several real scenarios. The first one is BGP routing, and we show, using up-to-date data reflecting the current BGP routing policy in the Internet, that a relative small number of less than 100 relay servers is sufficient to enable routing over shortest paths from a single source to all autonomous systems (ASs), reducing the average path length of inflated paths by 40%. We also demonstrate that the scheme is very useful for TCP performance improvement (results in an almost optimal placement of overlay nodes) and for Voice-over-IP (VoIP) applications where a small number of overlay nodes can significantly reduce the maximal peer-to-peer delay.

Improved approximation algorithms for computing $$k$$ k disjoint paths subject to two constraints

For a given graph $$G$$ G with distinct vertices $$s$$ s and $$t$$ t , nonnegative integral cost and delay on edges, and positive integral bound $$C$$ C and $$D$$ D on cost and delay respectively, the $$k$$ k bi-constraint path ( $$k$$ k BCP) problem is to compute $$k$$ k disjoint $$st$$ s t -paths subject to $$C$$ C and $$D$$ D . This problem is known to be NP-hard, even when $$k=1$$ k = 1 (Garey and Johnson, Computers and Intractability, 1979). This paper first gives a simple approximation algorithm with factor- $$(2,2)$$ ( 2 , 2 ) , i.e. the algorithm computes a solution with delay and cost bounded by $$2*D$$ 2 ? D and $$2*C$$ 2 ? C respectively. Later, a novel improved approximation algorithm with ratio $$(1+\beta ,\,\max \{2,\,1+\ln (1/\beta )\})$$ ( 1 + β , max { 2 , 1 + ln ( 1 / β ) } ) is developed by constructing interesting auxiliary graphs and employing the cycle cancellation method. As a consequence, we can obtain a factor- $$(1.369,\,2)$$ ( 1.369 , 2 ) approximation algorithm immediately and a factor- $$(1.567,\,1.567)$$ ( 1.567 , 1.567 ) algorithm by slightly modifying the algorithm. Besides, when $$\beta =0$$ β = 0 , the algorithm is shown to be with ratio $$(1,\, O(\ln n))$$ ( 1 , O ( ln n ) ) , i.e. it is an algorithm with only a single factor ratio $$O(\ln n)$$ O ( ln n ) on cost. To the best of our knowledge, this is the first non-trivial approximation algorithm that strictly obeys the delay constraint for the $$k$$ k BCP problem.

Polynomial-Time Algorithms for Energy Games with Special Weight Structures

Energy games belong to a class of turn-based two-player infinite-duration games played on a weighted directed graph. It is one of the rare and intriguing combinatorial problems that lie in NP∩co-NP, but are not known to be in P. The existence of polynomial-time algorithms has been a major open problem for decades and apart from pseudopolynomial algorithms there is no algorithm that solves any non-trivial subclass in polynomial time. In this paper, we give several results based on the weight structures of the graph. First, we identify a notion of penalty and present a polynomial-time algorithm when the penalty is large. Our algorithm is the first polynomial-time algorithm on a large class of weighted graphs. It includes several worst-case instances on which previous algorithms, such as value iteration and random facet algorithms, require at least sub-exponential time. Our main technique is developing the first non-trivial approximation algorithm and showing how to convert it to an exact algorithm. Moreover, we show that in a practical case in verification where weights are clustered around a constant number of values, the energy game problem can be solved in polynomial time. We also show that the problem is still as hard as in general when the clique-width is bounded or the graph is strongly ergodic, suggesting that restricting the graph structure does not necessarily help.

On the usefulness of predicates

Motivated by the pervasiveness of strong inapproximability results for Max-CSPs, we introduce a relaxed notion of an approximate solution of a Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to replace the constraints of an instance by some other (possibly real-valued) constraints, and then only needs to satisfy as many of the new constraints as possible.To be more precise, we introduce the following notion of a predicatePbeingusefulfor a (real-valued) objectiveQ: given an almost satisfiable Max-Pinstance, there is an algorithm that beats a random assignment on the corresponding Max-Qinstance applied to the same sets of literals. The standard notion of a nontrivial approximation algorithm for a Max-CSP with predicatePis exactly the same as saying thatPis useful forPitself.We say thatPis useless if it is not useful for anyQ. This turns out to be equivalent to the following pseudo-randomness property: given an almost satisfiable instance of Max-P, it is hard to find an assignment such that the induced distribution onk-bit strings defined by the instance is not essentially uniform.Under theunique games conjecture, we give a complete and simple characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is useless if and only if there is a pairwise independent distribution supported on the satisfying assignments of the predicate. It is natural to also consider the case when no negations are allowed in the CSP instance, and we derive a similar complete characterization (under the UGC) there as well. Finally, we also include some results and examples shedding additional light on the approximability of certain Max-CSPs.