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We consider the integrality gap of the subtour linear program (LP) relaxation of the traveling salesman problem (TSP) restricted to circulant instances. De Klerk and Dobre [Discrete Appl. Math., 159 (2011), pp. 1815--1826] conjectured that the value of the optimal solution to the subtour LP in these instances is equal to an entirely combinatorial lower bound from Van der Veen, Van Dal, and Sierksma [The Symmetric Circulant Traveling Salesman Problem, Research Memorandum 429, Institute of Economic Research, University of Groningen, 1991]. We prove this conjecture by giving an explicit optimal solution to the subtour LP. We then show that the integrality gap of the subtour LP is 2 on circulant instances, making such instances one of the few nontrivial classes of TSP instances for which the integrality gap of the subtour LP is exactly known. We also show that the degree constraints do not strengthen the subtour LP on circulant instances, mimicking the parsimonious property of metric, symmetric TSP instances shown in Goemans and Bertsimas [Math. Programming, 60 (1993), pp. 145--166] in a distinctly nonmetric set of instances. [ABSTRACT FROM AUTHOR]

Abstract The rank aggregation problem has received significant recent attention within the computer science community. Its applications today range far beyond the original aim of building metasearch engines to problems in machine learning, recommendation systems and more. Several algorithms have been proposed for these problems, and in many cases approximation guarantees have been proven for them. However, it is also known that some Markov chain based algorithms (MC1, MC2, MC3, MC4) perform extremely well in practice, yet had no known performance guarantees. We prove supra-constant lower bounds on approximation guarantees for all of them. Nevertheless, we show that in particular ways, MC4 can be seen as a generalization of Copeland score. [ABSTRACT FROM AUTHOR]

We study a semide finite programming relaxation of the traveling salesman problem introduced by de Klerk, Pasechnik, and Sotirov [SIAM J. Optim., 19 (2008), pp. 1559{1573] and show that their relaxation has an unbounded integrality gap. In particular, we give a family of instances such that the gap increases linearly with n. To obtain this result, we search for feasible solutions within a highly structured class of matrices; the problem of finding such solutions reduces to finding feasible solutions for a related linear program, which we do analytically. The solutions we find imply the unbounded integrality gap. Further, these solutions imply several corollaries that help us better understand the semidefinite program and its relationship to other TSP relaxations. Using the same technique, we show that a more general semidefinite program introduced by de Klerk, de Oliveira Filho, and Pasechnik [Handbook on Semidefinite, Conic and Polynomial Optimization, Springer, New York, 2012, pp. 171{199.] for the k-cycle cover problem also has an unbounded integrality gap. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that the Nash equilibrium in network routing games can have strictly higher cost than the optimum cost. In Stackelberg routing games, where a fraction of flow is centrally-controlled, a natural problem is to route the centrally-controlled flow such that the overall cost of the resulting equilibrium is minimized. We consider the scenario where the network administrator wants to know the minimum amount of centrally-controlled flow such that the cost of the resulting equilibrium solution is strictly less than the cost of the Nash equilibrium. We call this threshold the Stackelberg threshold and prove that for networks of parallel links with linear latency functions, it is equal to the minimum of the Nash flows on links carrying more optimum flow than Nash flow. Our approach also provides a simpler proof of characterization of the minimum fraction that must be centrally controlled to induce the optimum solution. [Copyright &y& Elsevier]

We give a simple primal algorithm for the generalized maximum flow problem that repeatedly finds and cancels generalized augmenting paths (GAPs). We use ideas of Wallacher (A generalization of the minimum-mean cycle selection rule in cycle canceling algorithms, 1991) to find GAPs that have a good trade-off between the gain of the GAP and the residual capacity of its arcs; our algorithm may be viewed as a special case of Wayne’s algorithm for the generalized minimum-cost circulation problem (Wayne in Math Oper Res 27:445–459, 2002). Most previous algorithms for the generalized maximum flow problem are dual-based; the few previous primal algorithms (including Wayne in Math Oper Res 27:445–459, 2002) require subroutines to test the feasibility of linear programs with two variables per inequality (TVPIs). We give an O( mn) time algorithm for finding negative-cost GAPs which can be used in place of the TVPI tester. This yields an algorithm with O( m log( mB/ε)) iterations of O( mn) time to compute an ε-optimal flow, or O( m 2 log ( mB)) iterations to compute an optimal flow, for an overall running time of O( m 3 nlog( mB)). The fastest known running time for this problem is $$\tilde{O}(m^2n\log B)$$ , and is due to Radzik (Theor Comput Sci 312:75–97, 2004), building on earlier work of Goldfarb et al. (Math Oper Res 22:793–802, 1997). [ABSTRACT FROM AUTHOR]

In a surprising result, Korupolu, Plaxton, and Rajaraman [13] showed that a simple local search heuristic for the capacitated facility location problem (CFLP) in which the service costs obey the triangle inequality produces a solution in polynomial time which is within a factor of 8+eof the value of an optimal solution. By simplifying their analysis, we are able to show that the same heuristic produces a solution which is within a factor of 6(1+e) of the value of an optimal solution. Our simplified analysis uses the supermodularity of the cost function of the problem and the integrality of the transshipment polyhedron.Additionally, we consider the variant of the CFLP in which one may open multiple copies of any facility. Using ideas from the analysis of the local search heuristic, we show how to turn anya-approximation algorithm for this variant into a polynomial-time algorithm which, at an additional cost of twice the optimum of the standard CFLP, opens at most one additional copy of any facility. This allows us to transform a recent 2-approximation algorithm of Mahdian, Ye, and Zhang [17] that opens many additional copies of facilities into a polynomial-time algorithm which only opens one additional copy and has cost no more than four times the value of the standard CFLP. [ABSTRACT FROM AUTHOR]

A number of recent papers on approximation algorithms have used the square roots of unity, −1 and 1, to represent binary decision variables for problems in combinatorial optimization, and have relaxed these to unit vectors in real space using semidefinite programming in order to obtain near optimum solutions to these problems. In this paper, we consider using the cube roots of unity, 1, ei2π/3, and ei4π/3, to represent ternary decision variables for problems in combinatorial optimization. Here the natural relaxation is that of unit vectors in complex space. We use an extension of semidefinite programming to complex space to solve the natural relaxation, and use a natural extension of the random hyperplane technique introduced by the authors in Goemans and Williamson (J. ACM 42 (1995) 1115–1145) to obtain near-optimum solutions to the problems. In particular, we consider the problem of maximizing the total weight of satisfied equations xu−xv≡c (mod 3) and inequations xu−xv≢c (mod 3), where xu∈{0,1,2} for all u. This problem can be used to model the Max-3-Cut problem and a directed variant we call Max-3-Dicut. For the general problem, we obtain a 0.793733-approximation algorithm. If the instance contains only inequations (as it does for Max-3-Cut), we obtain a performance guarantee of 7/12+3/4π2arccos2(−1/4)−ϵ>0.836008. This compares with proven performance guarantees of 0.800217 for Max-3-Cut (by Frieze and Jerrum (Algorithmica 18 (1997) 67–81) and 1/3+3/4π2arccos2(−1/4)−ϵ>0.836008. This compares with proven performance guarantees of 0.800217 for Max-3-Cut (by Frieze and Jerrum (Algorithmica 18 (1997) 67–81) and 1/3 arccos2(−1/4)−ϵ>0.836008. This compares with proven performance guarantees of 0.800217 for Max-3-Cut (by Frieze and Jerrum (Algorithmica 18 (1997) 67–81) and 1/3+10−8 for the general problem (by Andersson et al. (J. Algorithms 39 (2001) 162–204)). It matches the guarantee of 0.836008 for Max-3-Cut found independently by de Klerk et al. (On approximate graph colouring and Max-k-Cut algorithms based on the ϑ-function, Manuscript, October 2000). We show that all these algorithms are in fact equivalent in the case of Max-3-Cut, and that our algorithm is the same as that of Andersson et al. in the more general case. [Copyright &y& Elsevier]

In this survey, we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to NP-hard problems in combinatorial optimization. Because of parallels with the primal-dual method commonly used in combinatorial optimization, we call it the primal-dual method for approximation algorithms. We show how this technique can be used to derive approximation algorithms for a number of different problems, including network design problems, feedback vertex set problems, and facility location problems. [ABSTRACT FROM AUTHOR]

MAX SAT (the maximum satisfiability problem) is stated as follows: given a set of clauses with weights, find a truth assignment that maximizes the sum of the weights of the satisfied clauses. In this paper, we consider approximation algorithms for MAX SAT proposed by Goemans and Williamson and present a sharpened analysis of their performance guarantees. We also show that these algorithms, combined with recent approximation algorithms for MAX 2SAT, MAX 3SAT, and MAX SAT due to Feige and Goemans, Karloff and Zwick, and Zwick, respectively, lead to an improved approximation algorithm for MAX SAT. By using the MAX 2SAT and 3SAT algorithms, we obtain a performance guarantee of 0.7846, and by using Zwick''s algorithm, we obtain a performance guarantee of 0.8331, which improves upon the performance guarantee of 0.7977 based on Zwick''s conjecture. The best previous result for MAX SAT without assuming Zwick''s conjecture is a 0.770-approximation algorithm of Asano. Our best algorithm requires a new family of 3/4-approximation algorithms that generalize a previous algorithm of Goemans and Williamson. [Copyright &y& Elsevier]

In this note we give an alternate proof that a scheduling algorithm of Lawler [E.L. Lawler, Ann. Discrete Math., 2 (1978), pp. 75–90, E.L. Lawler and J.K. Lenstra, in Ordered Sets, I. Rival, ed., D. Reidel, 1982, pp. 655–675] finds the optimal solution for the scheduling problem 1¦prec¦ ∑jwjCj when the precedence constraints are series-parallel. We do this by using a linear programming formulation of 1¦prec¦ jwjCj introduced by Queyranne and Wang [Math. Oper. Res., 16 (1991), pp. 1–20]. Queyranne and Wang proved that their formulation completely describes the scheduling polyhedron in the case of series-parallel constraints; a by-product of our proof of correctness of Lawler's algorithm is an alternate proof of this fact. In the course of our proof it is helpful to use what might be called two-dimensional (2D) Gantt charts. We think these may find independent use, and to illustrate this we show that some recent work in the area becomes transparent using 2D Gantt charts. [ABSTRACT FROM AUTHOR]

We consider a 2-approximation algorithm for Euclidean minimum-cost perfect matching instances proposed by the authors in a previous paper. We present computational results for both random and real-world instances having between 1,000 and 131,072 vertices. The results indicate that our algorithm generates a matching within 2% of optimal in most cases. In over 1,400 experiments, the algorithm was never more than 4% from optimal. For the purposes of the study, we give a new implementation of the algorithm that uses linear space instead of quadratic space, and appears to run faster in practice. [ABSTRACT FROM AUTHOR]

The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an n-vertex tree T = (V , E) and an additional set L ⊆ V 2 of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links F such that every terminal pair that has arrived so far is 2-edge-connected in T ∪ F . This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network design problem. They gave a deterministic O (log 2 n) -competitive algorithm and showed an Ω (log n) lower bound on the competitive ratio of randomized algorithms. The case when T is a path is also interesting: it is exactly the online interval set cover problem, which also captures as a special case the parking permit problem studied by Meyerson (FOCS 2005). The contribution of this paper is to give tight results for online weighted tree and path augmentation problems. The main result of this work is a deterministic O (log n) -competitive algorithm for online WTAP, which is tight up to constant factors. [ABSTRACT FROM AUTHOR]

This paper considers the interplay between semidefinite programming, matrix rank, and graph coloring. Karger et al. (J ACM 45(2):246–265, 1998) give a vector program in which a coloring of a graph can be encoded as a semidefinite matrix of low rank. By complementary slackness conditions of semidefinite programming, if an optimal dual solution has high rank, any optimal primal solution must have low rank. We attempt to characterize graphs for which we can show that the corresponding dual optimal solution must have rank high enough that the primal solution encodes a coloring. In the case of the original Karger, Motwani, and Sudan vector program, we show that any graph which is a k-tree has sufficiently high dual rank, and we can extract the coloring from the corresponding low-rank primal solution. We can also show that if a graph is not uniquely colorable, then no sufficiently high rank dual optimal solution can exist. This allows us to completely characterize the planar graphs for which dual optimal solutions have sufficiently high dual rank, since it is known that the uniquely colorable planar graphs are precisely the planar 3-trees. We then modify the semidefinite program to have an objective function with costs, and explore when we can create an objective function such that the optimal dual solution has sufficiently high rank. We show that it is always possible to construct such an objective function given the graph coloring. The construction of the objective function gives rise to heuristics for 4-coloring planar graphs. We enumerated all maximal planar graphs with an induced K4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K_4$$\end{document} of up to 14 vertices; the heuristics successfully found a 4-coloring for 99.75% of them. Our research was motivated by trying to use semidefinite programming to prove the four-color theorem, which states that every planar graph can be colored with four colors. There is an intriguing connection of the Karger–Motwani–Sudan semidefinite program with the Colin de Verdière graph invariant (J Combin. Theory Ser B 50:11-21, 1990) (and a corresponding conjecture of Colin de Verdière), in which matrices that have some similarities to the dual feasible matrices of the semidefinite program must have high rank in the case that graphs are of a certain type; for instance, planar graphs have rank that would imply that the primal solution of the semidefinite program encodes a 4-coloring. [ABSTRACT FROM AUTHOR]

Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form L x = b , where L is the Laplacian matrix of a weighted graph with weights w (i , j) > 0 on the edges. The solution x of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge (i, j) is 1/w(i, j). Kelner et al. (in: Proceedings of the 45th Annual ACM Symposium on the Theory of Computing, pp 911–920, 2013. https://doi.org/10.1145/2488608.2488724) give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles (cycle toggling). In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by cut toggling: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a near-linear number of iterations. We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vector–matrix-vector problem, which has been conjectured to be difficult for dynamic algorithms (Henzinger et al., in: Proceedings of the 47th Annual ACM Symposium on the Theory of Computing, pp 21–30, 2015. https://doi.org/10.1145/2746539.2746609). The conjecture implies that the data structure does not have an O (n 1 - ϵ) time algorithm for any ϵ > 0 , and thus a straightforward implementation of the cut-toggling algorithm requires essentially linear time per iteration. To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an O ~ (m 1.5) time cut-toggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to O (m 1 + ϵ) for any ϵ > 0 . Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart. [ABSTRACT FROM AUTHOR]

In this paper, we study a very general type of online network design problem, and generalize two different previous algorithms, one for an online network design problem due to Berman and Coulston (Proceedings of the 29th annual ACM symposium on theory of computing, pp 344-353, 1997) and one for (offline) general network design problems due to Goemans and Williamson (SIAM J Comput 24:296-317, 1995); we give an O(logk)-competitive algorithm, where k is the number of nodes that must be connected. We also consider a further generalization of the problem that allows us to pay penalties in exchange for violating connectivity constraints; we give an online O(logk)-competitive algorithm for this case as well. [ABSTRACT FROM AUTHOR]

We study simple algorithms for the balanced MAX 2SAT problem, where we are given weighted clauses of length one and two with the property that for each variable x the total weight of clauses that x appears in equals the total weight of clauses for x¯. We show that such instances have a simple structural property in that any optimal solution can satisfy at most the total weight of the clauses minus half the total weight of the unit clauses. Using this property and a novel analysis of the computation tree, we are able to show that a large class of greedy algorithms, including Johnson’s algorithm, gives a 34-approximation algorithm for balanced MAX 2SAT; a similar statement is false for general MAX 2SAT instances. We further give a spectral 0.81-approximation algorithm for balanced MAX E2SAT instances (in which each clause has exactly 2 literals) by a reduction to a spectral algorithm of Trevisan for the maximum colored cut problem. We provide experimental results showing that this spectral algorithm performs well and is slightly better than Johnson’s algorithm and the Goemans–Williamson semidefinite programming algorithm on balanced MAX E2SAT instances. [ABSTRACT FROM AUTHOR]

We give a simple, randomized greedy algorithm for the maximum satisfiability problem (MAX SAT) that obtains a ¾ -approximation in expectation. In contrast to previously known ¾ -approximation algorithms, our algorithm does not use flows or linear programming. Hence we provide a positive answer to a question posed byWilliamson in 1998 on whether such an algorithm exists. Moreover, we show that Johnson's greedy algorithm cannot guarantee a ¾ -approximation, even if the variables are processed in a random order. Thereby we partially solve a problem posed by Chen, Friesen, and Zheng in 1999. In order to explore the limitations of the greedy paradigm, we use the model of priority algorithms of Borodin, Nielsen, and Rackoff. Since our greedy algorithm works in an online scenario where the variables arrive with their set of undecided clauses, we wonder if a better approximation ratio can be obtained by further fine-tuning its random decisions. For a particular information model we show that no priority algorithm can approximate Online MAX SAT within ¾ + ε (for any ε > 0). We further investigate the strength of deterministic greedy algorithms that may choose the variable ordering. Here we show that no adaptive priority algorithm can achieve approximation ratio ¾ . We propose two ways in which this inapproximability result can be bypassed. First we show that if our greedy algorithm is additionally given the variable assignments of an optimal solution to the canonical LP relaxation, then we can derandomize its decisions while preserving the overall approximation guarantee. Second we give a simple, deterministic algorithm that performs an additional pass over the input. We show that this 2-pass algorithm satisfies clauses with a total weight of at least ¾OPTLP, where OPTLP is the objective value of the canonical linear program. Moreover, we demonstrate that our analysis is tight and detail how each pass can be implemented in linear time. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we take a new approach to the very old problem of aggregating preferences of multiple agents. We define the notion of popular ranking: a ranking of a set of elements is popular if there exists no other permutation of the elements that a majority of the voters prefer. We show that such a permutation is unlikely to exist: we show that a necessary but not sufficient condition for the existence of a popular ranking is Condorcet’s paradox not occurring. In addition, we show that if Condorcet’s paradox does not occur, then we can efficiently compute a permutation, which may or may not be popular, but for which the voters will have to solve an NP-hard problem to compute a permutation that a majority of them prefer. [Copyright &y& Elsevier]

We give a 7.18-approximation algorithm for the minimum latency problem that uses only O(n log n) calls to the prize-collecting Steiner tree (PCST) subroutine of Goemans and Williamson. This improves the previous best algorithms in both performance guarantee and running time. A previous algorithm of Goemans and Kleinberg for the minimum latency problem requires an approximation algorithm for the k-minimum spanning tree (k-MST) problem which is called as a black box for each value of k. Their algorithm can achieve an approximation factor of 10.77 while making O(n(n + log C) log n) PCST calls, a factor of 8.98 using O(n³ (n + log C) log n) PCST calls, or a factor of 7.18 + ϵ using nO(1/ϵ) log C PCST calls, via the k-MST algorithms of Garg, Arya and Ramesh, and Arora and Karakostas, respectively. Here n denotes the number of nodes in the instance, and C is the largest edge cost in the input. In all cases, the running time is dominated by the PCST calls. Since the PCST subroutine can be implemented to run in O(n²) time, the overall running time of our algorithm is O(n³ log n). We also give a faster randomized version of our algorithm that achieves the same approximation guarantee in expectation, but uses only O(log² n) PCST calls, and derandomize it to obtain a deterministic algorithm with factor 7.18 + ϵ, using O( 1/ϵ log² n) PCST calls. The basic idea for our improvement is that we do not treat the k-MST algorithm as a black box. This allows us to take advantage of some special situations in which the PCST subroutine delivers a 2-approximate k-MST. We are able to obtain the same approximation ratio that would be given by Goemans and Kleinberg if we had access to 2-approximate k-MSTs for all values of k, even though we have them only for some values of k that we are not able to specify in advance. We also extend our algorithm to a weighted version of the minimum latency problem. [ABSTRACT FROM AUTHOR]

Abstract: Enumerative approaches to solving optimization problems, such as branch and bound, require a subroutine that produces a lower bound on the value of the optimal solution. In the domain of scheduling problems the requisite lower bound has typically been derived from either the solution to a linear-programming (LP) relaxation of the problem or the solution to a combinatorial relaxation. In this paper we investigate, from a theoretical perspective, the relationship between several LP-based lower bounds and combinatorial lower bounds for three scheduling problems in which the goal is to minimize the average weighted completion time of the jobs scheduled. We establish a number of facts about the relationship between these different sorts of lower bounds, including the equivalence of certain LP-based lower bounds for these problems to combinatorial lower bounds used in successful branch-and-bound algorithms. As a result, we obtain the first worst-case analysis of the quality of the lower bounds delivered by these combinatorial relaxations. [Copyright &y& Elsevier]

Abstract: The survivable network design problem (SNDP) is the following problem: given an undirected graph and values for each pair of vertices i and j, find a minimum-cost subgraph such that there are at least disjoint paths between vertices i and j. In the edge connected version of this problem (EC-SNDP), these paths must be edge-disjoint. In the vertex connected version of the problem (VC-SNDP), the paths must be vertex disjoint. The element connectivity problem (ELC-SNDP, or ELC) is a problem of intermediate difficulty. In this problem, the set of vertices is partitioned into terminals and nonterminals. The edges and nonterminals of the graph are called elements. The values are only specified for pairs of terminals , and the paths from i to j must be element disjoint. Thus if elements fail, terminals i and j are still connected by a path in the network. These variants of SNDP are all known to be NP-hard. The best known approximation algorithm for the EC-SNDP has performance guarantee of 2 and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primal-dual -approximation algorithm, where . Since this work first appeared as an extended abstract, it has been shown that it is hard to approximate VC-SNDP to factor . In this paper we investigate applying iterative rounding to ELC and VC-SNDP. We show that iterative rounding will not yield a constant factor approximation algorithm for general VC-SNDP. On the other hand, we show how to extend the analysis of iterative rounding applied to EC-SNDP to yield 2-approximation algorithms for both general ELC, and for the case of VC-SNDP when . The latter result improves on an existing 3-approximation algorithm. The former is the first constant factor approximation algorithm for a general survivable network design problem that allows node failures. [Copyright &y& Elsevier]

Garg [10] gives two approximation algorithms for the minimum-cost tree spanning k vertices in an undirected graph. Recently Jain and Vazirani [15] discovered primal-dual approximation algorithms for the metric uncapacitated facility location and k-median problems. In this paper we show how Garg’s algorithms can be explained simply with ideas introduced by Jain and Vazirani, in particular via a Lagrangean relaxation technique together with the primal-dual method for approximation algorithms. We also derive a constant factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques. [ABSTRACT FROM AUTHOR]

A number of basic models for VLSI layout are based on the construction of node- disjoint paths between terminals on a multilayer grid. In this setting, one is interested in minimizing both the number of layers required and the area of the underlying grid. Building on work of Cutler and Shiloach [Networks, 8 (1978), pp. 253-278], Aggarwal et al. [Proc. 26th IEEE Symposium on Foundations of Computer Science, Portland, OR, 1985; Algorithmica, 6 (1991), pp. 241-255], and Aggarwal, Klawe, and Shor [Algorithmica, 6 (1991), pp. 129-151], we prove an upper-bound trade-off between these two quantities in a general multilayer grid model. As a special case of our main result, we obtain significantly improved bounds for the problem of routing a full permutation on the mesh using node-disjoint paths; our new bound here is within polylogarithmic factors of the bisection bound. Our algorithms involve some new techniques for analyzing the structure of node-disjoint paths in planar graphs and indicate some respects in which this problem, at least in the planar case, is fundamentally different from its edge-disjoint counterpart. [ABSTRACT FROM AUTHOR]

The element connectivity problem falls in the category of survivable network design problems—it is intermediate to the versions that ask for edge-disjoint and vertex-disjoint paths. The edge version is by now well understood from the view-point of approximation algorithms [Williamson et al., Combinatorica 15 (1995) 435–454; Goemans et al., in: SODA ''94, 223–232; Jain, Combinatorica 21 (2001) 39–60], but very little is known about the vertex version. In our problem, vertices are partitioned into two sets: terminals and nonterminals. Only edges and nonterminals can fail—we refer to them as elements—and only pairs of terminals have connectivity requirements, specifying the number of element-disjoint paths required. Our algorithm achieves an approximation guarantee of factor 2Hk, where k is the largest requirement and Hn=1+1/2+⋯+1/n+⋯+1/n. Besides providing possible insights for solving the vertex-disjoint paths version, the element connectivity problem is of independent interest, since it models a realistic situation. [Copyright &y& Elsevier]

We study optimization problems that may be expressed as "Boolean constraint satisfaction problems." An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Different computational problems arise from constraint satisfaction problems depending on the nature of the "underlying" constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints (minimizing the number of unsatisfied constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used. Tight bounds on the approximability of every problem in Max CSP were obtained by Creignou [J. Comput. System Sci., 51 (1995), pp. 511-522]. In this work we determine tight bounds on the "approximability" (i.e., the ratio to within which each problem may be approximated in polynomial time) of every problem in Max Ones, Min CSP, and Min Ones. Combined with the result of Creignou, this completely classifies all optimization problems derived from Boolean constraint satisfaction. Our results capture a diverse collection of optimization problems such as MAX 3-SAT, Max Cut, Max Clique, Min Cut, Nearest Codeword, etc. Our results unify recent results on the (in-)approximability of these optimization problems and yield a compact presentation of most known results. Moreover, these results provide a formal basis to many statements on the behavior of natural optimization problems that have so far been observed only empirically. [ABSTRACT FROM AUTHOR]

We present a linear programming-based method for finding "gadgets," i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by Bellare, Goldreich, and Sudan [SIAM J. Comput., 27 (1998), pp. 804-915] of how to prove the optimality of gadgets: linear programming duality gives such proofs. The new gadgets, when combined with recent results of Håstad [Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 1-10], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 16/17+ ∊ and 12/13+ ∊ , respectively, is NP-hard for every ∊ > 0. Prior to this work, the best-known inapproximability thresholds for both problems were 71/72 (M. Bellare, O. Goldreich, and M. Sudan [SIAM J. Comput., 27 (1998), pp. 804-915]). Without using the gadgets from this paper, the best possible hardness that would follow from Bellare, Goldreich, and Sudan and Håstad is 18/19. We also use the gadgets to obtain an improved approximation algorithm for MAX3 SAT which guarantees an approximation ratio of .801. [ABSTRACT FROM AUTHOR]