A recent breakthrough in Edmonds' problem showed that the noncommutative rank can be computed in deterministic polynomial time, and various algorithms for it were devised. However, only quite complicated algorithms are known for finding a so-called shrunk subspace, which acts as a dual certificate for the value of the noncommutative rank. In particular, the operator Sinkhorn algorithm, perhaps the simplest algorithm to compute the noncommutative rank with operator scaling, does not find a shrunk subspace. Finding a shrunk subspace plays a key role in applications, such as separation in the Brascamp-Lieb polytope, one-parameter subgroups in the null-cone membership problem, and primal-dual algorithms for matroid intersection and fractional matroid matching. In this paper, we provide a simple Sinkhorn-style algorithm to find the smallest shrunk subspace over the complex field in deterministic polynomial time. To this end, we introduce a generalization of the operator scaling problem, where the spectra of the marginals must be majorized by specified vectors. Then we design an efficient Sinkhorn-style algorithm for the generalized operator scaling problem. Applying this to the shrunk subspace problem, we show that a sufficiently long run of the algorithm also finds an approximate shrunk subspace close to the minimum exact shrunk subspace. Finally, we show that the approximate shrunk subspace can be rounded if it is sufficiently close. Along the way, we also provide a simple randomized algorithm to find the smallest shrunk subspace. As applications, we design a faster algorithm for fractional linear matroid matching and efficient weak membership and optimization algorithms for the rank-2 Brascamp-Lieb polytope. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.08311

We consider the bin packing problem with d different item sizes s i and item multiplicities a i , where all numbers are given in binary encoding. This problem formulation is also known as the one-dimensional cutting stock problem . In this work, we provide an algorithm that, for constant d , solves bin packing in polynomial time. This was an open problem for all d\ge 3 . In fact, for constant d our algorithm solves the following problem in polynomial time: Given two d -dimensional polytopes P and Q , find the smallest number of integer points in P whose sum lies in Q . Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times, and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.

We consider incremental combinatorial optimization problems, in which a solution is constructed incrementally over time, and the goal is to optimize not the value of the final solution but the average value over all timesteps. We consider a natural algorithm of moving towards a global optimum solution as quickly as possible. We show that this algorithm provides an approximation guarantee of (9+sqrt(21))/15 > 0.9 for a large class of incremental combinatorial optimization problems defined axiomatically, which includes (bipartite and non-bipartite) matchings, matroid intersections, and stable sets in claw-free graphs. Furthermore, our analysis is tight.

In this paper, we consider the minimum unweighted Vertex Cover problem with Hard Capacity constraints (VCHC) on multigraphs and hypergraphs. Given a graph, the objective of VCHC is to find a smallest multiset of vertices that cover all edges, under the constraints that each vertex can only cover a limited number of incident edges, and the number of available copies of each vertex is bounded. This problem generalizes the classical unweighted vertex cover problem. Here we restrict our attention to unweighted instances, since the weighted version of VCHC is as hard as the set cover problem, as shown by Chuzhoy and Naor (FOCS 2002). We obtain improved approximation algorithms for VCHC on multigraphs and hypergraphs. This problem has first been studied by Saha and Khuller (ICALP 2012). They proposed a 38-approximation for multigraphs, and a max {6 f, 65}-approximation for hypergraphs, where f is the size of the largest hyperedge. In this paper, we significantly improve these approximation ratios to and 2 f respectively. In the case of multigraphs, our approximation ratio is very close to the longstanding bound of 2 for the classical vertex cover problem. Our algorithms consist of a two-step process, each based on rounding an appropriate linear program. In particular, for multigraphs, the analysis in the second step relies on identifying a matching structure within any extreme point solution. Furthermore, we consider the partial VCHC problem in which one only needs to cover all but ℓ edges. We propose a generic reduction from partial VCHC on f-hypergraphs to VCHC on (f + 1)-hypergraphs, with a small loss in the approximation factor. In particular, we present a (2f + 2)(1 + ∊)-approximation algorithm for partial VCHC on f-hypergraphs.

We consider the bin packing problem with d different item sizes si and item multiplicities ai, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem. In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d ≥ 3. In fact, for constant d our algorithm solves the following problem in polynomial time: given two d-dimensional polytopes P and Q, find the smallest number of integer points in P whose sum lies in Q. Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.

We present an efficient algorithm to find nonempty minimizers of a symmetric submodular function $f$ over any family of sets ${\cal I}$ closed under inclusion. Our algorithm makes $O(n^3)$ oracle calls to $f$ and ${\cal I}$, where $n$ is the cardinality of the ground set. In contrast, the problem of minimizing a general submodular function under a cardinality constraint is known to be inapproximable within $o(\sqrt{n/\log n})$ [Z. Svitkina and L. Fleischer, in Proceedings of the $49$th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Washington, DC, 2008, pp. 697--706]. We also present two extensions of the above algorithm. The first extension reports all nontrivial inclusionwise minimal minimizers of $f$ over ${\cal I}$ using $O(n^3)$ oracle calls, and the second reports all extreme subsets of $f$ using $O(n^4)$ oracle calls. Our algorithms are similar to a procedure by Nagamochi and Ibaraki [Inform. Process. Lett., 67 (1998), pp. 239--244] that finds all nontrivial inclusionwise minimal minimizers of a symmetric submodular function over a set of size $n$ using $O(n^3)$ oracle calls. Their procedure in turn is based on Queyranne's algorithm [M. Queyranne, Math. Program., 82 (1998), pp. 3--12] to minimize a symmetric submodular function by finding pendent pairs. Our results extend to any class of functions for which we can find a pendent pair whose head is not a given element.

Until recently, LP relaxations have only played a very limited role in the design of approximation algorithms for the Steiner tree problem. In particular, no (efficiently solvable) Steiner tree relaxation was known to have an integrality gap bounded away from 2, before Byrka et al. [3] showed an upper bound of ~1.55 of a hypergraphic LP relaxation and presented a ln(4)+ε ~1.39 approximation based on this relaxation. Interestingly, even though their approach is LP based, they do not compare the solution produced against the LP value. We take a fresh look at hypergraphic LP relaxations for the Steiner tree problem---one that heavily exploits methods and results from the theory of matroids and submodular functions---which leads to stronger integrality gaps, faster algorithms, and a variety of structural insights of independent interest. More precisely, along the lines of the algorithm of Byrka et al.[3], we present a deterministic ln(4)+ε approximation that compares against the LP value and therefore proves a matching ln(4) upper bound on the integrality gap of hypergraphic relaxations.

We consider the Asymmetric Traveling Salesman problem for costs satisfying the triangle inequality.We derive a randomized algorithm which delivers a solution within a factor O(log n/ log log n) of the optimum with high probability.

Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization. In this paper, we consider the problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a function such that, for every set S, (S) approximates f(S) within a factor α(n), where for rank functions of matroids and for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids. Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than , even for rank functions of a matroid.

Abstract The smallest k‐ECSS problem is, given a graph along with an integer k, find a spanning subgraph that is k‐edge connected and contains the fewest possible number of edges. We examine a natural approximation algorithm based on rounding an LP solution. A tight bound on the approximation ratio is 1 + 3/k for undirected graphs with k > 1 odd, 1 + 2/k for undirected graphs with k even, and 1 + 2/k for directed graphs with k arbitrary. Using iterated rounding improves the first upper bound to 1 + 2/k. On the hardness side we show that for some absolute constant c > 0, for any integer k ≥ 2 (k ≥ 1), a polynomial‐time algorithm approximating the smallest k‐ECSS on undirected (directed) multigraphs to within ratio 1 + c/k would imply P = NP. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

We consider a generalization of edge coloring bipartite graphs in which every edge has a weight in $[0,1]$ and the coloring of the edges must satisfy that the sum of the weights of the edges incident to a vertex v of any color must be at most 1. For unit weights, König's theorem says that the number of colors needed is exactly the maximum degree. For this generalization, we show that $2.557 n + o(n)$ colors are sufficient, where n is the maximum total weight adjacent to any vertex, improving the previously best bound of $2.833n+O(1)$ due to Du et al. Our analysis is interesting on its own and involves a novel decomposition result for bipartite graphs and the introduction of an associated continuous one-dimensional bin packing instance which we can prove allows perfect packing. This question is motivated by the question of the rearrangeability of 3-stage Clos networks. In that context, the corresponding parameter n of interest in the edge coloring problem is the maximum over all vertices of the number of unit-sized bins needed to pack the weights of the incident edges. In that setting, we are able to improve the bound to $2.5480 n + o(n)$, also improving a bound of $2.5625n+O(1)$ of Du et al. We also consider the online version of this problem in which edges have to be colored as soon as they are revealed. In this context, we can show that $5n$ colors are enough. This contrasts with the best known lower bound of $3n-2$ by Tsai, Wang, and Hwang but improves upon the previous best upper bound of $5.75n$ obtained by Gao and Hwang. Additionally, we show several improved bounds for more restricted versions of the problem. These online bounds are achieved by simple and easy-to-implement algorithms, inspired by the first fit heuristic for bin packing.

We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k+2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. This is almost best possible. The approach uses a sequence of simple algebraic, polyhedral and combinatorial arguments. It illustrates many techniques and ideas in combinatorial optimization as it involves polyhedral characterizations, uncrossing, matroid intersection, and graph orientations (or packing of spanning trees). The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree. It also gives a better understanding of the subtour relaxation for both the symmetric and asymmetric traveling salesman problems. The generalization to l-edge-connected subgraphs is briefly discussed

In third-generation (3G) wireless data networks, repeated requests for popular data items can exacerbate the already scarce wireless spectrum. In this paper, we propose an architectural and protocol framework that allows 3G service providers to host efficient content distribution services. We offload the spectrum intensive task of content distribution to an ad hoc network. Less mobile users (resident subscribers) are provided incentives to cache popular data items, while mobile users (transit subscribers) access this data from resident subscribers through the ad hoc network. Since the participants of this data distribution network act as selfish agents, they may collude to maximize their individual payoff. Our proposed protocol discourages potential collusion scenarios. In this architecture, the goal (social function) of the 3G service provider is to have the selfishly motivated resident subscribers service as many data requests as possible. However, the choice of which set of items to cache is left to the individual user. The caching activity among the different users can be modeled as a market sharing game. In this work, we study the Nash equilibria of market sharing games and the performance of such equilibria in terms of a social function. These games are a special case of congestion games that have been studied in the economics literature. In particular, pure strategy Nash equilibria for this set of games exist. We give a polynomial-time algorithm to find a pure strategy Nash equilibrium for a special case, while it is NP-hard to do so in the general case. As for the performance of Nash equilibria, we show that the price of anarchy-the worst case ratio between the social function at any Nash equilibrium and at the social optimum-can be upper bounded by a factor of 2. When the popularity follows a Zipf distribution, the price of anarchy is bounded by 1.45 in the special case where caching any item has a positive reward for all players. We prove that the selfish behavior of computationally bounded agents converges to an approximate Nash equilibrium in a finite number of improvements. Furthermore, we prove that, after each agent computes its response function once using a constant factor approximation algorithm, the outcome of the game is within a factor of O(logn) of the optimal social value, where n is the number of agents. Our simulation scenarios show that the price of anarchy is 30% better than that of the worst case analysis and that the system quickly (1 or 2 steps) converges to a Nash equilibrium.

We introduce the concept of a sink equilibrium. A sink equilibrium is a strongly connected component with no outgoing arcs in the strategy profile graph associated with a game. The strategy profile graph has a vertex set induced by the set of pure strategy profiles; its arc set corresponds to transitions between strategy profiles that occur with nonzero probability. (Here our focus will just be on the special case in which the strategy profile graph is actually a best response graph; that is, its arc set corresponds exactly to best response moves that result from myopic or greedy behaviour). We argue that there is a natural convergence process to sink equilibria in games where agents use pure strategies. This leads to an alternative measure of the social cost of a lack of coordination, the price of sinking, which measures the worst case ratio between the value of a sink equilibrium and the value of the socially optimal solution. We define the value of a sink equilibrium to be the expected social value of the steady state distribution induced by a random walk on that sink. We illustrate the value of this measure in three ways. Firstly, we show that it may more accurately reflects the inefficiency of uncoordinated solutions in competitive games when the use of pure strategies is the norm. In particular, we give an example (a valid-utility game) in which the game converges to solutions which are a factor n worse than socially optimal. The price of sinking is indeed n, but the price of anarchy is close to 1. Secondly, sink equilibria always exist. Thus, even in games in which pure strategy Nash equilibria (PSNE) do not exist, we can still calculate the price of sinking. Thirdly, we show that bounding the price of sinking can have important implications for the speed of convergence to socially good solutions in games where the agents make best response moves in a random order. We present two examples to illustrate our ideas. (i) Unsplittable selfish routing (and weighted congestion games):we prove that the price of sinking for the weighted unsplittable flow version of the selfish routing problem (for bounded-degree polynomial latency functions) is at most O(2/sup 2d/ d/sup 2d + 3/). In comparison, we give instances of these games without any PSNE. Moreover, our proof technique implies fast convergence to socially good (approximate) solutions. This is in contrast to the negative result of Fabrikant, Papadimitriou, and Talwar (2004) showing the existence of exponentially long best-response paths. (ii) Valid-utility games: we show that for valid-utility games the price of sinking is at most n+1; thus the worst case price of sinking in a valid-utility game is between it and n+1. We use our proof to show fast convergence to constant factor approximate solutions in basic-utility games. In addition, we present a hardness result which shows that, in general, there might be states that are exponentially far from any sink equilibrium in valid-utility games. We prove this by showing that the problem of finding a sink equilibrium (or a PSNE) in valid-utility games is PLS-complete.

The traveling salesman problem comes in two variants. The symmetric version (STSP) assumes that the cost c/sub ij/ of going to city i to city j is equal to c/sub ji/, while the more general asymmetric version (ATSP) does not make this assumption. In both cases, it is usually assumed that we are in the metric case, i.e., the costs satisfy the triangle inequality: c/sub ij/ + c/sub jk/ /spl ges/ c/sub ik/ for all i, j, k. In this assumption, we improve the lower bound on the integrality ratio of the Held-Karp bound for asymmetric TSP (with triangle inequality) from 4/3 to 2.

We consider a stochastic variant of the NP-hard 0/1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known, arbitrary distributions. Items are placed in the knapsack sequentially, and the act of placing an item in the knapsack instantiates its size. Our goal is to compute a solution "policy" that maximizes the expected value of items placed in the knapsack, and we consider both non-adaptive policies (that designate a priori a fixed sequence of items to insert) and adaptive policies (that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far). We show that adaptivity provides only a constant-factor improvement by demonstrating a greedy non-adaptive algorithm that approximates the optimal adaptive policy within a factor of 7. We also design an adaptive polynomial-time algorithm which approximates the optimal adaptive policy within a factor of 5 + /spl epsiv/, for any constant /spl epsiv/ > 0.

We consider a type of on-line, traffic scheduling problem in input queued, crossbar switches. The input to a problem, at each time step, is a set of desired traffic rates. These traffic rates in general cannot be exactly achieved since they assume arbitrarily small fractions of packets can be transmitted at each time step. The goal of the traffic scheduling problem is to closely approximate the given sequence of traffic rates by a sequence of switch uses in which only whole packets are sent. The focus of this paper is bounding the costs incurred in using such an approximation, in terms of the additional buffer size required. We establish universal bounds on the additional buffer size due to sending only whole packets; these bounds do not depend on the particular distribution of the input traffic, require no speedup, and guarantee 100% throughput. Specifically, for an N /spl times/ N input queued, crossbar switch, an on-line, packetizing algorithm is presented that guarantees 100% throughput with a buffer requirement of (N + 1)/sup 2//4 packets per input port with no speedup. The algorithm can he improved to run in O(N log N) time, using a fast algorithm for edge-coloring bipartite multigraphs. In the reverse direction, it is shown for an N /spl times/ N input queued, crossbar switch, that any on-line, packetizing algorithm with no speedup requires a buffer size of N/e - 2 packets per input port. We also extend the main packetizing algorithm in this paper to a general class of switch architectures.

In third generation (3G) wireless data networks, repeated requests for popular data items can exacerbate the already scarce wireless spectrum. In this paper we propose an architectural and protocol framework that allows 3G service providers to host efficient content distribution services. We offload the spectrum intensive task of content distribution to an ad-hoc network. Less mobile users (resident subscribers) are provided incentives to cache popular data items while mobile users (transit subscribers) access this data from resident subscribers through the ad-hoc network. Since the participants of this data distribution network act as selfish agents, they may collude to maximize their individual payoff. Our proposed protocol discourages potential collusion scenarios. In this architecture the goal (social function) of the 3G service provider is to have the selfishly motivated resident subscribers service as many data requests as possible. However, the choice of which set of items to cache is left to the individual user. The caching activity among the different users can be modeled as a market sharing game. In this work, we study the Nash equilibria of market sharing games and the performance of such equilibria in terms of a social function. These games are a special case of congestion games that have been studied in the economics literature. In particular, pure strategy Nash equilibria for this set of games exist. We give a polynomial-time algorithm to find a pure strategy Nash equilibrium for a special case while it is is NP-Hard to do so in the general case. As for the performance of Nash equilibria, we show that the price of anarchy -- the worst-case ratio between the social function at any Nash equilibrium and at the social optimum -- can be upper bounded by a factor of 2. When the popularity follows a Zipf distribution, the price of anarchy is bounded by 1.45 in the special case where caching any item has a positive reward for all players. We prove that the selfish behavior of computationally bounded agents converges to an approximate Nash equilibrium in a finite number of improvements. Furthermore, we show that even with one improvement by each player, an O(log n) approximate solution can be obtained. Our simulation scenarios show that the price of anarchy is 30% better than that of the worst-case analysis and that the system quickly (1 or 2 steps) converges to a Nash equilibrium.

We consider a generalization of edge coloring bipartite graphs in which every edge has a weight in [0,1] and the coloring of the edges must satisfy that the sum of the weights of the edges incident to a vertex v of any color must be at most 1. For unit weights, König's theorem says that the number of colors needed is exactly the maximum degree. For this generalization, we show that 2. 557 n + o(n) colors are sufficient where n is the maximum total weight adjacent to any vertex, improving the previously best bound of 2. 833n+O(1) due to Du et al. This question is motivated by the question of the rearrangeability of 3-stage Clos networks. In that context, the corresponding parameter n of interest in the edge coloring problem is the maximum over all vertices of the number of unit-sized bins needed to pack the weights of the incident edges. In that setting, we are able to improve the bound to 2. 5480 n + o(n), also improving a bound of 2. 5625n+O(1) of Du et al. Our analysis is interesting in its own and involves a novel decomposition result for bipartite graphs and the introduction of an associated continuous one-dimensional bin packing instance which we can prove allows perfect packing.

Abstract We consider the problem of selecting a core node in a network under two potentially competing criteria, one being the sum of the distances to a set of terminals, the other being the cost of connecting this core node and the terminals with a Steiner tree. We characterize the worst‐case trade‐off between approximation ratios for the two objectives. Our results, for example, show the existence of a core node in which both objectives are simultaneously within 1.37 times their optimum value (if we were to disregard the other objective). We also consider the problem of minimizing a weighted sum of the two criteria and perform a worst‐case analysis of a simple and fast heuristic, which does not need to enumerate possible core locations. This study was motivated by multimedia applications such as videoconferences or multiplayer games in which user‐dependent information has to be sent from the users to a core node to be chosen (at a cost proportional to the sum of the distances from the core node), and then global information has to be multicast back from the core node to all users (at a cost proportional to the Steiner tree cost). © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 179–186 2004

We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a time-indexed formulation, the other on a completion-time formulation. We show their equivalence by proving that a O(n log n) greedy algorithm leads to optimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, a concept which enhances our understanding of these LP relaxations. Based on the greedy solution, we describe two simple randomized approximation algorithms, which are guaranteed to deliver feasible schedules with expected objective function value within factors of 1.7451 and 1.6853, respectively, of the optimum. They are based on the concept of common and independent $\alpha$-points, respectively. The analysis implies in particular that the worst-case relative error of the LP relaxations is at most 1.6853, and we provide instances showing that it is at least $e/(e-1) \approx 1.5819$. Both algorithms may be derandomized; their deterministic versions run in O(n 2 ) time. The randomized algorithms also apply to the on-line setting, in which jobs arrive dynamically over time and one must decide which job to process without knowledge of jobs that will be released afterwards.

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 optimal solutions to these problems. In this paper, we consider using the cube roots of unity, 1, ei2π/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 [8] to obtain near-optimal solutions to the problems.

We show that, in any undirected graph, splitting-off can be performed while preserving all cuts of value at most 4/3 times the minimum value, and this is the best possible. This generalizes a classical splitting-off result of Lovász.

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 $1 | prec | \sum_j w_j C_j$ when the precedence constraints are series-parallel. We do this by using a linear programming formulation of $1 | prec | \sum_j w_j C_j$ 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.

Let G=(V,E) be a capacitated directed graph with a source s and k terminals t/sub i/ with demands d/sub i/, 1/spl les/i/spl les/k. We would like to concurrently route every demand on a single path from s to the corresponding terminal without violating the capacities. There are several interesting and important variations of this unsplittable flow problem. If the necessary cut condition is satisfied, we show how to compute an unsplittable flow satisfying the demands such that the total flow through any edge exceeds its capacity by at most the maximum demand. For graphs in which all capacities are at least the maximum demand, we therefore obtain an unsplittable flow with congestion at most 2, and this result is best possible. Furthermore, we show that all demands can be routed unsplittable in 5 rounds, i.e., all demands can be collectively satisfied by the union of 5 unsplittable flows. Finally, we show that 22.6% of the total demand can be satisfied unsplittably. These results are extended to the case when the cut condition is not necessarily satisfied. We derive a 2-approximation algorithm for congestion, a 5-approximation algorithm for the number of rounds and a 4.43=1/0.226-approximation algorithm for the maximum routable demand.

Let vc(G) denote the minimum size of a vertex cover of a graph G=(V,E). It is well known that one can approximate vc(G) to within a factor of 2 in polynomial time; and despite considerable investigation, no $(2 - \varepsilon)$-approximation algorithm has been found for any $\varepsilon > 0$. Because of the many connections between the independence number $\alpha(G)$ and the Lovász theta function $\vartheta(G)$, and because vc(G) = |V| - \alpha(G)$, it is natural to ask how well |V| - \vartheta(G)$ approximates vc(G). It is not difficult to show that these quantities are within a factor of 2 of each other ($|V| - \vartheta(G)$ is never less than the value of the canonical linear programming relaxation of vc(G)); our main result is that vc(G) can be more than $(2 - \varepsilon)$ times $|V| - \vartheta(G)$ for any $\varepsilon > 0$. We also investigate a stronger lower bound than $|V|- \vartheta(G)$ for vc(G).

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.

We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856times the optimal value.These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation.This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem.The best previously known approximation algorithms for these problems had perfc~rmance guarantees of ~for MAX CUT and ~for MAX 2SAT.Slight extensions of our analysis lead to a .79607-approximationalgorithm for the maximum directed cut problem (MAX DICUT) and a .758-approximationalgorithm for MAX SAT, where the best previously known approxim ation algorithms had performance guarantees of ~and ~, respectively.Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of :semidefinite programming in the design of approximation algorithms.

We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum-cost spanning tree, minimum-weight perfect matching, traveling salesman, and Steiner tree problems. Our technique produces approximation algorithms that run in $O(n^{2} \log n)$ time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of $O(n^{2} \log n)$ time compares favorably with the best strongly polynomial exact algorithms running in $O(n^{3})$ time for dense graphs. A similar result is obtained for the 2-matching problem and its variants. We also derive the first approximation algorithms for many NP-complete problems, including the nonfixed point-to-point connection problem, the exact path partitioning problem, and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413–4201.

We present randomized approximation algorithms for the MAX CUT and MAX 2SAT problems that always deliver solutions of expected value at least .87856times the optimal value.These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation.This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem.We then show how to derandomize the algorithm to obtain approximation algorithms with the same performance guarantee of .87856.The previous best-known approximation algorithms for these problems had performance guarantees of ~for MAX CUT and ~for MAX 2SAT.A slight extension of our analysis leads to a .79607-approximationalgorithm for the maximum directed cut problem, where a & approximation algorithm was the previous best-known algorithm.Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and, to the best of our knowledge, represents the first use of semidefinite programming in the design of approximation algorithms.

Yannakakis recently presented the first $\frac{3}{4}$-approximation algorithm for the Maximum Satisfiability Problem (MAX SAT). His algorithm makes nontrivial use of solutions to maximum flow problems. New, simple $\frac{3}{4}$-approximation algorithms that apply the probabilistic method/randomized rounding to the solution to a linear programming relaxation of MAX SAT are presented. It is shown that although standard randomized rounding does not give a good approximate result, the best solution of the two given by randomized rounding and a well-known algorithm of Johnson is always within $\frac{3}{4}$ of the optimal solution. It is further shown that an unusual twist on randomized rounding also yields 4-approximation algorithms. As a by-product of the analysis, a tight worst-case analysis of the relative duality gap of the linear programming relaxation is obtained.

We present the first polynomial-time approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut.This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem.If k is the maximum cut requirement of the problem, our solution comes within a factor of 2k of optimal.Our algorithm is primal-dual and shows the importance of this technique in designing approximation algorithms.1

Abstract We present some existing and some new formulations for the Steiner tree and Steiner arborescence problems. We show the equivalence of many of these formulations. In particular, we establish the equivalence between the classical bidirected dicut relaxation and two vertex weighted undirected relaxations. The motivation behind this study is a characterization of the feasible region of the dicut relaxation in the natural space corresponding to the Steiner tree problem. © 1993 by John Wiley & Sons, Inc.