Non-trivial Approximation Algorithm Research Articles

On some network design problems with degree constraints

We study several network design problems with degree constraints. For the minimum-cost Degree-Constrained 2-Node-Connected Subgraph problem, we obtain the first non-trivial bicriteria approximation algorithm, with 5b(v)+3 violation for the degrees and a 4-approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Feder, Motwani, and Zhu (2006). Then we consider the problem of finding an arborescence with at least k terminals and with minimum maximum outdegree. We show that the natural LP-relaxation has a gap of Ω(k) or Ω(n1/4) with respect to the multiplicative degree bound violation. We overcome this hurdle by a combinatorial O((klogk)/Δ⁎)-approximation algorithm, where Δ⁎ denotes the maximum degree in the optimum solution. We also give an Ω(logn) lower bound on approximating this problem. Then we consider the undirected version of this problem, however, with an extra diameter constraint, and give an Ω(logn) lower bound on the approximability of this version. We also consider a closely related Prize-Collecting Degree-Constrained Edge-Connectivity Survivable Network problem, and obtain several results in this direction by reducing the prize-collecting variant to the regular one. Finally, we consider the Terminal Steiner Tree problem, which is a simple variant of the Degree-Constrained Steiner Tree problem, when some terminals are required to be leaves. We show that this seemingly simple problem is equivalent to the Group Steiner Tree problem.

Approximating minimum-cost connectivity problems via uncrossable bifamilies

We give approximation algorithms for the Survivable Network problem. The input consists of a graph G = ( V,E ) with edge/node-costs, a node subset S ⊆ V , and connectivity requirements { r ( s,t ): s,t ∈ T ⊆ V }. The goal is to find a minimum cost subgraph H of G that for all s,t ∈ T contains r ( s,t ) pairwise edge-disjoint st -paths such that no two of them have a node in S ∖ { s,t } in common. Three extensively studied particular cases are: Edge-Connectivity Survivable Network ( S = ∅), Node-Connectivity Survivable Network ( S = V ), and Element-Connectivity Survivable Network ( r ( s,t ) = 0 whenever s ∈ S or t ∈ S ). Let k = max s,t ∈ T r ( s,t ). In Rooted Survivable Network, there is s ∈ T such that r ( u,t ) = 0 for all u ≠ s , and in the Subset k -Connected Subgraph problem r ( s,t ) = k for all s,t ∈ T . For edge-costs, our ratios are O ( k log k ) for Rooted Survivable Network and O ( k 2 log k ) for Subset k -Connected Subgraph. This improves the previous ratio O ( k 2 log n ), and for constant values of k settles the approximability of these problems to a constant. For node-costs, our ratios are as follows. — O ( k log | T |) for Element-Connectivity Survivable Network, matching the best known ratio for Edge-Connectivity Survivable Network. — O ( k 2 log | T |) for Rooted Survivable Network and O ( k 3 log | T |) for Subset k -Connected Subgraph, improving the ratio O ( k 8 log 2 | T |). — O ( k 4 log 2 | T |) for Survivable Network; this is the first nontrivial approximation algorithm for the node-costs version of the problem.

Assignment problem in content distribution networks

In a Content Distribution Network (CDN) , there are m servers storing the data; each of them has a specific bandwidth. All the requests from a particular client should be assigned to one server because of the routing protocol used. The goal is to minimize the total cost of these assignments—cost of each is proportional to the distance between the client and the server as well as the request size—while the load on each server is kept below its bandwidth limit. When each server also has a setup cost, this is an unsplittable hard-capacitated facility location problem . As much attention as facility location problems have received, there has been no nontrivial approximation algorithm when we have hard capacities (i.e., there can only be one copy of each facility whose capacity cannot be violated) and demands are unsplittable (i.e., all the demand from a client has to be assigned to a single facility). We observe it is NP-hard to approximate the cost to within any bounded factor in this case. Thus, for an arbitrary constant ϵ>0, we relax the capacities to a 1+ϵ factor. For the case where capacities are almost uniform , we give a bicriteria O (log n , 1+ϵ)-approximation algorithm for general metrics and a (1+ϵ, 1+ϵ)-approximation algorithm for tree metrics. A bicriteria (α,β)-approximation algorithm produces a solution of cost at most α times the optimum, while violating the capacities by no more than a β factor. We can get the same guarantees for nonuniform capacities if we allow quasipolynomial running time. In our algorithm, some clients guess the facility they are assigned to, and facilities decide the size of the clients they serve. A straightforward approach results in exponential running time. When costs do not satisfy metricity, we show that a 1.5 violation of capacities is necessary to obtain any approximation. It is worth noting that our results generalize bin packing (zero connection costs and facility costs equal to one), knapsack (single facility with all costs being zero), minimum makespan scheduling for related machines (all connection costs being zero), and some facility location problems.

An $O(k^3\log n)$-Approximation Algorithm for Vertex-Connectivity Survivable Network Design

In the Survivable Network Design problem (SNDP), we are given an undirected graph $G(V, E)$ with costs on edges, along with a connectivity requirement $r(u, v)$ for each pair $u, v$ of vertices. The goal is to find a minimum-cost subset $E^*$ of edges, that satisfies the given set of pairwise connectivity requirements. In the edge-connectivity version we need to ensure that there are $r(u, v)$ edge-disjoint paths for every pair $u, v$ of vertices, while in the vertex-connectivity version the paths are required to be vertex-disjoint. The edge-connectivity version of SNDP is known to have a 2-approximation. However, no non-trivial approximation algorithm has been known so far for the vertex version of SNDP, except for special cases of the problem. We present an extremely simple algorithm to achieve an $O(k^3 \log |T|)$-approximation for this problem, where $k$ denotes the maximum connectivity requirement, and $T$ is the set of vertices that participate in one or more pairs with non-zero connectivity requirements. We also give a simple proof of the recently discovered $O(k^2 \log |T|)$-approximation algorithm for the single-source version of vertex-connectivity SNDP. Our results establish a natural connection between vertex-connectivity and a well-understood generalization of edge-connectivity, namely, element-connectivity, in that, any instance of vertex-connectivity can be expressed by a small number of instances of the element-connectivity problem.

An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem

We consider the minimum maximal matching problem, which is NP -hard (Yannakakis and Gavril (1980) [18]). Given an unweighted simple graph G = ( V , E ) , the problem seeks to find a maximal matching of minimum cardinality. It was unknown whether there exists a non-trivial approximation algorithm whose approximation ratio is less than 2 for any simple graph. Recently, Z. Gotthilf et al. (2008) [5] presented a ( 2 − c log | V | | V | ) -approximation algorithm, where c is an arbitrary constant. In this paper, we present a ( 2 − 1 χ ′ ( G ) ) -approximation algorithm based on an LP relaxation, where χ ′ ( G ) is the edge-coloring number of G. Our algorithm is the first non-trivial approximation algorithm whose approximation ratio is independent of | V | . Moreover, it is known that the minimum maximal matching problem is equivalent to the edge dominating set problem. Therefore, the edge dominating set problem is also ( 2 − 1 χ ′ ( G ) ) -approximable. From edge-coloring theory, the approximation ratio of our algorithm is 2 − 1 Δ ( G ) + 1 , where Δ ( G ) represents the maximum degree of G. In our algorithm, an LP formulation for the edge dominating set problem is used. Fujito and Nagamochi (2002) [4] showed the integrality gap of the LP formulation for bipartite graphs is at least 2 − 1 Δ ( G ) . Moreover, χ ′ ( G ) is Δ ( G ) for bipartite graphs. Thus, as far as an approximation algorithm for the minimum maximal matching problem uses the LP formulation, we believe our result is the best possible.

Buy-at-Bulk Network Design with Protection

We consider approximation algorithms for buy-at-bulk network design, with the additional constraint that demand pairs be protected against a single edge or node failure in the network. In practice, the most popular model used in high-speed telecommunication networks for protection against failures is the so-called 1 + 1 model. In this model, two-edge or node-disjoint paths are provisioned for each demand pair. We obtain the first nontrivial approximation algorithms for buy-at-bulk network design in the 1 + 1 model for both edge and node-disjoint protection requirements. Our results are for the single-cable cost model, which is prevalent in optical networks. More specifically, we present a constant-factor approximation for the single-sink case and a polylogarithmic factor approximation for the multicommodity case. These results are of interest for practical applications and also suggest several new challenging theoretical problems.

Approximating Steiner Networks with Node-Weights

The (undirected) Steiner Network problem is as follows: given a graph $G=(V,E)$ with edge/node-weights and edge-connectivity requirements $\{r(u,v):u,v\in U\subseteq V\}$, find a minimum-weight subgraph H of G containing U so that the $uv$-edge-connectivity in H is at least $r(u,v)$ for all $u,v\in U$. The seminal paper of Jain [Combinatorica, 21 (2001), pp. 39–60], and numerous papers preceding it, considered the Edge-Weighted Steiner Network problem, with weights on the edges only, and developed novel tools for approximating minimum-weight edge-covers of several types of set functions and families. However, for the Node-Weighted Steiner Network (NWSN) problem, nontrivial approximation algorithms were known only for $0,1$ requirements. We make an attempt to change this situation by giving the first nontrivial approximation algorithm for NWSN with arbitrary requirements. Our approximation ratio for NWSN is $r_{\max}\cdot O(\ln|U|)$, where $r_{\max}=\max_{u,v\in U}r(u,v)$. This generalizes the result of Klein and Ravi [J. Algorithms, 19 (1995), pp. 104–115] for the case $r_{\max}=1$. We also give an $O(\ln|U|)$-approximation algorithm for the node-connectivity variant of NWSN (when the paths are required to be internally disjoint) for the case $r_{\max}=2$. Our results are based on a much more general approximation algorithm for the problem of finding a minimum node-weighted edge-cover of an uncrossable set-family. Finally, we give evidence that a polylogarithmic approximation ratio for NWSN with large $r_{\max}$ might not exist even for $|U|=2$ and unit weights.

Approximating k-generalized connectivity via collapsing HSTs

An instance of the k -generalized connectivity problem consists of an undirected graph G=(V,E), whose edges are associated with non-negative costs, and a collection ${\mathcal{D}}=\{(S_{1},T_{1}),\ldots,(S_{d},T_{d})\}$ of distinct demands, each of which comprises a pair of disjoint vertex sets. We say that a subgraph ??G connects a demand (S i ,T i ) when it contains a path with one endpoint in S i and the other in T i . Given an integer parameter k, the goal is to identify a minimum cost subgraph that connects at least k demands in ${\mathcal{D}}$ . Alon, Awerbuch, Azar, Buchbinder and Naor (SODA '04) seem to have been the first to consider the generalized connectivity paradigm as a unified machinery for incorporating multiple-choice decisions into network formation settings. Their main contribution in this context was to devise a multiplicative-update online algorithm for computing log-competitive fractional solutions, and to propose provably-good rounding procedures for important special cases. Nevertheless, approximating the generalized connectivity problem in its unconfined form, where one makes no structural assumptions about the underlying graph and collection of demands, has remained an open question up until a recent O(log?2 nlog?2 d) approximation due to Chekuri, Even, Gupta and Segev (SODA '08). Unfortunately, the latter result does not extend to connecting a pre-specified number of demands. Furthermore, even the simpler case of singleton demands has been established as a challenging computational task, when Hajiaghayi and Jain (SODA '06) related its inapproximability to that of dense k -subgraph. In this paper, we present the first non-trivial approximation algorithm for k-generalized connectivity, which is derived by synthesizing several techniques originating in probabilistic embeddings of finite metrics, network design, and randomization. Specifically, our algorithm constructs, with constant probability, a feasible subgraph whose cost is within a factor of O(n 2/3?polylog(n,k)) of optimal. We believe that the fundamental approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.

Approximate k-Steiner Forests via the Lagrangian Relaxation Technique with Internal Preprocessing

An instance of the k -Steiner forest problem consists of an undirected graph G=(V,E), the edges of which are associated with non-negative costs, and a collection $\mathcal{D}=\{(s_{1},t_{1}),\ldots,(s_{d},t_{d})\}$of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest ℱ⊆G connects a demand (s i ,t i ) when it contains an s i -t i path. Given a profit k i for each demand (s i ,t i ) and a requirement parameter k, the goal is to find a minimum cost forest that connects a subset of demands whose combined profit is at least k. This problem has recently been studied by Hajiaghayi and Jain (SODA ’06), whose main contribution in this context was to relate the inapproximability of k-Steiner forest to that of the dense k -subgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research. In this paper, we present the first non-trivial approximation algorithm for the k-Steiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of $O(\min \{n^{2/3},\sqrt{d}\}\cdot \log d)$of optimal, where n is the number of vertices in the input graph and d is the number of demands. We believe that the approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.

Tight approximation algorithm for connectivity augmentation problems

The S-connectivity λ G S ( u , v ) of ( u , v ) in a graph G is the maximum number of uv-paths that no two of them have an edge or a node in S − { u , v } in common. The corresponding Connectivity Augmentation ( CA ) problem is: given a graph G 0 = ( V , E 0 ) , S ⊆ V , and requirements r ( u , v ) on V × V , find a minimum size set F of new edges (any edge is allowed) so that λ G 0 + F S ( u , v ) ⩾ r ( u , v ) for all u , v ∈ V . Extensively studied particular choices of S are the edge- CA (when S = ∅ ) and the node- CA (when S = V ). A. Frank gave a polynomial algorithm for undirected edge- CA and observed that the directed case even with rooted { 0 , 1 } -requirements is at least as hard as the Set-Cover problem (in rooted requirements there is s ∈ V − S so that if r ( u , v ) > 0 then: u = s for directed graphs, and u = s or v = s for undirected graphs). Both directed and undirected node- CA have approximation threshold Ω ( 2 log 1 − ε n ) . The only polylogarithmic approximation ratio known for CA was for rooted requirements— O ( log n ⋅ log r max ) = O ( log 2 n ) , where r max = max u , v ∈ V r ( u , v ) . No nontrivial approximation algorithms were known for directed CA even for r ( u , v ) ∈ { 0 , 1 } , nor for undirected CA with S arbitrary. We give an approximation algorithm for the general case that matches the known approximation thresholds. For both directed and undirected CA with arbitrary requirements our approximation ratio is: O ( log n ) for S ≠ V arbitrary, and O ( r max ⋅ log n ) for S = V .

APPROXIMATE BLOCK SORTING

We consider the problem BLOCK-SORTING: Given a permutation, sort it by using a minimum number of block moves, where a block is a maximal substring of the permutation which is also a substring of the identity permutation, and a block move repositions the chosen block so that it merges with another block. Although this problem has recently been shown to be NP-hard [3], nothing better than a trivial 3-approximation was known. We present here the first non-trivial approximation algorithm to this problem. For this purpose, we introduce the following optimization problem: Given a set of increasing sequences of distinct elements, merge them into one increasing sequence with a minimum number of block moves. We show that the merging problem has a polynomial time algorithm. Using this, we obtain an O(n3) time 2-approximation algorithm for BLOCK-SORTING. We also observe that BLOCK-SORTING, as well as sorting by transpositions, are fixed-parameter-tractable in the framework of [6].

On network design problems: fixed cost flows and the covering steiner problem

Network design problems, such as generalizations of the Steiner Tree Problem, can be cast as edge-cost-flow problems. An edge-cost flow problem is a min-cost flow problem in which the cost of the flow equals the sum of the costs of the edges carrying positive flow.We prove a hardness result for the Minimum Edge Cost Flow Problem (MECF). Using the one-round two-prover scenario, we prove that MECF does not admit a 2 log 1-ε n -ratio approximation, for every constant ε > 0, unless NP ⊆ DTIME ( n polylogn ).A restricted version of MECF, called Infinite Capacity MECF (ICF), is defined. The ICF problem is defined as follows: (i) all edges have infinite capacity, (ii) there are multiple sources and sinks, where flow can be delivered from every source to every sink, (iii) each source and sink has a supply amount and demand amount, respectively, and (iv) the required total flow is given as part of the input. The goal is to find a minimum edge-cost flow that meets the required total flow while obeying the demands of the sinks and the supplies of the sources. This problem naturally arises in practical scheduling applications, and is equivalent to the special case of single source MECF, with all edges not touching the source or the sink having infinite capacity.The directed ICF generalizes the Covering Steiner Problem in directed and undirected graphs. The undirected version of ICF generalizes several network design problems, such as: Steiner Tree Problem, k -MST, Point-to-point Connection Problem, and the generalized Steiner Tree Problem.An O (log x )-approximation algorithm for undirected ICF is presented. We also present a bi-criteria approximation algorithm for directed ICF. The algorithm for directed ICF finds a flow that delivers half the required flow at a cost that is at most O ( n ε /ε 4 ) times bigger than the cost of an optimal flow. The running time of the algorithm is O ( x 2/ε ċ n 1+1/ε ), where x denotes the required total flow.Randomized approximation algorithms for the Covering Steiner Problem in directed and undirected graphs are presented. The algorithms are based on a randomized reduction to a problem called 1/2-Group Steiner. In undirected graphs, the approximation ratio matches the approximation ratio of Konjevod et al. [2002]. However, our algorithm is much simpler. In directed graphs, the algorithm is the first nontrivial approximation algorithm for the Covering Steiner Problem. Deterministic algorithms are obtained by derandomization.

Greedy approximation algorithms for directed multicuts

AbstractThe Directed Multicut (DM) problem is: given a simple directed graph G = (V, E) with positive capacities ue on the edges, and a set K ⊆ V × V of ordered pairs of nodes of G, find a minimum capacity K‐multicut; C ⊆ E is a K‐multicut if in G − C there is no (s, t)‐path for any (s, t) ⫅ K. In the uncapacitated case (UDM) the goal is to find a minimum size K‐multicut. The best approximation ratio known for DM is $O(\min\{\sqrt{n},opt\})$ by Gupta, where n = |V|, and opt is the optimal solution value. All known nontrivial approximation algorithms for the problem solve large linear programs. We give the first combinatorial approximation algorithms for the problem. Our main result is an Õ(n2/3/opt1/3)‐approximation algorithm for UDM, which improves the $O(\min\{opt,\sqrt{n}\})$‐approximation for opt = Ω(n1/2+ϵ). Combined with the article of Gupta, we get that UDM can be approximated within better than $O(\sqrt n)$, unless $opt={\tilde \Theta}(\sqrt n)$. We also give a simple and fast O(n2/3)‐approximation algorithm for DM. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(4), 214–217 2005

Is constraint satisfaction over two variables always easy?

AbstractBy the breakthrough work of Håstad [J ACM 48(4) (2001), 798–859], several constraint satisfaction problems are now known to have the following approximation resistance property: Satisfying more clauses than what picking a random assignment would achieve is NP‐hard. This is the case for example for Max E3‐Sat, Max E3‐Lin, and Max E4‐Set Splitting. A notable exception to this extreme hardness is constraint satisfaction over two variables (2‐CSP); as a corollary of the celebrated Goemans‐Williamson algorithm [J ACM 42(6) (1995), 1115–1145], we know that every Boolean 2‐CSP has a nontrivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables. An intriguing question then is whether this is also the case for 2‐CSPs over larger, non‐Boolean domains. This question is still open, and is equivalent to whether the generalization of Max 2‐SAT to domains of size d, can be approximated to a factor better than (1 − 1/d2). In an attempt to make progress towards this question, in this paper we prove, first, that a slight restriction of this problem, namely, a generalization of linear inequations with two variables per constraint, is not approximation resistant, and, second, that the Not‐All‐Equal Sat problem over domain size d with three variables per constraint, is approximation resistant, for every d ≥ 3. In the Boolean case, Not‐All‐Equal Sat with three variables per constraint is equivalent to Max 2‐SAT and thus has a nontrivial approximation algorithm; for larger domain sizes, Max 2‐SAT can be reduced to Not‐All‐Equal Sat with three variables per constraint. Our approximation algorithm implies that a wide class of 2‐CSPs called regular 2‐CSPs can all be approximated beyond their random assignment threshold. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004

An Approximation Algorithm for the Fault Tolerant Metric Facility Location Problem

We consider a fault tolerant version of the metric facility location problem in which every city, j, is required to be connected to r j facilities. We give the first non-trivial approximation algorithm for this problem, having an approximation guarantee of 3 · H k , where k is the maximum requirement and H k is the kth harmonic number. Our algorithm is along the lines of [2] for the generalized Steiner network problem. It runs in phases, and each phase, using a generalization of the primal–dual algorithm of [5] for the metric facility location problem, reduces the maximum residual requirement by one.

A New Approximation Algorithm for Finding Heavy Planar Subgraphs

We provide the first nontrivial approximation algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH, the NP-hard problem of finding a heaviest planar subgraph in an edge-weighted graph G . This problem has applications in circuit layout, facility layout, and graph drawing. No previous algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH had a performance ratio exceeding 1/3 , which is obtained by any algorithm that produces a maximum weight spanning tree in G . Based on the Berman—Ramaiyer Steiner tree algorithm, the new algorithm has performance ratio at least 1/3+1/72 and at most 5/12 . We also show that if G is complete and its edge weights satisfy the triangle inequality, then the performance ratio is at least 3/8 . Furthermore, we derive the first nontrivial performance ratio (7/12 instead of 1/2 ) for the NP-hard SC MAXIMUM WEIGHT OUTERPLANAR SUBGRAPH problem.

Approximation algorithms for classification problems with pairwise relationships

In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects , in a way that is consistent with some observed data that we have about the problem. An active line of research in this area is concerned with classification when one has information about pairwise relationships among the objects to be classified; this issue is one of the principal motivations for the framework of Markov random fields, and it arises in areas such as image processing, biometry, and document analysis. In its most basic form, this style of analysis seeks to find a classification that optimizes a combinatorial function consisting of assignment costs ---based on the individual choice of label we make for each object---and separation costs ---based on the pair of choices we make for two "related" objects.We formulate a general classification problem of this type, the metric labeling problem ; we show that it contains as special cases a number of standard classification frameworks, including several arising from the theory of Markov random fields. From the perspective of combinatorial optimization, our problem can be viewed as a substantial generalization of the multiway cut problem, and equivalent to a type of uncapacitated quadratic assignment problem .We provide the first nontrivial polynomial-time approximation algorithms for a general family of classification problems of this type. Our main result is an O (log k log log k )-approximation algorithm for the metric labeling problem, with respect to an arbitrary metric on a set of k labels, and an arbitrary weighted graph of relationships on a set of objects. For the special case in which the labels are endowed with the uniform metric ---all distances are the same---our methods provide a 2-approximation algorithm.

Lower bounds for approximate polygon decomposition and minimum gap

We consider the problem of decomposing polygons (with holes) into various types of simpler polygons. We focus on the problem of partitioning a rectilinear polygon, with holes, into rectangles, and show an Ω(n logn) lower bound on the time-complexity. The result holds for any decomposition, optimal or approximative. The bound matches the complexity of a number of algorithms in the literature, proving their optimality and settling the complexity of approximate polygon decomposition in these cases. As a related result we show that any non-trivial approximation algorithm for the minimum gap problem requires Ω(n logn) time.

The Loading Time Scheduling Problem

In this paper we study precedence constrained scheduling problems, where the tasks can only be executed on a specified subset of the set of machines. Each machine has a loading time that is incurred only for the first task that is scheduled on the machine in a particular run. This basic scheduling problem arises in the context of machining on numerically controlled machines and query optimization in databases and in other artificial intelligence applications. We give the first nontrivial approximation algorithm for this problem. We also prove nontrivial lower bounds on best possible approximation ratios for these problems. These improve on the nonapproximability results that are implied by the nonapproximability results for the shortest common supersequence problem. We use the same algorithm technique to obtain approximation algorithms for a problem arising in the context of code generation for parallel machines and for the weighted shortest common supersequence problem.

On bounded occurrence constraint satisfaction

An approximation algorithm for a constraint satisfaction problem is said to be nontrivial if its performance ratio is strictly superior to the expected performance of the algorithm which simply chooses a random assignment. We prove that any constraint satisfaction problem where each variable appears a bounded number of times admits a nontrivial polynomial time approximation algorithm.