This paper is motivated by real-life applications of bi-objective optimization. Having many non dominated solutions, one wishes to cluster the Pareto front using Euclidian distances. The p-center problems, both in the discrete and continuous versions, are proven solvable in polynomial time with a common dynamic programming algorithm. Having @math points to partition in @math clusters, the complexity is proven in @math (resp @math ) time and @math memory space for the continuous (resp discrete) @math -center problem. @math -center problems have complexities in @math . To speed-up the algorithm, parallelization issues are discussed. A posteriori, these results allow an application inside multi-objective heuristics to archive partial Pareto Fronts.
The HSS problem, maximizing the representativity of @math solutions among a PF of size @math , is known to be NP-hard in dimension 3 (and greater) since @cite_10 . An exact algorithm in @math and a polynomial-time approximation scheme for any constant dimension @math are also provided in @cite_10 . The 2d case is solvable in polynomial time thanks to a DP algorithm with a complexity in @math time and @math space provided in @cite_31 . The time complexity of the DP algorithm was improved in @math by @cite_26 and in @math by @cite_14 .
@cite_14 @cite_31 @cite_26 @cite_10
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par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . The traveling salesman problem (TSP) is a canonical NP-complete problem which is proved by Trevisan [SIAM J. Comput., 30 (2000), pp. 475--485] to be MAX-SNP hard even on high-dimensional Euclidean metrics. To circumvent this hardness, researchers have been developing approximation schemes for „simpler” instances of the problem. For instance, the algorithms of Arora and of Talwar show how to approximate TSP on low-dimensional metrics (for different notions of metric dimension). However, a feature of most current notions of metric dimension is that they are „local”: the definitions require every local neighborhood to be well-behaved. In this paper, we define a global notion of dimension that generalizes the popular notion of doubling dimension, but still allows some small dense regions; e.g., it allows some metrics that contain cliques of size @math . Given a metric with global dimension @math , we give a @math -approximation algorithm that runs in subexponential time, i.e., in $ (O(n^... Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. Let p > 1 be any fixed real. We show that assuming NP n RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in e p norm within a constant factor. Under the stronger assumption NP n RTIME(2poly(log n)), we show that there is no polynomial-time algorithm with approximation ratio 2(log n)1 2−e where n is the dimension of the lattice and e > 0 is an arbitrarily small constant.We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product, a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2(log n)1 2-e.