Apart from these particular cases, the construction of the optimal design relies on numerical **optimization** procedures. The case of the determinant—which corresponds to the choice q = 0, i.e., the D-optimality—is studied for example **in** [Wyn70] and [VBW98]. Another criterion based on matrix conditioning—referred to as G-optimality—is developed **in** [MYZ15]. **In** the latter paper, the construction of an optimal design is performed **in** two steps. **In** the ﬁrst step one only deals with an **optimization** problem on the set of all possible information matrices, while **in** the second step, one wishes to identify a possible probability distribution associated with the optimal information matrix. General **optimization** algorithms are discussed **in** [Fed10] and [ADT07]. A general **optimization** frame on measure sets including gradient descent methods is considered **in** [MZ04]. **In** the frame of ﬁxed given support points, eﬃcient SDP based algorithms are proposed and studied **in** [Sag11] and [SH15]. Let us mention the paper [VBW98] which is one of the original motivations to develop SDP solvers, especially for Max-Det-Problems—corresponding to D-optimal design—and the so-called problem of analytical centering.

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Keywords: min-max, min-max regret, **approximation**, fptas, shortest path, min- imum spanning tree.
1 Introduction
The definition of an instance of a combinatorial **optimization** problem requires to specify parameters, **in** particular objective function coefficients, which may be uncertain or imprecise. Uncertainty/imprecision can be structured through the concept of scenario which corresponds to an assignment of plausible values to parameters. There exist two natural ways of describing the set of all possible scenarios. **In** the interval data case, each numerical parameter can take any value between a lower and an upper bound. **In** the discrete scenario case, which is considered here, the scenario set is described explicitly. Kouvelis and Yu [6] proposed the min-max and min-max regret criteria, stemming from decision theory, to construct solutions hedging against parameters variations. The min- max criterion aims at constructing solutions having a good performance **in** the worst case. The min-max regret criterion, less conservative, aims at obtaining a solution minimizing the maximum deviation, over all possible scenarios, of the value of the solution from the optimal value of the corresponding scenario.

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M(e’/-’, 1/2) = 1/8 and M(eI/8, 1/3) =0if~=2 (mod 3); 1/18 otherwise.
Then, what is the condition such that ,Jt~l (9, ~) - 0 holds? It seems
that a non-constant **polynomial** **in** a quasi-periodic part influences whether
M (0, Ø) = 0 or not. So, we consider the cases each quasi-periodic form

major breakthrough has been obtained **in** [2] where the Hamiltonian cycle problem is solved **in** O ∗ (1.66 n ).
Dealing with **polynomial** space, the best running time known so far for traveling salesman problem is O ∗ (4 n n log n ) reached by the algorithm **in** [16] (see [3]).
**In** this article, we study **in** Section 2 the possibility to get ratios arbitrary close to 1 with a running time better than the one of exact computation. We handle min steiner tree and **some** versions of traveling salesman problem . **In** both cases, the basic idea is to find a small part of the instance verifying **some** suitable properties, then to solve the instance on the remaining part and to build finally a global solution. **In** Section 3, we show how one can take advantage of the possible existence of **some** polytime r-**approximation** algorithm **in** order to reach interesting (though exponential) running times for ratios slightly better than r.

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the ` p summability of (j β kψ
j k L ∞ ) j≥1 for **some** β > 1
2 , which still represents an improvement over
the condition **in** [19]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen-Loève basis for the representation of b might be suboptimal compared to other representations, **in** terms of the resulting summability properties of (ku ν k V ) ν∈F . While we focus on the diffusion equation, our analysis applies to

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VIII. C ONCLUSIONS
This article addressed the stability analysis for a class of complementarity systems using the method of Lyapunov functions. Questions pertaining to the existence of cone-copositive Lyapunov functions were answered **in** the affirmative for exponentially stable systems. **Some** refinements of this result, under certain conditions on the vector field **in** the system dynamics, allow us to restrict our search for cone-copositive Lyapunov functions within the class of homogeneous and rational polynomials. These statements indeed bring tractability to the numerical methods that have been proposed **in** this paper for computing Lyapunov functions. **In** particular, two hierarchies of convex **optimization** problems are obtained using the methods based on discretization and SOS **approximation**, respectively. Several immediate questions of interest emerge from our work which require further investigation. The first one among those is to extend our results to broader classes of complementarity systems. Systems of the form (C-Sys) are one particular class of relative degree one systems, but **in** applications, one sees more complex complementarity systems of the form studied **in** [50]. **In** such a wider class of systems, one sees different kinds of constraints on the state trajectories. Moreover, the constraints may vary with time **in** which case one has to consider the possibility of time-varying Lyapunov functions. It would be interesting to consider converse questions for this broader class of systems.

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a k+1 k − ε standard **approximation** for Min-Max b n k cTSP(1,2) could decide (k − 4)PP **in**
**polynomial** time.
6 Min-Sum EkTSP
Bellmore and Hong [3] showed that when the constraint p = k is replaced by p ≤ k, then Min-Sum kTSP is standard equivalent to TSP on an extended graph. This is true even for the directed version of the problem and when there is a cost associated with activating a salesman. For our case the transformation simply involves replacing the depot vertex 0 by k vertices of zero distance. Also, the metric case of the p ≤ k version is not of interest since the solution is just a single cycle (thus, we deal with the case p = k and Min-Sum EkTSP denote this problem).

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paschos@lamsade.dauphine.fr October 29, 2007
Abstract
The fact that it is very unlikely that a **polynomial** time algorithm could ever be devised for optimally solving NP-hard problems, strongly motivates both researchers and practitioners **in** trying to heuristically solving such problems, by making a trade-off between computational time and solution’s quality. **In** other words, heuristic computation consists of trying to find **in** reasonable time, not the best solution but one solution which is “near” the optimal one. Among classes of heuristic methods for NP-hard problems, the **polynomial** **approximation** algorithms aim at solving a given NP-hard problem **in** **polynomial** time by computing feasible solutions that are, under **some** predefined criterion, as near as possible to the optimal ones. The **polynomial** **approximation** theory deals with the study of such algorithms. This survey presents and analyzes **in** a first time **approximation** algorithms for **some** classical examples of NP -hard problems. **In** a second time, it shows how classical notions and tools of complexity theory, such as **polynomial** reductions, can be matched with **polynomial** **approximation** **in** order to devise structural results for NP-hard **optimization** problems. Finally, it presents a quick description of what it is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled.

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Although both approximations ˜ ̺ k (y) and ˆ ̺ k (y) differ **in** their expressions, they fundamentally
relate to the same notion that parameters from other models can be ignored when conditioning on the model index M . This approach is therefore bypassing the simultaneous exploration of several parameters spaces and restricts the simulation to marginal samplers on each separate model. This feature is very appealing since it cuts most of the complexity from the **schemes** both of Carlin and Chib (1995) and of Green (1995). We however question the foundations of those approximations as presented **in** both Scott (2002) and Congdon (2006, 2007) and advance below arguments that both authors are using incompatible versions of joint distributions on the collection of parameters that jeopardise the validity of the approximations.

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Keywords: Bsde, Monte-Carlo methods, branching process. MSC2010: Primary 65C05, 60J60; Secondary 60J85, 60H35.
1 Introduction
Since the seminal paper of Pardoux and Peng [22], the theory of Backward Stochastic Differential Equations (BSDEs hereafter) has been largely developed, and has lead to many applications **in** optimal control, finance, etc. (see e.g. El Karoui, Peng and Quenez [11]). Different approaches have been proposed during the last decade to solve them numerically, without relying on pure PDE based resolution methods. A first family of numerical **schemes**, based on a time discretization technique, has been introduced by Bally and Pagès [2],

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The numerical **approximation** of (1) has already been considered by sev- eral authors. It is not, however, a completely trivial matter. Indeed, this class of **schemes** has originally been devised for (steady) transport problems, based on a genuinely multidimensional upwind approach. Among the first contribu- tions, one may quote the work of P.L. Roe [21]. **Some** connections with more classical **schemes**, as the streamline diffusion method by Hughes, Johnson and co-authors, have soon been made [12]. However, the main problem is that, even though **some** residual distribution **schemes** can be recast as a particular class stabilized finite elements with emphasis on L ∞ stability, there is no clear general framework allowing to choose the test functions **in** order to recover a traditional variational statement. The main reason of this problem is related to the underlying formulation: everything is seen from a discrete point of view, and emphasis is put on the point-wise behavior of the residual (which is natural given the focus on L ∞ stability). Indeed the same remark applies to variants of the method not aiming at approximating point values of the solution, as **in** [7]. **In** this case, the local discrete (point-wise) residuals are replaced by residuals for **polynomial** coefficient sets for which once again a maximum principle is sought for.

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a factor n Ω(1)/ log log n (cf. [KST97], [DKS98], [DKRS00]), and this hardness ratio is **in** fact also valid for average dense instances. Only recently a **polynomial** time algorithm with the first sublinear **approximation** ratio O(n/logn) was designed for the general problem **in** [BK01]. Thus, the improvement **in** **approximation** ratio for the dense instances given by this paper seems to be the largest known for any NP-hard constraint satisfaction problem. This paper extends the density sampler technique for graphs developed **in** [BFK00] to k-uniform hypergraphs for k ≥ 3, as the main tool to attack the dense MIN-Ek-LIN2 problems, or equivalently, k-ary versions of the Nearest Codeword problems, and the dense MIN-EkSat problems. The paper is organized as follows. **In** Section 2 we give the pre- liminaries and prove NP-hardness **in** exact setting of all the dense minimum satisfaction problems considered **in** this paper. Section 3 contains our main result on sampling k-uniform hypergraphs crucial for the rest of the paper. **In** Section 4, we design a PTAS for dense MIN-Ek-LIN2 and **in** Section 5 a PTAS for dense MIN-EkSAT for any k.

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– van Dam and Sotirov give **some** new bounding procedures for a rather gen- eral graph partitioning problem, and show that the resulting bounds dominate previous known ones, yet remain tractable **in** practice.
– Pe˜ na et al. show that a wide family of PO problems can be reformulated as linear optimisation problems over completely positive cones. This extends a result of Burer, which applied only to certain mixed 0-1 quadratic programs. – de Klerk et al. analyse a known **polynomial**-time **approximation** scheme for

Sec. 5.2. Path-complete graphs and the joint spectral radius 117
theory. This is done **in** Section 5.2, where we define the notion of a path-complete graph (Definition 5.2) and prove that any such graph provides an **approximation** scheme for the JSR (Theorem 5.4). **In** Section 5.3, we give examples of families of path-complete graphs and show that many of the previously proposed techniques come from particular classes of simple path-complete graphs (e.g., Corollary 5.8, Corollary 5.9, and Remark 5.3.2). **In** Section 5.4, we characterize all the path- complete graphs with two nodes for the analysis of the JSR of two matrices. We determine how the approximations obtained from all of these graphs compare (Proposition 5.12). **In** Section 5.5, we study **in** more depth the **approximation** properties of a particular pair of “dual” path-complete graphs that seem to per- form very well **in** practice. Subsection 5.5.1 contains more general results about duality within path-complete graphs and its connection to transposition of ma- trices (Theorem 5.13). Subsection 5.5.2 gives an **approximation** guarantee for the graphs studied **in** Section 5.5 (Theorem 5.16), and Subsection 5.5.3 contains **some** numerical examples. **In** Section 5.6, we prove a converse theorem for the method of max-of-quadratics Lyapunov functions (Theorem 5.17) and an approx- imation guarantee for a new class of methods for proving stability of switched systems (Theorem 5.18). Finally, **some** concluding remarks and future directions are presented **in** Section 5.7.

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2.2 **Some** other NPO problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Notations 4 4 The s
ope of the paper and its main
ontributions 5 II Roughing out a new **approximation** framework 7 5 **Approximation**
hains: a generalization of the **approximation** algorithms 7 6 **Approximation** level 7 7 Convergen
e and hardness threshold 8 8 Fun
tional **approximation**-preserving redu
tions 10 III A
hieving **approximation** results **in** the new framework 11 9 Indu
ed hereditary subgraph maximization problems 12 10 Maximum independent set 14 10.1 A rst improvement of the **approximation** of the maximum-weight independent set via theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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Based upon semi-lo
al improvement, a renement of algorithm D_COLOR3 is devised **in** [24 ℄. The basi
idea remains the same: one greedily nds a
olle
tion of independent sets of up to a
ertain
onstant size k + 1 ,
olors any of them with a new
olor and removes all of them from the input graph; next she/he transforms the surviving graph into an instan
e of k SC as we have seen **in** se
tion 13 and approximately solves k SC **in** this instan
e. **In** [24 ℄, k = 6 . The algorithm for 6SC proposed **in**
ludes three phases. The rst one is totally greedy and
onsists of nding a maximal
olle
tion of disjoint 6-sets **in** the initial 6SC-instan
e (S, C) . The elements
overed by this
olle
tion are next removed from C and the remaining sets are updated. Of
ourse this update will eventually
reate **some** 1-sets. The se
ond phase is more restri
tive than the rst one. Here a maximal
olle
tion of disjoint 5-sets, then 4-sets is
onstru
ted but with the restri
tion that any su
h set is
hosen to make part of the
olle
tion only if its
hoi
e does not **in**
rease the number of 1-sets
reated during the rst phase. This is done greedily by
onsidering a set and by examining if the removal of its elements will
reate additional one sets. **In** what follows, we denote by STRICT_PHASE the pro
edure implementing the se
ond phase. The elements
overed by
olle
tion so-
onstru
ted are removed from C and the remaining sets are updated. Finally, the third and last phase, applied **in** the surviving 3SC-instan
e, is a semi-lo
al (2,1)-improvement. **In** what follows, we denote by SL_OPT21 the algorithm repeatedly applying semi-lo
al (2,1)- improvement until no su
h improvement is possible.

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Problem 1.1. Find n + 1 interpolation points 0 ≤ x 0 < x 1 < · · · < x n ≤ 1 such
that the **polynomial** interpolant p n ∈ P n defined by p n (x i ) = f (x i ) for all 0 ≤ i ≤ n
satisfies p n ∈ P n + .
Problem 1.1 has interest for pure numerical analysis purposes, and also because the question of having good characterization and convenient use of positive polyno- mials is central **in** applications and scientific computing. A non exhaustive list of references which reflects **some** of our own interests is: [1, 10] for cubic polynomials, [4] for automatization of the testing, [7, 5] with sum of squares characterization, [8, 9] on considerations on computer aided design with Bernstein and B´ezier curves, [15, 11, 14] for non negative numerical **approximation** **in** scientific computing for hyperbolic equa- tions and finally [5, 6] and therein for comprehensive references on **polynomial** theory. As Problem 1.1 is difficult to handle **in** full generality, it is convenient for theo-

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is bounded independently of ε then the sequence of Radon measures µ ϕ ε weakly- ∗
converges up to subsequence to the measure H n−1 x {ϕ 6= 1}. As already stated **in** the
introduction of this thesis, this fact is essential when approximating the Mumford - Shah functional as **in** the limit energy one aims at recovering the length of the jump set of **some** BV function which is contained **in** the set {ϕ 6= 1}. Later on **some** more general functionals have been introduced with different penalization of the jump set [BBB95] to model fractures. Recently [ABS99, DMOT16] other phase-field methods have dealt with the problem of efficiently approaching these energies. The main idea is that the limit ϕ rather then acquiring only the values {0, 1} should range over [0, +∞). **In** this chapter we follow this method to approximate any functional E h where h is a concave

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after a finite number of relaxations. The SDP solver used **in** this computation is mosek .
The table records the number of variables (v), the number of inequality and equality constraints (c), the maximum degree of the constraints and of the poly- nomial to minimize (d), the number of minimizer points (sol), the maximal order (o), the maximal number of parameters (p), the maximal size of the moment ma- trices (s) **in** the SDP problems, and the total CPU time **in** seconds (t).