On percolation and NP ‐hardness

The edge‐percolation and vertex‐percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational complexity of problems whose inputs are obtained by applying percolation to worst‐case instances. Specifically, we show that a number of classical NP ‐hard problems on graphs remain essentially as hard on percolated instances as they are in the worst‐case (assuming NP⊈BPP ). We also prove hardness results for other NP ‐hard problems such as Constraint Satisfaction Problems and Subset‐Sum, with suitable definitions of random deletions. Along the way, we establish that for any given graph G the independence number α(G) and the chromatic number χ(G) are robust to percolation in the following sense. Given a graph G, let G′ be the graph obtained by randomly deleting edges of G with some probability p∈(0,1) . We show that if α(G) is small, then α(G′) remains small with probability at least 0.99. Similarly, we show that if χ(G) is large, then χ(G′) remains large with probability at least 0.99. We believe these results are of independent interest.


INTRODUCTION
The theory of N P-hardness suggests that we are unlikely to find optimal solutions to N P-hard problems in polynomial time. This theory applies to the worst-case setting where one considers the worst running-time over all inputs of a given size. It is less clear whether these hardness results apply to "real-life" instances. One way to address this question is to examine to what extent known N Phardness results are stable under random perturbations, as it seems reasonable to assume that a given instance of a problem may be subjected to noise.
Several works have studied the effect of random perturbations of the input on the runtime of algorithms. For example, Spielman and Teng [42] introduced the idea of smoothed analysis to explain the superior performance of algorithms in practice compared with formal worst-case bounds. Roughly speaking, smoothed analysis studies the running time of an algorithm on a perturbed worst-case instance. In particular, they showed that subjecting the weights of an arbitrary linear program to Gaussian noise yields instances on which the simplex algorithm runs in expected polynomial time, despite the fact that there are pathological linear programs for which the simplex algorithm requires exponential time. Since then smoothed analysis has been applied to a number of other problems [16,43].
In contrast to smoothed analysis, we study when worst-case instances of problems remain hard under random perturbations. Specifically, we study to what extent N P-hardness results are robust when instances are subjected to random deletions. Previous work is mainly concerned with Gaussian perturbations of weighted instances. Less work has examined the robustness of hardness results of unweighted instances with respect to discrete noise.
We focus on two forms of percolation 1 on graphs. Given a graph G = (V , E) and a parameter p ∈ (0, 1), we define G p,e = (V , E � ) as the probability space of graphs on the same set of vertices, where each edge e ∈ E is contained in E � independently with probability p. We say that G p,e is obtained from G by edge percolation. We define G p,v = (V � , E � ) as the probability space of graphs, in which every vertex v ∈ V is contained in V � independently with probability p, and G p,v is the subgraph of G induced by the vertices V � . We say that G p,v is obtained from G by vertex percolation. We also study appropriately defined random deletions applied to instances of other N P-hard problems, such as 3-SAT and Subset-Sum.
Throughout we refer to instances that are subjected to random deletions as percolated instances. Our main question is whether such percolated instances remain hard to solve by polynomial-time algorithms assuming N P BPP.
The study of random discrete structures has resulted with a wide range of mathematical tools which have proven instrumental in proving rigorous results regarding such structures [7,22,24,32]. One reason for studying percolated instances is that it may offer the opportunity to apply these methods to a broader range of distributions of instances of N P-hard problems beyond random graphs and random formulas. Furthermore, percolated instances are studied in host of different disciplines and are frequently used to explain properties of the internet [1,10], hence it is of interest to understand the computational complexity of instances arising from percolated networks.
We demonstrate that a polynomial-time algorithm for deciding whether G � is 3-colorable is impossible assuming N P BPP. We show this by considering the following polynomial time reduction from the 3-Coloring problem to itself.
We will need the following definition.
Definition 1.1 Given a graph G, the R-blowup of G is a graph G � = (V � , E � ), where every vertex v is replaced by an independent set v of size R, which we call the cloud corresponding to v. We then connect the clouds u and v by a complete R × R bipartite graph (u, v) ∈ E.
Given an n-vertex graph H the reduction outputs a graph G that is an R-blow-up of H for R = C log(n), where C > 0 is large enough. It is easy to see that H is 3-colorable if and only if G is 3-colorable.
In fact, the foregoing reduction satisfies a stronger robustness property for random subgraphs G � of G obtained by deleting edges of G with probability 1 2 . Namely, if H is 3-colorable, then G is 3-colorable, and hence G � is also 3-colorable with probability 1. On the other hand, if H is not 3-colorable, then G is not 3-colorable, and with high probability G � is not 3-colorable either.
Indeed, for any edge (v 1 , v 2 ) in H let U 1 , U 2 be two clouds in G corresponding to v 1 and v 2 . Fixing two arbitrary sets U � 1 ⊆ U 1 and U � 2 ⊆ U 2 each of size R/3, the probability that there is no edge in G � connecting a vertex from U � 1 to a vertex in U � 2 is 2 −R 2 /9 . By union bounding over the |E| · R R/3 choices of U � 1 , U � 2 we get that there is at least one edge between U � 1 and U � 2 with high probability. When this holds we can decode any 3-coloring of G � to a 3-coloring of H by coloring each vertex v of H with the color that appears the largest number of times in the coloring of the corresponding cloud in G � , breaking ties arbitrarily. Therefore, a polynomial time algorithm for deciding the 3-colorability of G � implies a polynomial time algorithm for determining the 3-colorability of H with high probability, and hence unless N P ⊆ coRP there is no polynomial time algorithm that given a 3-colorable graph G finds a 3-coloring of a random subgraph of G. 2

Toward a stronger notion of robustness
The foregoing example raises the question of whether the blow-up described above is really necessary. Naïvely, one could hope for a stronger hardness result of the 3-Coloring problem, namely, that for any graph H that is not 3-colorable, with high probability a random subgraph H � of H is not 3-colorable either. However, this is not true in general, as H can be a 3-critical graph, that is, a 3-colorable graph such that deletion of any edge of H decreases its chromatic number (consider for example the case of an odd cycle). Nonetheless, if random deletions do not decrease the chromatic number of a graph by much, then one could use hardness of approximation results on the chromatic number to deduce hardness results for coloring percolated graphs. This naturally leads to the following question. 3 Question Let G be an arbitrary graph, and let G � be a random subgraph of G obtained from G by deleting each edge of G with probability 1/2. Is it true that if χ(G) is large, then χ(G � ) is also large with probability at least 0.99? 4 2 Note that in the foregoing example, if we start with a bounded degree graph H, we can reduce it to a bounded degree graph G by using an R × R bipartite expander instead of the complete bipartite graph. 3 See the Preliminaries Section For precise definition of the chromatic number χG and the independence number αG 4 We note that if instead of choosing the subgraph G � at random, we choose an arbitrary subgraph of G with |E|/2 edges, then it is possible that χ(G � ) is much smaller than χ(G). For example, consider the n-vertex graph G = (V , E) that consists of a clique of size n/3 and a complete bipartite graph with n/3 edges on each side. Then χ(G) = n/3, whereas if we remove all the edges of the n/3-clique, the graph becomes 2-colorable, while the number of removed edges is n/3 2 < n 2 /18 < |E|/2.
In this paper we give a positive answer to this question and show that in some sense the chromatic number of any graph is robust against random deletions. We also consider the question of robustness for other graph parameters. For independent sets we demonstrate that if the independence number of G is small, then with high probability the independence number of a random subgraph of G is small as well. Hardness results are derived for other graph-theoretic problems such as Minimum Vertex Cover and Hamiltonian Cycle. Similarly, we show that for CSP formulas that are sufficiently dense randomly deleting its clauses does not change the maximum possible fraction of clauses that can be satisfied simultaneously. In particular, this implies that these problems remain essentially as hard on percolated instances as they are on worst-case instances.
Remark It is worth noting that there are graph parameters for which hardness on percolated instances differs significantly from hardness on the original instance. For example, standard results in random graph theory imply that for every n-vertex graph G, with high probability the size of the largest clique in the graph G � obtained by edge percolation with p = 1 2 is O(log n). In particular, a maximum clique in G � can be found in time n O(log n) , which is significantly faster than the fastest known algorithm for finding a maximum clique in the worst-case.

Robustness of N P-hard problems under percolation
In proving hardness results for percolated instances we use the concept of robust reductions which we explain below. It will be convenient to consider promise problems. 5 We start by introducing the following definition.   5 Recall, that a promise problem is a generalization of a decision problem, where for the problem L there are two disjoint subsets L YES and L NO , such that an algorithm that solves L must accept all the inputs in L YES and reject all inputs in L NO . If the input does not belong to L YES ∪ L NO , there is no requirement on the output of the algorithm.
We use the term noise-robust to avoid confusion with other notions of robust reductions that have appeared in the literature. In order to ease readability, we will often write robust reductions instead, always referring to noise-robust reductions as defined above.
Note that in the last item of Definition 1.2 there is no reduction involved. Instead, we think of the problem A as a relaxation of B with A YES ⊆ B YES and A NO ⊆ B NO , and hence any algorithm that solves B in particular solves A. However, the relaxation is also robust: after applying noise to a YES-instance (resp. NO-instance) of A, it stays a YES-instance (resp. NO-instance) of B with high probability. In particular, it is not necessarily true that every problem is a noise-robust relaxation of itself. Proposition 1.3 Let L = (L YES , L NO ) be a promise problem, and for each y instance of L, let noise(y) be a distribution on instances of L that is samplable in time that is polynomial in |y|.
If L is N P-hard under a noise-robust reduction, then there is no polynomial time algorithm that when given an input y decides with probability 0.99 whether noise(y) ∈ L YES or noise(y) ∈ L NO , unless N P ⊆ BPP.
Proof We prove the contrapositive. Suppose that such an algorithm A exists. Then A(noise(y)) outputs YES if and only if y is a YES instance of L with probability greater than 0.99 in randomized polynomial time. Therefore, N P ⊆ BPP.
Indeed, the example given in Section 1.1 gives a noise-robust reduction from the 3-Coloring problem to itself, where noise refers to random deletions of the edges in a given graph (edge percolation). Therefore, the 3-Coloring problem is N P-hard under a noise-robust reduction.

Our results
In this paper we show that a number of N P-hard problems remain hard to solve even after random deletions, that is, they are N P-hard under noise-robust reductions. Furthermore, we show that some gap N P-hard problems are, in fact, noise-robust relaxation of the same problems with a larger gap. Specifically, we focus on showing these results for the gap versions of the maximum independent set and chromatic number problems. As technical tools, we prove a number of combinatorial results about the independence number and the chromatic number of percolated graphs that might be of independent interest.

Maximum independent set and percolation
We start with the following result, saying that for any graph G, if G � is obtained from G by p-edge percolation, then with high probability the ratio between the independence number of G � and the the independence number of G is upper bounded by O(log(np)/p).
We observe that, in general, the upper bound above cannot be improved, as it is well known that the independence number of G(n, p) is Ω log(np) p with high probability (see, eg, [7]). In the Coloring-vs-MIS(q, a) problem, given a graph G the goal is to distinguish between the YES-case where χ(G) ≤ q and the NO-case where α(G) ≤ a. By using Theorem 1.4 together with inapproximability results of Feige and Kilian [17] we obtain the following hardness result. Theorem 1.5 For any q, a the Coloring-vs-MIS(q, O a p log(np) ) problem is a noiserobust relaxation of Coloring-vs-MIS(q, a), where n denotes the number of vertices in the given graph, and noise is edge-percolation with probability p.
In particular, for any constant ε > 0, there is no polynomial time algorithm that given an n-vertex graph G approximates either α(G p,e ) or χ(G p,e ) within a 1 pn 1−ε (resp. pn 1−ε ) factor for any p > 1 n 1−ε unless N P ⊆ BPP.
Remark Note that for constant p > 0 (eg, p = 1/2) the theorem establishes inapproximability for the independence number of G p,e that matches the inapproximability for the worst case. Note also that for p > 1 n 1−ε (in fact, for p > log(n) n ) such random percolated graphs have maximal degree at most O(pn) with high probability. Therefore, such graphs G p,e can be colored efficiently using O(pn) colors. In particular, with high probability G p,e contains an independent set of size Ω(1/p) and hence, the independence number can be approximated within a factor of pn on p-percolated instances.
We also prove analogous theorems for vertex percolation. We show that approximating α(G) and χ(G) on vertex percolated instances with p > 1 n 1−δ is essentially as hard as on worst-case instances. Here n is the number of vertices in the graph and δ ∈ (0, 1) does not depend on n. We show this by relying again on the hardness of the gap problem Coloring-vs-MIS for percolated instances.

Theorem 1.6 The Coloring-vs-MIS(q, a) problem is a noise-robust relaxation of itself,
where noise is vertex percolation with any p > 0.
In particular, for any δ, ε > 0 unless N P ⊆ BPP there is no polynomial time algorithm that approximates either α(G p,v ) or χ(G p,v ) within a (n � ) 1−ε factor for any constant ε > 0, where n � denotes the number of vertices in G p,v , and any p > 1 n 1−δ .
Remark Note that in the case of vertex percolation, the graph obtained by p-vertex percolation contains O(pn) vertices with high probability, and the (in)approximability guarantee should depend on the number of vertices in the percolated graph G p,v , and not on the number of vertices in the original graph. In particular, for p = O log(n) n the graph G p,v contains n � = O(log(n)) vertices and can be trivially solved in time 2 O(n � ) = poly(n), and hence we cannot expect to prove hardness results for vertex percolation with p = O log(n) n .
The proof of Theorem 1.4 can be found in Section 2. The proofs of Theorem 1.5 and Theorem 1.6 can be found in Section 3.

1.3.2
Graph coloring and edge percolation Theorem 1.5 says that it is hard to approximate the chromatic number of a percolated graph within a n 1−ε factor, but says nothing about hardness of coloring percolated graphs with small (constant) chromatic number. We address this question below by proving lower bounds 6 on the chromatic number of percolated graphs. To do this we use results from additive combinatorics and discrete Fourier analysis.
We also obtain lower bounds on the chromatic numbers of G p,v and G p,e for p < 1 2 by composing the bounds in Theorems 1.7 and 1.8 �log 2 (1/p)� times.
In the Gap-Coloring(q, Q) problem we are given an n-vertex graph G and the goal is to distinguish between the YES-case where G is q-colorable, and the NO-case where the chromatic number of G is at least Q. There is a large body of work proving hardness results for this problem [23,25,28] including stronger results assuming variants of the Unique Games Conjecture [12,14]. The strongest N P-hardness result of this form, due to Huang [25], shows that Gap-Coloring(q, exp(Ω(q 1/3 ))) is N Phard. Combining this with Theorem 1.8 we obtain an analogous hardness result under noise-robust reductions for this problem. Theorem 1.9 For some fixed c > 0 and all q < cQ 1/3 the Gap-Coloring(q, cQ 1/3 ) problem is a noise-robust relaxation of Gap-Coloring(q, Q), and noise is 1 2 -edge-percolation applied to the graph.
In particular, for any sufficiently large constant q given a q-colorable graph G it is N P-hard to find a exp(o(q 1/3 ))-coloring of G 1 2 ,e with high probability.
We prove the theorem in Section 3.

Satisfiability and other CSPs
We prove hardness of approximating the value of percolated k-CSP instances. An instance Φ of k-CSP over some alphabet Σ (eg, Σ = {0, 1}) is a formula consisting of a collection of clauses C 1 , ..., C m over n variables x 1 , ..., x n taking values in Σ, where each clause is associated with some k-ary predicate f : Σ k → {0, 1} over variables x i 1 , . . . , x i k . An instance Φ is said to be simple if all clauses in Φ are distinct. Given an assignment σ : {x 1 , ..., x n } → Σ we say that the constraint C on the variables Given a formula Φ, and an assignment σ to its variables the value of Φ with respect to the assignment σ, denoted by val σ (Φ), is the fraction of constraints of Φ satisfied by σ. The value of Φ is defined as We are typically interested in CSP where constraints belong to some fixed family of predicates F. For example, in the k-SAT problem, the constraints are all of the form f (z 1 , . . . , We assume that k, the arity of the constraints, is some fixed constant that does not depend on the number of variables n. These definitions give rise to the following optimization problem. Given a CSP instance Φ find an assignment that maximizes the value of Φ. We refer to this maximization problem as Max-CSP-F, where F denotes the family of predicates that constraints are taken from. For 0 < s < c ≤ 1, Gap-CSP-F(c, s) is the promise problem where YES-instances are formulas Φ such that val(Φ) ≥ c, and NO-instances are formulas Φ such that val(Φ) ≤ s. Here we assume the constraints of CSP instances are restricted to be in the family F.
We study two models of percolation on instances of CSP, clause percolation and variable percolation. Given an instance Φ of CSP its clause percolation is a random formula Φ c p over the same set of variables, that is obtained from Φ by keeping each clause of Φ independently with probability p. Theorem 1.10 Let ε, δ ∈ (0, 1) be fixed constants. There is a polynomial time reduction such that given a simple unweighted instance Φ of Max-CSP-F outputs a simple unweighted instance Ψ of Max-CSP-F on n variables, such that val(Ψ) = val(Φ), and for any p > 1 n k−1−δ the following holds.
This immediately implies the following corollary.

Corollary 1.11 Let F be a collection of Boolean constraints of arity k, and sup-
with n denoting the number of variables in a given formula, and ε, δ > 0 arbitrary constants.
One ingredient of the proof of Theorem 1.10 that may be of independent interest is establishing that for an arbitrary small positive constant η > 0 the k-CSP instances with n k−η clauses are essentially as hard to approximate as arbitrary instances.
We also consider variable percolation. Given an instance Φ of CSP we consider a random formula Φ v p whose set of variables is a subset S of the variables of Φ, where each variable of Φ is in S independently with probability p ∈ (0, 1) and the clauses of Φ v p are all clauses of Φ induced by S. In other words, a clause C of Φ survives if and only if all variables of C are in S. Using ideas similar to those used for clause percolation we show that p-vertex percolated instance are essentially as hard as in the worst case for p > 1 n 1−δ for any δ ∈ (0, 1). For proofs, see Section 4.

Other problems
For the Minimum Vertex Cover problem, we prove that for any α and δ > 0 an algorithm that gives α approximation for percolated instances implies also a α − δ approximation algorithm for worst-case instances. Our results hold for both edge and vertex percolation, where the edges or the vertices of a given graph remain with probability p > 1 n 1−ε for some ε ∈ (0, 1). In particular, assuming the Unique Games Conjecture, the result of [29] implies that 2 − δ approximation for the Minimum Vertex Cover problem is N P-hard under a noise-robust reduction for any constant δ > 0. See Theorem 5.2 and Theorem 5.4 for the precise statements.
For the Hamiltonicity problem, we prove hardness results for percolated instances of both directed and undirected graphs with respect to edge percolation. We show that the problem where one needs to determine whether a graph contains a Hamiltonian cycle is also hard for percolated graphs, where each edge of a given graph is kept in the graph with probability p > 1 n 1−ε for any ε ∈ (0, 1). See Theorem 6.1 for the precise statement.
We also study percolation on instances of the Subset-Sum problem, where each item of the set is deleted with probability 1 − p. We show that the problem remains hard as long as p = Ω( 1 n 1/2−ε ), where ε ∈ (0, 1/2) is an arbitrary constant, and n is the number of items in the given instance. See Theorem 7.1 for the precise statement.

Related work
Our work is related to several streams of research which we survey next.

Algorithmic applications of random deletions
The idea to use random deletions in order to obtain "easier" instances has been applied to problems in network design such as finding minimum cuts and maximum flows in graphs. For example, in [6,26] it was shown that one can achieve faster running times for algorithms approximating cuts and flows in a given graph G by randomly sampling G p by choosing to keep in each edge of G with some, appropriately chosen, probability p, and applying known algorithms for sparse graphs. The analysis, then, claims that with high probability all the cut values in G p are essentially equal to the cut values in G multiplied by p, and hence the minimum cut value in G p can be used to approximate the minimum cut value of G. Other works have considered the case of deletion probabilities that may depend on the deleted edge or vertex. In particular, [6] used nonuniform sampling in order to improve upon the approximation algorithms of [26] for finding the minimum cut in a graph. More recently, sampling subgraphs has been used to find independent sets [19]. The deletion probability of a vertex in the algorithm in [19] is directly proportional to its degree, and there may be dependencies between deletions of different vertices.

Discrete smoothed analysis
Spielman and Teng [41] have considered adapting the framework of smoothed analysis to unweighted graphs. As they observed, randomly deleting every existing edge of the graph and randomly adding every non-edge with some small probability p can drastically change the optimum of the optimization problem. In fact, arbitrary graphs perturbed in this way resemble random graphs (eg, the clique number of an arbitrary graph that is perturbed in this manner will be O(log n) with high probability). To avoid this problem they define random property preserving perturbations, which are random perturbations of G conditioned on the perturbation not changing the optimum value of the original instance. For example, they consider perturbations that do not change the value of the minimum bisection of the perturbed graph G.
Our results indicate that if one considers only random deletions (as opposed to both deleting and adding random edges), then for certain problems, such as independent set or coloring, the resulting problem retains some of the properties of the original problem. Namely, while the optimum value may change, it is quite concentrated and can be predicted (up to small deviations) from the optimum of the original problem. This robustness property is one of the reasons why solving an optimization problem on a percolated instance is nearly as difficult as solving the original problem.

Semi-random models
Semi-random models are hybrid models that combine random and adversarial decisions. For example, the semi-random model for the independent set problem is defined in [18] as follows. Given a set V of n vertices, choose a set S ⊆ V of size αn uniformly at random (where α ∈ (0, 1)), and add every possible edge between S and V \ S to the graph independently with probability p. Next, an adversary is allowed to add arbitrary edges to G, provided that the added edges do not connect two vertices in S. The main result is that when p > (1 + �)αn/ log n, it is possible to recover S in polynomial time, whereas for p < (1 − �)αn/ log n it is N P-hard (under randomized reductions) to recover S. Further positive results are given to semi-random instances of the min-bisection problem with a small planted bisection. One motivation for studying these models is that algorithms that work well for random graphs generalize in some cases to semi-random instances as well.
Our percolation model differs from semi-random models, as we are interested in finding or approximating the optimal solution rather than recovering the "planted" independent set or a bisection (as observed in [18], the independent set S in the semi-random model may not be optimal after an adversary has changed G). For this reason, the algorithmic task of finding the optimal value in percolated instances seem to be significantly harder than recovering a "planted" solution in semi-random models. Finally, the proof ideas behind our hardness result for the independent set problem differ from those given in [18].

1.4.4
Additional results The effect of subsampling variables in mathematical relaxations of constraint satisfaction problems on the value of these relaxations was studied in [5]. In particular, [5] shows that for a certain kind of LP and SDP formulation of k-CSP, the value of these formulations for a random instance obtained by choosing randomly a subset S of variables of the original instance and considering the LP that only contains variables from S gives (with high probability) an additive approximation of � to the value of the LP/SDP's of the relaxation of the original instance. On the other hand, for other kinds of mathematical relaxations, such as the Lasserre Hierarchy, the value of subsampled mathematical relaxations may drastically differ from the value of the original instance. For example, for dense random k-XOR instances over n-variables, the value of the CSP as well as k-rounds of the Lasserre hierarchy is less than 1/2 (with high probability). On the other hand, for a random sub-instance obtained by choosing a random subset of δn (for an appropriately chosen δ) variables of the original instance, it is known that with high probability the value of the random sub-instance is with high probability at most 0.51, whereas the value of k-rounds of the Lasserre hierarchy is 1 [37]. Edge-percolated graphs have been also used to construct hard instances for specific algorithms. For example, Kučera [31] proved that the well-known greedy coloring algorithm performs poorly on the complete r-partite graph in which every edge is removed independently with probability 1/2 and r = n ε for ε > 0. Namely, for this graph G, even if vertices are considered in a random order by the greedy algorithm, with high probability Ω( n log n ) colors are used to color the percolated graph whereas χ(G) ≤ n ε .
Misra [33] studied edge percolated instances of the Max-Cut problem. He proves that assuming N P � = BPP there is no polynomial time algorithm that computes the size of the maximum cut in G p,e for any p > 1+ε d−1 in graphs of maximal degree d. This result is tight in the sense that it is easy to show that once p < 1−ε d−1 , with high probability G p,e breaks into connected components of logarithmic size, and hence can be solved optimally in polynomial time. The techniques used in [33] differ from ours and rely on a recent hardness result for counting independent sets in sparse graphs [40].
Recently there has been a lot of work generalizing the classical result on random graphs, asking about properties of random subgraphs of fixed graphs satisfying certain properties. For example, there have been several results studying the emergence of a giant component in G p when G is an expander graph [2,21,30], and relating it to the well studied Erdös-Rényi random model [15]. Bukh [9] has considered coloring edge-percolated graphs, and states the question of whether E[χ(G 1 2 ,e )] = Ω(χ(G)/ log(χ(G))) as an "interesting problem." Bukh observed that the chromatic number of G 1 2 ,e has the same distribution as the chromatic number of the complement of G 1 2 ,e , and therefore E[χ(G 1 2 ,e )] ≥ √ χ(G). However, it is not clear how to leverage the lower bound on the expectation to obtain a lower bound on χ(G 1 2 ,e ) with high probability, which is required for our noise-robust reductions. See also [38] for more progress on this problem.

Preliminaries
An independent set in a graph G = (V , E) is a set of vertices that spans no edges. The independence number α(G) denotes the maximum size of an independent set in G. A legal coloring of a graph G is an assignment of colors to vertices such that no two adjacent vertices share the same color. The chromatic number χ(G) denotes the minimum number of colors needed for a legal coloring of G. Note that in a legal coloring of G each color class forms an independent set, and hence χ(G) · α(G) ≥ n.
A vertex cover in a graph G = (V , E) is a set of vertices S ⊆ V such that every edge e ∈ E is incident to at least one vertex in S. Note that a subset of the vertices S ⊆ V is an independent set in G if and only if V \ S is a vertex cover. In particular, G contains a vertex cover of size k if and only if it contains an independent set of size n − k.
We will always measure the running time of algorithms in terms of the size of the percolated instance. Since G and G p,e have the same number of vertices, this generally does not affect the size of the instance by more than a polynomial factor. On the other hand, G p,v may be much smaller than G for very small values of p. However, in this work we will be only dealing with the case where p = 1 n 1−Ω(1) , hence with high probability the size of the vertex percolated and original graphs are polynomially related as well.
We will use the following version of the Chernoff bound (see, eg, [34, Theorems 4.4 and 4.5]).

MAXIMUM INDEPENDENT SET AND PERCOLATION
In this section we demonstrate the hardness of approximating α(G) and χ(G) in both edge percolated and vertex percolated graphs. We base our results on a theorem of Feige and Kilian, saying that for every fixed ε > 0 the problem Coloring-vs-MIS(n ε , n ε ) is N P-hard.

Edge percolation
Below we prove Theorem 1.4. We will use the following lemma, due to Turan (see, eg, [3]).
Lemma 2.2 Every graph H with l vertices and e edges contains an independent set of size at least l 2 2e+l .
As a corollary we observe that if a graph contains no large independent sets, then it also cannot contain large subsets of vertices that span a small number of edges. Proof Let H be a subgraph of G induced by l vertices, and suppose that H spans e edges. Then, by Lemma 2.2 we have α(H) ≥ l 2 2e+l . On the other hand, α(H) ≤ α(G) ≤ k, and hence l 2 2e+l ≤ k, as required.
We are now ready to prove Theorem 1.4 saying that for any n-vertex graph G = (V , E) it holds that with high probability α(G p,e ) ≤ O α(G) p log(np) .
Proof of Theorem 1.4 For a given graph G, let k = α(G) + 1 and let C > 0 be a large enough constant. By Corollary 2.3, every subset of size l = C α(G) p log(np) spans at least 2k edges in G. Hence, by taking union bound over all subsets of size l, the probability there exists a set of size l in G p,e that spans no edge is at most where the last inequality uses the choices of l and k, implying that en l l < (np) l and exp(−p l(l−k) 2k ) < exp(−Ω(l · log(np))) = (np) −Ω(l) . Therefore, with high probability α(G p,e ) ≤ C α(G) p log(np). The "in particular" part follows immediately from Theorem 2.1.

Vertex percolation
Next we show that Coloring-vs-MIS is robust to vertex percolation as claimed in Theorem 1.6. We show that approximating α(G) and χ(G) on vertex percolated instances with p > 1 n 1−δ is essentially as hard as on worst-case instances, where n is the number of vertices in the given graph, and δ ∈ (0, 1) is a arbitrary constant (that does not depend on n). We show this by relying again on the hardness of the gap problem Coloring-vs-MIS for percolated instances.
Recall that in the case of vertex percolation, the (in)approximability guarantee should depend on the number of vertices in the percolated graph G p,v , and not on the number of vertices in the original graph.
Proof of Theorem 1. 6 The strong robustness of Coloring-vs-MIS(q, a) is clear, since for any graph G if G � is a vertex induced subgraph of G, then χ(G � ) ≤ χ(G), and α(G � ) ≤ α(G), which is, in particular, true for G � ∼ G p,v .
Let G be an n-vertex graph, let G p,v be it vertex-percolated subgraph, and let n � be the number of vertices in G p,v . By the Chernoff bound in Lemma 1.12 with high probability we have |n � − pn| < 0.1pn. Therefore, we may assume from now on that n η < 2(n � ) ε .

GRAPH COLORING AND PERCOLATION
We present our results in terms of the maximum coverage problem (see, for example, [44]), which is a variant of the set cover problem, and show later how graph coloring is related to maximum coverage.

Maximum coverage
In the maximum coverage problem we are given a family of sets and a number c. The goal is to find c sets in F such the cardinality of the union of these c sets is as large as possible. We will make use of the representation of a set S in terms of its incidence vector x(S) ∈ {0, 1} n . In this way, we can reformulate the maximum coverage problem as follows. Given A ⊆ F n 2 , find elements y 1 , . . . , y c ∈ A that maximize �∨ c i=1 y i � 1 , the Hamming weight of the bitwise-OR of the vectors.
We will prove two existential results saying that if A is of constant density α > 0, then there exists a good cover using only 2 or 3 vectors.

Proof of Lemma 3.2 using Fourier analysis
We use an inequality from Fourier analysis to give a proof of Lemma 3.2 via the probabilistic method.
Definition 3.5 Given x ∈ F n 2 , define y ∼ N ρ (x) by letting each y i be equal to x i with probability 1+ρ 2 , and be equal to 1 − x i with probability 1−ρ 2 .
Let Uni(S) denote the uniform distribution on a set S, and let U n denote Uni(F n 2 ). The following lemma is a corollary of the reverse Bonami-Beckner inequality. [y ∈ B] ≥ α (1+ρ)/ (1−ρ) .

Coloring using subgraphs
We now show how to apply the results in the previous subsection to the graph coloring problem. Throughout this section we let G = (V , E) with n = |V | , m = |E|. We will identify the elements of [n] with vertices V in the vertex percolation case and the elements of [m] with edges E in the edge percolation case. Let G |U denote the subgraph of G induced by U ⊆ V .
Proof Assume that V 1 ∩V 2 = ∅ (if not, replace V 1 with V 1 \V 2 in the following argument).
Color G |V 1 with k 1 colors and color G |V 2 with k 2 fresh colors. Because G |V 1 and G |V 2 are colored with separate colors any edges between V 1 and V 2 have endpoints with distinct colors.
Proof Let c 1 be a coloring of G 1 with k 1 colors, and let c 2 be a coloring of G 2 with k 2 colors. We claim that the coloring c(v) = (c 1 (v), c 2 (v)) is a legal coloring of G with k 1 k 2 colors. Consider an edge e = (u, v) ∈ E. If e ∈ E 1 then c(u) differs from c(v) in the first coordinate. Otherwise e ∈ E 2 in which case c(u) differs from c(v) in the second coordinate.

Lower bounding the chromatic number
We now prove lower bounds on the chromatic number of percolated graphs. We will consider both vertex and edge percolation with p = 1 2 . This choice of p is important because G 1 2 ,v , G 1 2 ,e become the distributions of graphs induced by uniformly random subsets of V and E, respectively. However, it is easy to obtain bounds for p < 1 2 by composing the bounds for p = 1 2 . The idea will be to argue that if many subgraphs of a graph G are k-colorable, then G is colorable with f (k) colors for relatively small f (k). To see how this idea works, consider the following easy case. Suppose that Pr[χ(G 1 2 ,v ) ≤ k] > 1 2 . Then there exists V � ⊆ V such that G |V � and G |V � are both k-colorable. It follows that G is 2k-colorable by Lemma 3.7. Below we consider the case where the density of k colorable subgraphs is less than 1 2 .

Vertex percolation
We now prove Theorem 1.7, saying that if G is an n-vertex graph with k = χ(G), then for every λ > 0 it holds that Pr Next we define a martingale X 0 , X 1 , . . . , X k as follows. Fix some k-coloring V = V 1 ∪ · · · ∪ V k of G, and let U i ⊆ V i be a random subset chosen by adding each element of V i into U i with probability 0.5. Then G 1 2 ,v = G |U 1 ∪···∪U k , where for U ⊆ V , G |U denotes the subgraph of G induced by the vertices in U.
For each i = 1, . . . , k define the random variable X i by choosing U 1 ⊆ V 1 , . . . , U i ⊆ V i at random, and letting X i be the expected chromatic number of G 1 2 ,v , when the expectation is taken over U i+1 , . . . , U k . That is, we sample a random graph H = (V H , E H ) ∼ G 1 2 ,v by exposing the color class U i and all the edges induced by U 1 ∪ · · · ∪ U i in the ith step, and let Note that this martingale is a direct analog of the vertex exposure martingale typically used when analyzing random graphs, where in each step we use color classes in place of the vertices.

Edge percolation
For a random G 1 2 ,e let G 1 2 ,e be the graph obtained from G by removing all edges in G 1 2 ,e . By observing that G 1 2 ,e and G 1 2 ,e have the same distribution, and using Lemma 3.8 we get that analogous to the bound in (3) for vertex percolation. However, using the martingale as above Azuma's inequality implies Pr[χ(G 1 2 ,e ) ≤ (1 − λ) √ k] < e −λ 2 /2 , which is not enough to prove that χ(G 1 2 ,e ) is large with high probability.
Below we use alternative techniques to prove Theorem 1.8 asserting that a weaker bound on χ(G 1 2 ,e ) holds with probability 1 − α for any α > 0. To the best of our knowledge these techniques are new to this area, and may be of independent interest.
with k 3 colors. We then color the endpoints of the remaining E \ (E 1 ∪ E 2 ∪ E 3 ) edges using 8/α 3 new colors to achieve a (k 3 + 8/α 3 )-coloring of G.
The next lemma gives an unconditional upper bound on the chromatic number of a graph.

Lemma 3.10 Let G = (V , E) be a graph with |E| = m. Then χ(G)
The result follows by Lemma 3.7.
We use a variant of the same partitioning trick in the following lemma.
The YES-case is clear, since removing edges can only decrease the chromatic number. The NOcase follows from Theorem 1.8. Therefore, the Gap-Coloring(q, Ω(Q 1/3 )) problem is a noise-robust relaxation of Gap-Coloring(q, Q). The "in particular" part of the theorem follows from the result of Huang [25], who showed that Gap-Coloring(q, exp(Ω(q 1/3 ))) is N P-hard.

CONSTRAINT SATISFACTION PROBLEMS AND PERCOLATION
In this section we deal with percolation in Constraint Satisfaction Problems (CSP).

Clause percolation
We show that for k-CSP the problem of approximating the optimal value on p-percolated instances is essentially as hard as approximating it on a worst-case instance as long as p > 1 n k−1−δ for any constant δ > 0. Recall that a simple instance of a k-CSP is an instance in which no clause appears more than once. For non simple CSP's, proving hardness results of percolated instances (with respect to clause percolation) is easy, as we can simply duplicate every clause many times.
To prove Theorem 1.10 we start with the following lemma. Proof The first item is clear, as any assignment that satisfies Φ will also satisfy Φ c p . For the second item, let m � be the number of clauses in Φ c p . By the concentration bounds in Lemma 1.12 we have where Ω(·) hides some absolute constant. Fix an assignment σ to the variables of Φ, and let s = val σ (Φ). That is, the number of clauses in Φ satisfied by σ is sm. We claim that Note that this is clearly true for s = 0. Let us assume that s > 0, and let S σ denote the number of clauses in Φ c p satisfied by σ. Since we pick each clause with probability p independently, and recalling that p > Cn ε 2 m and s ≤ 1 we get thus implying (5). Denoting by E the event that |m � − pm| > εpm, by (4) and (5) we get Suppose now that val(Φ) = s. If σ is an optimal assignment to Φ, that is, val σ (Φ) = s, then we immediately have by the argument above that val σ (Φ c p ) > s − ε with high probability. On the other hand, for any assignment σ � it holds that Pr[val σ � (Φ c p ) > s + ε] < e −Ω(C·n) for some sufficiently large C > 0, and by taking union bound over all assignments σ we get where the last inequality holds by the assumption that C = Θ(log(|Σ|)). This completes the proof of the lemma.
Next, we show a polynomial time reduction which, given a Max-CSP-F instance Φ, outputs a Max-CSP-F instance Ψ with n variables and n k−ε clauses such that val(Ψ) = val(Φ). We use similar ideas to those used in [11] who proved that unweighted instances of CSP problems are as hard to approximate as weighted ones.

Lemma 4.2 For any δ ∈ (0, 1) there is a polynomial time reduction which, given a simple unweighted Max-CSP-F instance Φ, outputs a simple Max-CSP-F instance Ψ with n variables and at least n k−δ clauses such that val(Ψ) = val(Φ).
Proof The reduction works as follows. Let R be a parameter to be chosen later. Given an instance Φ of k-CSP with M clauses over the variables x 1 , . . . , x N the reduction creates the following instance Ψ. For each variable x i of Φ, the instance Ψ will have a set of R corresponding variables X i = {x i,j : j ∈ [R]}, where we think of each variable in X i as a copy of x i . For each clause C of Φ we add to Ψ the R k clauses obtained by taking the same constraint over each combination of the variables from the corresponding X i 's. We call this set of R k clauses the cloud corresponding to C. So, Ψ has n = NR variables and m = M · R k clauses. Therefore, if R > N k/δ , then m > n k−δ .
Next we claim that val(Φ) = val(Ψ). Clearly, we have val(Φ) ≤ val(Ψ), as any assignment σ : {x 1 , . . . , x N } ∈ Σ to Φ can be extended to the assignment τ to Ψ by letting In the other direction, let τ be an assignment to the variables of Ψ. 7 For each i ∈ [N] and a ∈ Σ let p a i = |{j∈[R]:τ(x i,j )=a}| R be the fraction of elements of X i that are assigned the value a. Construct an assignment σ to the variables of Φ randomly, by setting σ(x i ) = a with probability p a i independently for each x i . Equivalently we can choose one of the R copies of x i in Ψ uniformly at random and assign to x i the value assigned by τ to the variable chosen. Then for each clause C of Φ, the probability that σ satisfies C is equal to the fraction of the clauses in Ψ in the cloud corresponding to C that are satisfied by τ. Denote by SAT σ (C i ) the number of clauses that are satisfied by σ in the cloud corresponding to C i . Since each clause of Φ corresponds to the same number of clauses in Ψ, it follows that the expected value of Φ under the assignment σ is Hence, there exists an assignment σ to the variables of Φ such that val σ (Φ) ≥ val τ (Ψ), and thus val(Φ) ≥ val(Ψ), as required.
Theorem 1.10 follows immediately from Lemmas 4.1 and 4.2.
We observe that it is unlikely that Lemma 4.2 could also hold for Max-CSP-F instances with-arity k and Ω(n k ) constraints, as, for example, the value of a 3-SAT formula with Ω(n 3 ) clauses, can be (1 − δ)-approximated for every δ ∈ (0, 1) in polynomial time [4].

Variable percolation
Next we show that Max-CSP-F is also hard under variable percolation. We prove below that for p that is no too small, with high probability Max-CSP-F is hard to approximate on percolated instances within the same factor as in the worst-case setting.
The following corollary is the analogue of Corollary 1.11 for variable percolation.

Corollary 4.4
Let F be a collection of Boolean constraints of arity k, and suppose that for some 0 < s < c ≤ 1 the problem Gap-CSP-F(c, s) is N P-hard. Then Gap-CSP-F(c − ε, s + ε) is N P-hard under a robust reduction with respect to vertex percolation with any parameter p > 1 n 1−δ , where n denotes the number of variables in a given formula, and ε, δ > 0 are arbitrary constants.
Proof of Theorem 4.3 The reduction is the same reduction as in the proof of Theorem 1.10. Namely, given a simple unweighted instance Φ with N variables and M clauses the reduction replaces each variable x i of Φ, with a set of R corresponding variables X i = {x i,j : j ∈ [R]}, and replaces each clause of Φ with a cloud of R k corresponding clauses, by taking all possible combinations of the variables from the corresponding X i 's. That is, the output of the reduction Ψ has n = NR variables and m = M · R k clauses. We For each i ∈ [N] let X � i be variables from X i that remain in Ψ v p after variable percolation. By the Chernoff bound in Lemma 1.12, it follows that for p > 1 N 1−δ with high probability . We assume from now on that this is indeed the case. For a constraint C i of Φ let x i 1 , . . . , x i k be the variables appearing in C i . Then, the number of clauses in the cloud corresponding to C i in Ψ v p is equal to k j=1 |X � i j |, and the total number of clauses in Ψ v p is M i=1 k j=1 |X � i j |. By Lemma 4.2 we have val(Ψ) = val(Φ). In particular, if Φ is satisfiable, then so is Ψ, as any assignment that satisfies Ψ also satisfies any subformula of Ψ, which implies that Ψ v p is also satisfiable with probability 1. Suppose now that val(Φ) < 1. We claim that with high probability |val( To prove that val(Ψ v p ) ≥ val(Φ) − ε, let σ be an optimal assignment to Φ. Extend σ to an assignment τ to Ψ v p by letting τ(x i,j ) = σ(x i ) for all 1 ≤ i ≤ R. Note that for each constraint C i of Φ if C i is satisfied by σ, then in Ψ v p all clauses in the corresponding cloud are satisfied, and otherwise no clause in the corresponding cloud is satisfied. Denoting by SAT τ (C i ) the number of clauses that are satisfied by τ in the cloud corresponding to C i we have By the choice of R we get for large enough N Next, we prove that val(Φ) ≥ val(Ψ v p ) − ε. Given an assignment τ to the variables of Ψ v p we decode it into an assignment to Φ using the same decoding as in the proof of Lemma 4.2. Namely, we choose a random assignment σ to the variables of Φ by setting σ(x i ) = a with probability p a i independently between i's, where p a i = i be the set of clauses in C i that belong to Ψ v p . Let SAT τ (C � i ) be the number of clauses that are satisfied by τ in C � i , it follows that the expected value of Φ under the assignment σ is On the other hand we have Now, using the assumption that for all i ∈ [n] it holds that ||X � i | − pR| < pR log n, we get that both (6) and (7) are between A simple computation reveals that the difference between the two quantities is at most O( log N pR ), and hence This completes the proof of Theorem 4.3.

MINIMUM VERTEX COVER AND PERCOLATION
In the Minimum Vertex Cover problem given a graph G, the goal is to find a vertex cover of G of minimum size. There is a simple 2-approximation algorithm for the Minimum Vertex Cover problem [44]. On the hardness side, the problem is N P-hard to approximate within a factor of 1.3606 [13], and assuming the Unique Games Conjecture is known to be N P-hard to approximate within a (2 − ε) factor for any constant ε > 0 [29]. We prove that the same hardness results are percolation robust.

Edge percolation
We have the following simple lemma regarding independent sets in edge percolated subgraph of K R,R .

Lemma 5.1 Consider the complete bipartite graph G = K R,R with bipartition A, B.
Then, the probability that there is an independent set I in G p,e such that |I ∩ A| = |I ∩ B| = C log(R)/p is at most R −3 , where C is a large enough constant independent of n or p.
Proof For fixed sets S A ⊆ A and S B ⊆ B each of size C log(R)/p the probability that S A and S B span no edge is (1 − p) (C log(R)/p) 2 . Therefore, by union bound over all S A and S B the probability that there is is an independent set I in G p,e with |I ∩ A| = |I ∩ B| = C log(R)/p is at most Consider the following Gap-Vertex-Cover(c, s) problem where the YES-instances are graphs that have a vertex cover of size cn, and NO-instances are all graphs whose minimum vertex cover is larger than sn, where n is the number of vertices in G. Note that, equivalently, the YES-instances are graphs that contain an independent set of size α(G) ≥ (1 − c)n, the NO-instances are graphs whose maximal independent set is of size α(G) ≤ (1 − s)n.
We remark that the result of Khot and Regev [29] proves that assuming the Unique Games Conjecture the problem Gap-Vertex-Cover( 1 2 + ε, 1 − ε) if N P-hard for all constant ε > 0. We use this to show hardness of approximation for this problem on edge-percolated instances.
Theorem 5.2 Let ε, δ ∈ (0, 1) be fixed constants. Assuming the Unique Games Conjecture, Gap-Vertex-Cover( 1 2 + ε, 1 − ε) is N P-hard under a noise-robust reduction, where noise is edge percolation with parameter p for any p > 1 n 1−δ and n denotes the number of vertices in the given graph.
In particular, assuming the Unique Games Conjecture (2 − ε)-approximation of the Vertex Cover problem is hard on edge percolated instances.
Proof By [29] assuming the Unique Games Conjecture, for any ε > 0 the problem Gap-Vertex-Cover( 1 2 + ε, 1 − ε) is N P-hard. Equivalently, given an N-vertex graph G it is N P-hard to distinguish between the case that α(G) > (1/2 − ε)N and the case that α(G) < εN. We show a reduction from this problem to itself (with slightly larger parameter ε) that is robust for edge percolation.
Consider the reduction that given a graph G outputs the R-blowup of G, which we denote by H, with R to be chosen later. That is the graph H is a graph on n = NR vertices, and it is clear that α(H) = α(G) · R. Therefore, this is indeed a reduction from the Gap-Vertex-Cover( 1 2 + ε, 1 − ε) to itself. We show below that in fact the reduction is robust for edge percolation. In order to do it we prove that with high probability where H ∼ H p,e denotes the edge percolation of H with parameter p. Indeed, the left inequality is clear because For the right inequality, by Lemma 5.1 with probability at least (1 − N 2 /R 3 ) the following holds: for every edge (u, v) of G the corresponding clouds u and v in H are such that there is no independent set I in H, such that |I ∩ u| ≥ C log(R)/p and |I ∩ v| ≥ C log(R)/p. Therefore, if I is an independent set that intersects some clouds on more than C log(R)/p, then the vertices corresponding to these clouds must form an independent set in G. Thus, with probability at least Next we choose the parameter R such that the reduction above is indeed a robust reduction for edge percolation with parameter p. For the parameter p let c = log(pn) log(n) so that p = 1 n 1−c , and let R = N 2/c (where N is the number of vertices in the original graph). Now, if α(G) > (1/2 − ε)N, then by (8) we have α( H) ≥ α(G) · R > (1/2 − ε)NR = (1/2 − ε)n, and hence H contains a vertex cover of size (1/2 + ε)n On the other hand, we claim that if α(G) < εN, then with high probability α( H) < 2εn. Indeed, by the choice of R we have p = 1 . Therefore, by the right inequality of (8)

Vertex percolation
We now proceed with vertex percolation. Note that when considering vertex percolation, the percolation parameter p depends on the number of vertices in the given (worst-case instance) graph, while the performance of the algorithm is measured with respect to the number of vertices in the percolated graph, which is close to pn with high probability. We will need the following concentration bound, which is an immediate corollary of the Chernoff bound in Lemma 1.12.
Proof By Lemma 1.12 for each j ∈ [m] it holds that Pr | n i=1 X where C > 0 is some absolute constant. By taking union bound over we get Pr ∃j ∈ [m] : We can now deal with vertex percolation and vertex-cover.
Theorem 5.4 Let ε, δ ∈ (0, 1) be fixed constants. Assuming the Unique Games Conjecture, Gap-Vertex-Cover(1 − ε, 1/2 + ε) is N P-hard under a robust reduction with respect to vertex percolation with parameter p, for any p > 1 n 1−δ , where n is the number of vertices in the starting graph.
In particular, assuming the Unique Games Conjecture (2 − ε)-approximation of the Vertex Cover problem is hard on vertex percolated instances.
Proof The reduction is the same as in the proof of Theorem 5.2. For parameters N, p and ε let n = NR and R = ( N ε 2 ) 1/c , where c = log(pn) log(n) is such that p = 1 n 1−c . Given an N-vertex graph G the reduction produces the R-blowup of G, which we denote by H. Then H is a graph on n = NR vertices.
Let H = H p,v denote the vertex percolation of H with parameter p. By Corollary 5.3, with high probability the number of vertices in H ∼ H p,v , which we denote by m is between pNR − C pNR log(NR) and pNR + C pNR log(NR), and the number of vertices in every cloud of H is between pR−C pR log N and pR+C pR log N, for some absolute constant C > 0 independent of N or p.
Clearly any independent set I in H gives rise to an independent set in G by taking all vertices v of G such that I intersects the corresponding cloud v. This implies that with high probability it holds (for N large enough) that By the choice of R we have R > C 2 lg(N) ε 2 p , and hence |α( H) − α(G)pR| ≤ ε · α(G)pR.

HAMILTONICITY AND PERCOLATION
Recall that an Hamiltonian cycle in a graph is a cycle that visits every vertex exactly once. Deciding if a graph (whether directed or undirected) contains a Hamiltonian cycle is a classical N P-hard problem, which we denote by HamCycle. A Hamiltonian path, is a simple path that traverses all vertices in the graph.
In this section we prove that unless N P = coRP, there is no polynomial time algorithm that given an n-vertex graph G decides with high probability whether G p,e contains a Hamiltonian cycle for any p > 1 n 1−� where � ∈ (0, 1). A natural approach in proving that deciding the Hamiltonicity of percolated instances is hard, is to "blow up" edges. Namely to replace each edge (u, v) by a clique of size k and connect both endpoints of the edges to all vertices of the clique. The idea is that when k is large enough, there is a Hamiltonian path with high probability between all pairs of distinct vertices of the clique even after percolation. Hence with high probability, we can connect u and v after percolating the edges, by a path that traverses all the vertices of the percolated clique.
The problem with this idea, is that the resulting graph after this blowup operation may not be Hamiltonian (even if the starting graph is) as there is a new set of vertices for every edge in the original graph that needs to be traversed by an Hamiltonian cycle. For directed graphs, we overcome this problem by adding to each vertex v a large clique C, adding a directed edge (v, c) for every c ∈ C and adding a directed edge (c, u) for every c ∈ C and u ∈ N(v) (where N(v) is the set of all vertices having a directed edge from v). Combining this reduction with the standard N P-hardness reduction from directed to undirected Hamiltonian cycle (given in Karp's original paper [27]) yields a similar result for undirected graphs. We omit the details. Theorem 6.1 Let ε ∈ (0, 1) be a fixed constant. The HamCycle is N P-hard under a noise-robust reduction, where noise is the edge percolation with probability p > 1 n 1−ε .
We will need the following claim. Proof Let p 0 ∈ (0, 1), and consider the random graph H p 0 ,e Note that for R > 10 p 0 with probability at least q : are both edges of H p 0 ,e . Conditioning on these specific v 1 , v R ∈ U, we show that with high probability there is a Hamiltonian path from v 1 to v R in the subgraph of H p 0 ,e induced by U.
By a result of [20,Theorem 1.3] if D is a p 0 -edge percolation of the complete directed graph with R vertices with p 0 = p 3 log(R) = 10 log 4 (R) R , then with high probability every edge of D is contained in some Hamiltonian cycle in D. Note that the probability that H p 0 ,e contains a Hamiltonian path from v 1 to v R is equal to the probability that H p 0 ,e contains a Hamiltonian cycle that goes through the edge (v 1 → v R ), conditioned on the event that (v 1 → v R ) is an edge in H p 0 ,e . Therefore, since the distribution of the subgraph of H p 0 ,e induced by U is distributed like D, it follows that with high probability the subgraph H p 0 ,e induced by U contains a Hamiltonian path from v 1 to v R , and hence H p 0 ,e contains a Hamiltonian path from s to t with probability at least 1 2 . Next, let � = 3 log(R) so that p = � · p 0 . We claim that the graph H p,e contains an Hamiltonian path from s to t with probability at least 1− 1 independent copies of H p 0 ,e , then the probability that none of the H � i contains a Hamiltonian path from s to t is at most (1/2) � < 1 R 3 . Therefore, since 1 − (1 − p 0 ) � ≤ p it follows that H dominates ∪ � i=1 H i , and hence H p,e contains an Hamiltonian path from s to t with probability at least 1 − 1 R 3 , as required.
Proof of Theorem 6.1 In order to prove the theorem, we show a reduction that given a directed graph G = (V , E) produces a directed graph G � = (V � , E � ) such that • If G contains a Hamiltonian cycle, then G � contains a Hamiltonian cycle, and with high probability G � p,e contains a Hamiltonian cycle. • If G does not contain a Hamiltonian cycle, then neither G � nor G � p,e contains a Hamiltonian cycle.
The reduction works as follows. Let V = [N] be the vertices of G, and let R be a parameter to be chosen later. The vertices of G � will be That is, we turn the graph G into G � by adding a clique U i for each vertex v i ∈ V , and letting all edges outgoing from v i go through this clique. This completes the description of the reduction. Let us first show that that G contains a Hamiltonian cycle if and only if G � contains a Hamiltonian cycle. Indeed, suppose that C = (σ 1 , . . . , σ N ) is a Hamiltonian cycle in G. Then C � = (σ 1 , u is a Hamiltonian cycle in G � . In the other direction, suppose that G � contains a Hamiltonian cycle C � . It is easy to see that any i ∈ V appearing in C � must be followed immediately by a permutation of all R vertices in U i . Therefore, by restricting C � to the vertices in V we get a Hamiltonian cycle in G. Next we show that the reduction above is robust to edge percolation. LetG � = G � p,e be the edge percolation of G � . Clearly if G � does not contain a Hamiltonian cycle, then neither doesG � . Therefore, it is only left to show that if G � contain a Hamiltonian cycle C, then with high probabilityG � also contains a Hamiltonian cycle. As explained above a Hamiltonian cycle in G � is given by a permutation σ = (σ 1 , . . . σ N ) ∈ S N and some ordering of the vertices in each U i , that is, Finally, we specify the choice of the parameter R. The obtained graph H has n = NR vertices, and the constraints we have are p > log 4 R R and R 3 � N. Therefore, in order to prove the theorem for p > 1 n 1−ε with ε ∈ (0, 1) it is enough to take R = N 1/c , where c = log(pn) log(n) > ε such that p = 1 n 1−c .

SUBSET-SUM AND PERCOLATION
In this section we consider the Subset-Sum problem. In the Subset-Sum problem we are given a set items {a i } n i=1 which are positive integers, and a target integer S. The goal is to decide whether there is a subset of a i 's whose sum is S.
Given an instance I = ({a i } n i=1 ; S) of the Subset-Sum problem, we define p-percolation on I with probability p to be a random instance I p , where each item a i is included in I p with probability p independently, with the target of I p being the same as the target of I.
It is known the Subset-Sum problem is N P-hard. Below we prove hardness of the Subset-Sum problem with respect to the above percolation.
Theorem 7.1 The Subset-Sum problem is N P-hard under a noise-robust reduction, where noise is p-percolation with p > 1 n 1/2−ε , where n is the number of items in a given instance, and ε > 0 is any fixed constant.
Proof In order to prove the theorem, we show a reduction that given an instance I = ({a i } N i=1 ; S) of the Subset-Sum problem with all a i > 0, produces an instance I � on n variables such that the following two properties are satisfied.
Let us assume that the number of items in I is even. (If N is odd, then, add an item to I that is equal to zero). Let R be a parameter to be chosen later, let M = 2RN, L 0 = 6RM · where S � = S · M + N i=1 L i . Clearly, if R is not too large, this is a polynomial time reduction that outputs a Subset-Sum instance with n = 2N(2R + 1) items.
We show first that I ∈ Subset-Sum if and only if I � ∈ Subset-Sum. Indeed, suppose that for some subset T ⊆ [N] it holds that i∈T a i = S. Consider the following subset of items of I � . For each i ∈ T take the item from J i that corresponds to k = 0, and for i ∈ [N] \ T take the item from J � i that corresponds to k = 0. Summing these items we get In the other direction, suppose that I � ∈ Subset-Sum. Then, there is some subset Claim 7.2 For each i ∈ [N] there is a unique t i ∈ {0, 1} and a unique k i ∈ {−R, . . . , R} such that (i, t i , k i ) ∈ T � . Furthermore, N i=1 k i = 0 and N i=1 a i · t i = S.
Proof Note first that for each i ∈ [N] there are at most 2 · (2R + 1) < 6R terms (i, t, k) in T � . Therefore, (i,t,k)∈T � (a i · M · t + k) < 6R · N i=1 (a i M + R) < L 0 . Considering the sum in the LHS of (9) modulo L 0 (and recalling that L i = L 0 · (6R) i ) we conclude that and By the choice of L i , and using again the fact that for each i ∈ [N] there are at most 2 · (2R + 1) < 6R terms (i, t, k) in T � , it is now easy to see that each L i term in the LHS of (10) must appear exactly once, that is, for each i ∈ [N] there is a unique t i ∈ {0, 1} and a unique k i ∈ {−R, . . . , R} such that (i, t i , k i ) ∈ T � . To see that N i=1 k i = 0, note that (i,t i ,k i )∈T � k i < RN < M. Now, since in (9) we have L i ≡ 0 (mod M) and S � ≡ 0 (mod M) it follows that N i=1 k i = 0. This concludes the proof. Therefore, by defining T = {i ∈ [N] : t i = 1} we get that i∈T a i = S, and so I ∈ Subset-Sum.
Next, we claim that the reduction above is in fact robust to noise. Indeed, consider the percolated instance I � p for some p ∈ (0, 1). Note that if I / ∈ Subset-Sum, then I � / ∈ Subset-Sum, and hence I � p / ∈ Subset-Sum with probability 1. Therefore, it remains to show that if I ∈ Subset-Sum, then with high probability I � p ∈ Subset-Sum. The proof relies on the following claim. Claim 7.3 Let N ∈ N be even, and let R ∈ N. Let A 1 , . . . , A n ⊆ {−R, . . . , R} be random sets chosen by letting each k ∈ {−R, . . . , R} be in A i with probability p independently. Then, with probability ≥ 1 − N/2 · (1 − p 2 ) 2R for each i ∈ [n] there is k i ∈ A i such that Proof Note that for each odd i ∈ [N], the probability for a fixed element k ∈ {−R, . . . , R} that both k ∈ A i and −k ∈ A i+1 hold is p 2 . Therefore, Pr[� ∃k ∈ {−R, . . . , R} : k ∈ A i and − k ∈ A i+1 ] = (1 − p 2 ) 2R+1 .
Hence, by taking the union bound over all pairs (i, i + 1) with odd values of i we get that with probability at least 1 − N/2 · (1 − p 2 ) 2R+1 , for all odd i's there is k i ∈ A i such that −k i ∈ A i+1 . Suppose now that I ∈ Subset-Sum, that is, for some subset T ⊆ [N] it holds that i∈T a i = S. Note that the percolated instance I � p is obtained from I � by taking random subsets of J i and J � i independently of each other. For i ∈ [N] define A i to be the ppercolated subsets of J i if i ∈ T , and define A i to be the p-percolated subsets of Note that if R > C log(N) , then the conclusion of Claim 7.3 holds with probability at least 1 − 1/N. hence, in the percolated instance I � p by taking the items from A i 's that correspond to k i ∈ A i 's from Claim 7.3 we get i∈T (L i + a i · M + k i ) + i∈[N]\T (L i + k i ) = i∈ [N] L i + i∈T a i · M + i∈ [N] k i = i∈ [N] L i + S · L + 0 = S � . Therefore, with high probability I � p ∈ Subset-Sum as required. Finally, note that the reduction works as long as R > C log(N) p 2 , or equivalently p > Ω log N R . It is easy to verify that if we set R = N 1/c , with c = log(pn) log(n) − 1 2 such that p = 1 n 1/2−c , then the foregoing reduction is indeed a robust reduction with respect to percolation with parameter p > 1 n 1/2−ε for any constant ε > 0, where n is the number of items in I � .

CONCLUSION
We have examined the complexity of percolated instances of several well studied N P-hard problems and established the hardness of solving these problems on such instances. It might be of interest to study the hardness of percolated instances of other N P-hard problems, and of other classes of problems such as counting, fixed parameter tractability (eg, W [1]-hard problems), and parallel computation.
Below we mention three specific problems that we find interesting.
1. Is there a polynomial time algorithm that solves 3-SAT on clause-percolated instances over n-variable formulas for p = O(1/n 2 )?
2. Is the HamCycle problem N P-hard under a robust reduction with respect to vertex percolation. In this paper we showed that the problem is hard with respect to edge percolation, and it could be the case that the problem is not N P-hard under a vertex percolation robust reduction. Proving such a result (if true) could be very interesting.
3. Is there a polynomial time algorithm that solves the Maximum Independent Set (MIS) problem exactly on percolated instances of planar graphs, or no such algorithm exists (assuming N P � = BPP, or some other reasonable hardness assumption)? Recall, that the maximum independent set in planar graphs can be approximated within a multiplicative factor of 1 − δ for arbitrary δ ∈ (0, 1). We note that our proof methods for showing hardness on percolated instances break down when considering planar graphs.

ACKNOWLEDGMENTS
We thank Itai Benjamini, Uri Feige, and Sam Hopkins for useful discussions. We are grateful to the anonymous referees for helpful comments and turning our attention to [9].