Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming.Making this connection precise, we show the following result : If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains.Using this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP.Unconditionally, for all 2-CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.

The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a d-regular graph, the edge expansion/conductance Φ(S) of a subset S ⊆ V is defined as Φ(S) = |E(S,V \S)| d|S| . Approximating the conductance of small linear sized sets (size δn) is a natural optimization question that is a variant of the well-studied Sparsest Cut problem. However, there are no known algorithms to even distinguish between almost complete edge expansion (Φ(S) = 1 − ε), and close to 0 expansion.In this work, we investigate the connection between Graph Expansion and the Unique Games Conjecture. Specifically, we show the following:-We show that a simple decision version of the problem of approximating small set expansion reduces to Unique Games. Thus if approximating edge expansion of small sets is hard, then Unique Games is hard. Alternatively, a refutation of the UGC will yield better algorithms to approximate edge expansion in graphs.This is the first non-trivial "reverse" reduction from a natural optimization problem to Unique Games.-Under a slightly stronger UGC that assumes mild expansion of small sets, we show that it is UG-hard to approximate small set expansion.-On instances with sufficiently good expansion of small sets, we show that Unique Games is easy by extending the techniques of [4].

We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's).More concretely, we show for every 2-CSP instance ℑ a rounding algorithm for r rounds of the Lasserre SDP hierarchy for ℑ that obtains an integral solution that is at most ε worse than the relaxation's value (normalized to lie in [0, 1]), as long as *

This work is concerned with approximating constraint satisfaction problems (CSPs) with an additional global cardinality constraints. For example, Max Cut is a boolean CSP where the input is a graph G = (V, E) and the goal is to find a cut S ∪S = V that maximizes the number of crossing edges, |E(S ,S )|. The Max Bisection problem is a variant of Max Cut with an additional global constraint that each side of the cut has exactly half the vertices, i.e., |S | = |V|/2. Several other natural optimization problems like Min Bisection and approximating Graph Expansion can be formulated as CSPs with global constraints.In this work, we formulate a general approach towards approximating CSPs with global constraints using SDP hierarchies. To demonstrate the approach we present the following results: -Using the Lasserre hierarchy, we present an algorithm that runs in time O(n poly(1/ε) ) that given an instance of Max Bisection with value 1 − ε, finds a bisection with value 1 − O( √ ε). This approximation is near-optimal (up to constant factors in O()) under the Unique Games Conjecture.-By a computer-assisted proof, we show that the same algorithm also achieves a 0.85-approximation for Max Bisection, improving on the previous bound of 0.70 (note that it is Unique Games hard to approximate better than a 0.878 factor). The same algorithm also yields a 0.92-approximation for Max 2-Sat with cardinality constraints.-For every CSP with a global cardinality constraints, we present a generic conversion from integrality gap instances for the Lasserre hierarchy to a dictatorship test whose soundness is at most integrality gap. Dictatorship testing gadgets are central to hardness results for CSPs, and a generic conversion of the above nature lies at the core of the tight Unique Games based hardness result for CSPs.[Rag08]

We introduce a method for proving lower bounds on the ecacy of semide nite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2 n δ , for some constant δ > 0. This result yields the rst super-polynomial lower bounds on the semide nite extension complexity of any explicit family of polytopes.Our results follow from a general technique for proving lower bounds on the positive semide nite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sumof-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for 3-.

The Small-Set Expansion Hypothesis (Raghavendra, Steurer, STOC 2010) is a natural hardness assumption concerning the problem of approximating the edge expansion of small sets in graphs. This hardness assumption is closely connected to the Unique Games Conjecture (Khot, STOC 2002). In particular, the Small-Set Expansion Hypothesis implies the Unique Games Conjecture (Raghavendra, Steurer, STOC 2010).Our main result is that the Small-Set Expansion Hypothesis is in fact equivalent to a variant of the Unique Games Conjecture. More precisely, the hypothesis is equivalent to the Unique Games Conjecture restricted to instance with a fairly mild condition on the expansion of small sets. Alongside, we obtain the first strong hardness of approximation results for the Balanced Separator and Minimum Linear Arrangement problems. Before, no such hardness was known for these problems even assuming the Unique Games Conjecture.These results not only establish the Small-Set Expansion Hypothesis as a natural unifying hypothesis that implies the Unique Games Conjecture, all its consequences and, in addition, hardness results for other problems like Balanced Separator and Minimum Linear Arrangement, but our results also show that the Small-Set Expansion Hypothesis problem lies at the combinatorial heart of the Unique Games Conjecture.The key technical ingredient is a new way of exploiting the structure of the Unique Games instances obtained from the Small-Set Expansion Hypothesis via (Raghavendra, Steurer, 2010). This additional structure allows us to modify standard reductions in a way that essentially destroys their local-gadget nature. Using this modification, we can argue about the expansion in the graphs produced by the reduction without relying on expansion properties of the underlying Unique Games instance (which would be impossible for a local-gadget reduction).

In recent years the framework of learning from label proportions (LLP) has been gaining importance in machine learning. In this setting, the training examples are aggregated into subsets or bags and only the average label per bag is available for learning an example-level predictor. This generalizes traditional PAC learning which is the special case of unit-sized bags. The computational learning aspects of LLP were studied in recent works [21,22] which showed algorithms and hardness for learning halfspaces in the LLP setting. In this work we focus on the intractability of LLP learning Boolean functions. Our first result shows that given a collection of bags of size at most 2 which are consistent with an OR function, it is NP-hard to find a CNF of constantly many clauses which satisfies any constant-fraction of the bags. This is in contrast with the work of [21] which gave a (2/5)-approximation for learning ORs using a halfspace. Thus, our result provides a separation between constant clause CNFs and halfspaces as hypotheses for LLP learning ORs.Next, we prove the hardness of satisfying more than 1/2 + o(1) fraction of such bags using a t-DNF (i.e. DNF where each term has ≤ t literals) for any constant t. In usual PAC learning such a hardness was known [15] only for learning noisy ORs. We also study the learnability of parities and show that it is NP-hard to satisfy more than (q/2 q−1 + o(1))-fraction of q-sized bags which are consistent with a parity using a parity, while a random parity based algorithm achieves a (1/2 q−2 )-approximation.

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