Improved Hardness for Cut, Interdiction, and Firefighter Problems

We study variants of the classic s-t cut problem and prove the following improved hardness results assuming the Unique Games Conjecture (UGC). * For Length-Bounded Cut and Shortest Path Interdiction, we show that both problems are hard to approximate within any constant factor, even if we allow bicriteria approximation. If we want to cut vertices or the graph is directed, our hardness ratio for Length-Bounded Cut matches the best approximation ratio up to a constant. Previously, the best hardness ratio was 1.1377 for Length-Bounded Cut and 2 for Shortest Path Interdiction. * For any constant k >= 2 and epsilon > 0, we show that Directed Multicut with k source-sink pairs is hard to approximate within a factor k - epsilon. This matches the trivial k-approximation algorithm. By a simple reduction, our result for k = 2 implies that Directed Multiway Cut with two terminals (also known as s-t Bicut} is hard to approximate within a factor 2 - epsilon, matching the trivial 2-approximation algorithm. * Assuming a variant of the UGC (implied by another variant of Bansal and Khot), we prove that it is hard to approximate Resource Minimization Fire Containment within any constant factor. Previously, the best hardness ratio was 2. For directed layered graphs with b layers, our hardness ratio Omega(log b) matches the best approximation algorithm. Our results are based on a general method of converting an integrality gap instance to a length-control dictatorship test for variants of the s-t cut problem, which may be useful for other problems

Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere

For an $n$-variate order-$d$ tensor $A$, define $A_{\max} := \sup_{\| x \|_2 = 1} \langle A , x^{\otimes d} \rangle$ to be the maximum value taken by the tensor on the unit sphere. It is known that for a random tensor with i.i.d $\pm 1$ entries, $A_{\max} \lesssim \sqrt{n\cdot d\cdot\log d}$ w.h.p. We study the problem of efficiently certifying upper bounds on $A_{\max}$ via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include: - When $A$ is a random order-$q$ tensor, we prove that $q$ levels of SoS certifies an upper bound $B$ on $A_{\max}$ that satisfies $$B ~~~~\leq~~ A_{\max} \cdot \biggl(\frac{n}{q^{\,1-o(1)}}\biggr)^{q/4-1/2} \quad \text{w.h.p.}$$ Our upper bound improves a result of Montanari and Richard (NIPS 2014) when $q$ is large. - We show the above bound is the best possible up to lower order terms, namely the optimum of the level-$q$ SoS relaxation is at least $$A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ .$$ - When $A$ is a random order-$d$ tensor, we prove that $q$ levels of SoS certifies an upper bound $B$ on $A_{\max}$ that satisfies $$B ~~\leq ~~ A_{\max} \cdot \biggl(\frac{\widetilde{O}(n)}{q}\biggr)^{d/4 - 1/2} \quad \text{w.h.p.}$$ For growing $q$, this improves upon the bound certified by constant levels of SoS. This answers in part, a question posed by Hopkins, Shi, and Steurer (COLT 2015), who established the tight characterization for constant levels of SoS

Towards a Characterization of Approximation Resistance for Symmetric CSPs

A Boolean constraint satisfaction problem (CSP) is called approximation resistant if independently setting variables to 1 with some probability achieves the best possible approximation ratio for the fraction of constraints satisfied. We study approximation resistance of a natural subclass of CSPs that we call Symmetric Constraint Satisfaction Problems (SCSPs), where satisfaction of each constraint only depends on the number of true literals in its scope. Thus a SCSP of arity k can be described by a subset of allowed number of true literals. For SCSPs without negation, we conjecture that a simple sufficient condition to be approximation resistant by Austrin and Hastad is indeed necessary. We show that this condition has a compact analytic representation in the case of symmetric CSPs (depending only on the gap between the largest and smallest numbers in S), and provide the rationale behind our conjecture. We prove two interesting special cases of the conjecture, (i) when S is an interval and (ii) when S is even. For SCSPs with negation, we prove that the analogous sufficient condition by Austrin and Mossel is necessary for the same two cases, though we do not pose an analogous conjecture in general

Understanding the Correlation Gap For Matchings

Given a set of vertices V with |V| = n, a weight vector w in (R^+ cup {0})^{binom{V}{2}}, and a probability vector x In [0, 1]^{binom{V}{2}} in the matching polytope, we study the quantity (E_{G}[ nu_w(G)])/(sum_(u, v) in binom{V}{2} w_{u, v} x_{u, v}) where G is a random graph where each edge e with weight w_e appears with probability x_e independently, and let nu_w(G) denotes the weight of the maximum matching of G. This quantity is closely related to correlation gap and contention resolution schemes, which are important tools in the design of approximation algorithms, algorithmic game theory, and stochastic optimization. We provide lower bounds for the above quantity for general and bipartite graphs, and for weighted and unweighted settings. The best known upper bound is 0.54 by Karp and Sipser, and the best lower bound is 0.4. We show that it is at least 0.47 for unweighted bipartite graphs, at least 0.45 for weighted bipartite graphs, and at least 0.43 for weighted general graphs. To achieve our results, we construct local distribution schemes on the dual which may be of independent interest

Clustering affine subspaces: hardness and algorithms

We study a generalization of the famous k-center problem where each object is an affine subspace of dimension Δ, and give either the first or significantly improved algorithms and hardness results for many combinations of parameters. This generalization from points (Δ = 0) is motivated by the analysis of incomplete data, a pervasive challenge in statistics: incomplete data objects in ℝd can be modeled as affine subspaces. We give three algorithmic results for different values of k, under the assumption that all subspaces are axis-parallel, the main case of interest because of the correspondence to missing entries in data tables. 1) k = 1: Two polynomial time approximation schemes which runs in poly (Δ, 1/∊)nd. 2) k = 2: O(Δ1/4)-approximation algorithm which runs in poly(n, d, Δ) 3) General k: Polynomial time approximation scheme which runs in We also prove nearly matching hardness results; in both the general (not necessarily axis-parallel) case (for k ≥ 2) and in the axis-parallel case (for k ≥ 3), the running time of an approximation algorithm with any approximation ratio cannot be polynomial in even one of k and Δ, unless P = NP. Furthermore, assuming that the 3-SAT problem cannot be solved sub-exponentially, the dependence on both k and Δ must be exponential in the general case (in the axis-parallel case, only the dependence on k drops to . The simplicity of the first and the third algorithm suggests that they might be actually used in statistical applications. The second algorithm, which demonstrates a theoretical gap between the axis-parallel and general case for k = 2, displays a strong connection between geometric clustering and classical coloring problems on graphs and hypergraphs, via a new Helly-type theorem

Optimal Online Contention Resolution Schemes via Ex-Ante Prophet Inequalities

Online contention resolution schemes (OCRSs) were proposed by Feldman, Svensson, and Zenklusen [Moran Feldman et al., 2016] as a generic technique to round a fractional solution in the matroid polytope in an online fashion. It has found applications in several stochastic combinatorial problems where there is a commitment constraint: on seeing the value of a stochastic element, the algorithm has to immediately and irrevocably decide whether to select it while always maintaining an independent set in the matroid. Although OCRSs immediately lead to prophet inequalities, these prophet inequalities are not optimal. Can we instead use prophet inequalities to design optimal OCRSs? We design the first optimal 1/2-OCRS for matroids by reducing the problem to designing a matroid prophet inequality where we compare to the stronger benchmark of an ex-ante relaxation. We also introduce and design optimal (1-1/e)-random order CRSs for matroids, which are similar to OCRSs but the arrival order is chosen uniformly at random

LP/SDP Hierarchy Lower Bounds for Decoding Random LDPC Codes

Random (dv,dc)-regular LDPC codes are well-known to achieve the Shannon capacity of the binary symmetric channel (for sufficiently large dv and dc) under exponential time decoding. However, polynomial time algorithms are only known to correct a much smaller fraction of errors. One of the most powerful polynomial-time algorithms with a formal analysis is the LP decoding algorithm of Feldman et al. which is known to correct an Omega(1/dc) fraction of errors. In this work, we show that fairly powerful extensions of LP decoding, based on the Sherali-Adams and Lasserre hierarchies, fail to correct much more errors than the basic LP-decoder. In particular, we show that: 1) For any values of dv and dc, a linear number of rounds of the Sherali-Adams LP hierarchy cannot correct more than an O(1/dc) fraction of errors on a random (dv,dc)-regular LDPC code. 2) For any value of dv and infinitely many values of dc, a linear number of rounds of the Lasserre SDP hierarchy cannot correct more than an O(1/dc) fraction of errors on a random (dv,dc)-regular LDPC code. Our proofs use a new stretching and collapsing technique that allows us to leverage recent progress in the study of the limitations of LP/SDP hierarchies for Maximum Constraint Satisfaction Problems (Max-CSPs). The problem then reduces to the construction of special balanced pairwise independent distributions for Sherali-Adams and special cosets of balanced pairwise independent subgroups for Lasserre. Some of our techniques are more generally applicable to a large class of Boolean CSPs called Min-Ones. In particular, for k-Hypergraph Vertex Cover, we obtain an improved integrality gap of $k-1-\epsilon$ that holds after a \emph{linear} number of rounds of the Lasserre hierarchy, for any k = q+1 with q an arbitrary prime power. The best previous gap for a linear number of rounds was equal to $2-\epsilon$ and due to Schoenebeck.Comment: 23 page