Tight Hardness Research Articles

Pushing the Online Boolean Matrix-vector Multiplication conjecture off-line and identifying its easy cases

Henzinger et al. posed the so-called Online Boolean Matrix-vector Multiplication (OMv) conjecture and showed that it implies tight hardness results for several basic dynamic or partially dynamic problems [STOC'15]. We first show that the OMv conjecture is implied by a simple off-line conjecture that we call the MvP conjecture. We then show that if the definition of the OMv conjecture is generalized to allow individual (i.e., it might be different for different matrices) polynomial-time preprocessing of the input matrix, then we obtain another conjecture (called the OMvP conjecture) that is in fact equivalent to our MvP conjecture. On the other hand, we demonstrate that the OMv conjecture does not hold in restricted cases where the rows of the matrix or the input vectors are clustered, and develop new efficient randomized algorithms for such cases. Finally, we present applications of our algorithms to answering graph queries.

Tight Hardness Results for Consensus Problems on Circular Strings and Time Series

Consensus problems for strings and sequences appear in numerous application contexts, ranging among bioinformatics, data mining, and machine learning. Closing some gaps in the literature, we show that several fundamental problems in this context are NP-and W[1]-hard, and that the known (including some brute-force) algorithms are close to optimality assuming the Exponential Time Hypothesis. Among our main contributions is to settle the complexity status of computing a mean in dynamic time warping spaces which, as pointed out by Brill et al. [DMKD 2019], suffered from many unproven or false assumptions in the literature. We prove this problem to be NP-hard and additionally show that a recent dynamic programming algorithm is essentially optimal. In this context, we study a broad family of circular string alignment problems. This family also serves as a key for our hardness reductions, and it is of independent (practical) interest in molecular biology. In particular, we show tight hardness and running time lower bounds for Circular Consensus String; notably, the corresponding non-circular version is easily linear-time solvable.

Tight Hardness of the Non-Commutative Grothendieck Problem

$\newcommand{\eps}{\varepsilon} $We prove that for any $\eps > 0$ it is $\textsf{NP}$-hard to approximate the non-commutative Grothendieck problem to within a factor $1/2 + \eps$, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of $\ell_2$ into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates. We also observe that one can obtain a tight $\textsf{NP}$-hardness result for the commutative Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture (Khot and Naor, Mathematika 2009).

Minimizing the number of switch instances on a flexible machine in polynomial time

We revisit the tool switching problem on a flexible manufacturing machine. We present a polynomial algorithm for the problem of finding a switching plan that minimizes the number of tool switch instances on the machine, given a fixed job sequence. We prove tight hardness results for the variable sequence case with the same objective function, as well as a new objective function naturally arising in multi-feeder mailroom inserting systems.

Towards a tight hardness–randomness connection between permanent and arithmetic circuit identity testing

We make progress towards proving an equivalence between the problem of derandomization of arithmetic circuit identity testing (ACIT), and the arithmetic circuit complexity of the permanent defined by pern=∑σ∈Sn∏i=1nxiσ(i). We develop an ACIT-based derandomization hypothesis, and show this is a necessary condition for proving that permanent has super-polynomial arithmetic circuits over fields of characteristic zero. Informally, this hypothesis poses the existence of a subexponential size hitting set computable by subexponential size uniform TC0 circuits against size n arithmetic circuits with multilinear output. Assuming the Generalized Riemann Hypothesis (GRH), we show that this hypothesis is sufficient for proving that either permanent does not have polynomial size (nonuniform) arithmetic circuits, or that the Boolean circuit class uniform TC0 is strictly contained in uniform NC2.

Finding common structured patterns in linear graphs

A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting ( ⊏ ) or crossing ( ≬ ). Given a family of linear graphs, and a non-empty subset R ⊆ { < , ⊏ , ≬ } , we are interested in the Maximum Common Structured Pattern ( MCSP) problem: find a maximum size edge-disjoint graph, with edge pairs all comparable by one of the relations in R , that occurs as a subgraph in each of the linear graphs of the family. The MCSP problem generalizes many structure-comparison and structure-prediction problems that arise in computational molecular biology. We give tight hardness results for the MCSP problem for { < , ≬ } -structured patterns and { ⊏ , ≬ } -structured patterns. Furthermore, we prove that the problem is approximable within ratios: (i) 2 ℋ ( k ) for { < , ≬ } -structured patterns, (ii) k 1 / 2 for { ⊏ , ≬ } -structured patterns, and (iii) O ( k log k ) for { < , ⊏ , ≬ } -structured patterns, where k is the size of the optimal solution and ℋ ( k ) = ∑ i = 1 k 1 / i is the k th harmonic number. Also, we provide combinatorial results concerning different types of structured patterns that are of independent interest in their own right.

Identifying Codes and Covering Problems

The identifying code problem for a given graph involves finding a minimum set of vertices whose neighborhoods uniquely overlap at any given graph vertex. Initially introduced in 1998, this problem has demonstrated its fundamental nature through a wide variety of applications, such as fault diagnosis, location detection, and environmental monitoring, in addition to deep connections to information theory, superimposed and covering codes, and tilings. This work establishes efficient reductions between the identifying code problem and the well-known set-covering problem, resulting in a tight hardness of approximation result and novel, provably tight polynomial-time approximations. The main results are also extended to r -robust identifying codes and analogous set (2r+1)-multicover problems. Finally, empirical support is provided for the effectiveness of the proposed approximations, including good constructions for well-known topologies such as infinite two-dimensional grids.

Optimal Inapproximability Results for MAX‐CUT and Other 2‐Variable CSPs?

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX‐CUT to within a factor of $\alpha_{\text{\tiny{GW}}} + \epsilon$ for all $\epsilon > 0$; here $\alpha_{\text{\tiny{GW}}} \approx .878567$ denotes the approximation ratio achieved by the algorithm of Goemans and Williamson in [J. Assoc. Comput. Mach., 42 (1995), pp. 1115–1145]. This implies that if the Unique Games Conjecture of Khot in [Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775] holds, then the Goemans–Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans–Williamson algorithm might be intrinsic to the MAX‐CUT problem. Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21–30]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21–30] contains a proof of an asymptotic version of it. Our techniques extend to several other two‐variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX‐2SAT, MAX‐q‐CUT, and MAX‐2LIN(q). For MAX‐2SAT we show approximation hardness up to a factor of roughly $.943$. This nearly matches the $.940$ approximation algorithm of Lewin, Livnat, and Zwick in [Proceedings of the 9th Annual Conference on Integer Programming and Combinatorial Optimization, Springer‐Verlag, Berlin, 2002, pp. 67–82]. Furthermore, we show that our .943... factor is actually tight for a slightly restricted version of MAX‐2SAT. For MAX‐q‐CUT we show a hardness factor which asymptotically (for large q) matches the approximation factor achieved by Frieze and Jerrum [Improved approximation algorithms for MAX k‐CUT and MAX BISECTION, in Integer Programming and Combinatorial Optimization, Springer‐Verlag, Berlin, pp. 1–13], namely $1 - 1/q + 2({\rm ln}\,q)/q^2$. For MAX‐2LIN(q) we show hardness of distinguishing between instances which are $(1-\epsilon)$‐satisfiable and those which are not even, roughly, $(q^{-\epsilon/2})$‐satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 205–214]. The hardness result holds even for instances in which all equations are of the form $x_i - x_j = c$. At a more qualitative level, this result also implies that $1-\epsilon$ vs. ε hardness for MAX‐2LIN(q) is equivalent to the Unique Games Conjecture.

Fair versus Unrestricted Bin Packing

Abstract. We consider the on-line Dual Bin Packing problem where we have nunit size bins and a sequence of items. The goal is to maximize the number of items that are packed in the bins by an on-line algorithm. We investigate unrestrictedalgorithms that have the power of performing admission control on the items, i.e., rejecting items while there is enough space to pack them, versus fairalgorithms that reject an item only when there is not enough space to pack it. We show that by performing admission control on the items, we get better performance compared with the performance achieved on the fair version of the problem. Our main result shows that with an unfair variant of First-Fit, we can pack approximately two-thirds of the items for sequences for which an optimal off-line algorithm can pack all the items. This is in contrast to standard First-Fit where we show an asymptotically tight hardness result: if the number of bins can be chosen arbitrarily large, the fraction of the items packed by First-Fit comes arbitrarily close to five-eighths.