Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems

Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH

The k-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over F_2, which can be stated as follows: given a generator matrix A and an integer k, determine whether the code generated by A has distance at most k. Here, k is the parameter of the problem. The question of whether k-Even Set is fixed parameter tractable (FPT) has been repeatedly raised in literature and has earned its place in Downey and Fellows\u27 book (2013) as one of the "most infamous" open problems in the field of Parameterized Complexity. In this work, we show that k-Even Set does not admit FPT algorithms under the (randomized) Gap Exponential Time Hypothesis (Gap-ETH) [Dinur\u2716, Manurangsi-Raghavendra\u2716]. In fact, our result rules out not only exact FPT algorithms, but also any constant factor FPT approximation algorithms for the problem. Furthermore, our result holds even under the following weaker assumption, which is also known as the Parameterized Inapproximability Hypothesis (PIH) [Lokshtanov et al.\u2717]: no (randomized) FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only 0.99-satisfiable (where the parameter is the number of variables). We also consider the parameterized k-Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer k, and the goal is to determine whether the norm of the shortest vector (in the l_p norm for some fixed p) is at most k. Similar to k-Even Set, this problem is also a long-standing open problem in the field of Parameterized Complexity. We show that, for any p > 1, k-SVP is hard to approximate (in FPT time) to some constant factor, assuming PIH. Furthermore, for the case of p = 2, the inapproximability factor can be amplified to any constant

Parameterized Inapproximability of the Minimum Distance Problem over All Fields and the Shortest Vector Problem in All ℓpNorms

Funding Information: M. Cheraghchi’s research was partially supported by the National Science Foundation under Grants No. CCF-2006455 and CCF-2107345. V. Guruswami’s research was supported in part by NSF grants CCF-2228287 and CCF-2210823 and a Simons Investigator award. J. Ribeiro’s research was supported by NOVA LINCS (UIDB/04516/2020) with the financial support of FCT - Fundação para a Ciência e a Tecnologia and by the NSF grants CCF-1814603 and CCF-2107347 and the following grants of Vipul Goyal: the NSF award 1916939, DARPA SIEVE program, a gift from Ripple, a DoE NETL award, a JP Morgan Faculty Fellowship, a PNC center for financial services innovation award, and a Cylab seed funding award. Publisher Copyright: © 2023 ACM.We prove that the Minimum Distance Problem (MDP) on linear codes over any fixed finite field and parameterized by the input distance bound is W[1]-hard to approximate within any constant factor. We also prove analogous results for the parameterized Shortest Vector Problem (SVP) on integer lattices. Specifically, we prove that SVP in the p norm is W[1]-hard to approximate within any constant factor for any fixed p >1 and W[1]-hard to approximate within a factor approaching 2 for p=1. (We show hardness under randomized reductions in each case.) These results answer the main questions left open (and explicitly posed) by Bhattacharyya, Bonnet, Egri, Ghoshal, Karthik C. S., Lin, Manurangsi, and Marx (Journal of the ACM, 2021) on the complexity of parameterized MDP and SVP. For MDP, they established similar hardness for binary linear codes and left the case of general fields open. For SVP in p norms with p > 1, they showed inapproximability within some constant factor (depending on p) and left open showing such hardness for arbitrary constant factors. They also left open showing W[1]-hardness even of exact SVP in the 1 norm.publishersversionpublishe

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

Cryptography from tensor problems

We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler

On the Closest Vector Problem with a Distance Guarantee

We present a substantially more efficient variant, both in terms of running time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky, and Micciancio for solving CVPP (the preprocessing version of the Closest Vector Problem, CVP) with a distance guarantee. For instance, for any $\alpha < 1/2$, our algorithm finds the (unique) closest lattice point for any target point whose distance from the lattice is at most $\alpha$ times the length of the shortest nonzero lattice vector, requires as preprocessing advice only $N \approx \widetilde{O}(n \exp(\alpha^2 n /(1-2\alpha)^2))$ vectors, and runs in time $\widetilde{O}(nN)$. As our second main contribution, we present reductions showing that it suffices to solve CVP, both in its plain and preprocessing versions, when the input target point is within some bounded distance of the lattice. The reductions are based on ideas due to Kannan and a recent sparsification technique due to Dadush and Kun. Combining our reductions with the LLM algorithm gives an approximation factor of $O(n/\sqrt{\log n})$ for search CVPP, improving on the previous best of $O(n^{1.5})$ due to Lagarias, Lenstra, and Schnorr. When combined with our improved algorithm we obtain, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve CVPP with (the only slightly worse) approximation factor of O(n).Comment: An early version of the paper was titled "On Bounded Distance Decoding and the Closest Vector Problem with Preprocessing". Conference on Computational Complexity (2014