Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to non-federal websites. Their policies may differ from this site.
-
Assuming the Unique Games Conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the MAX CUT problem is αCUT ≃ 0.87856, obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. The currently best approximation algorithm for MAX DI-CUT, i.e., the MAX CUT problem in directed graphs, achieves a ratio of about 0.87401, leaving open the question whether MAX DI-CUT can be approximated as well as MAX CUT. We obtain a slightly improved algorithm for MAX DI-CUT and a new UGC-hardness result for it, showing that 0.87446 ≤ αDI-CUT ≤ 0.87461, where αDI-CUT is the best approximation ratio that can be obtained in polynomial time for MAX DI-CUT under UGC. The new upper bound separates MAX DI-CUT from MAX CUT, resolving a question raised by Feige and Goemans. A natural generalization of MAX DI-CUT is the MAX 2-AND problem in which each constraint is of the form z1∧z2, where z1 and z2 are literals, i.e., variables or their negations (In MAX DI-CUT each constraint is of the form \neg{x1}∧x2, where x1 and x2 are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that α2AND < 0.87435 and conjectured that MAX 2-AND and MAX DI-CUT have the same approximation ratio. Our new lower bound on MAX DI-CUT refutes this conjecture, completing the separation of the three problems MAX 2-AND, MAX DI-CUT and MAX CUT. We also obtain a new lower bound for MAX 2-AND, showing that 0.87414 ≤ α2AND ≤ 0.87435. Our upper bound on MAX DI-CUT is achieved via a simple, analytical proof. The lower bounds on MAX DI-CUT and MAX 2-AND (the new approximation algorithms) use experimentally-discovered distributions of rounding functions which are then verified via computer-assisted proofs.more » « less
-
Promise Constraint Satisfaction Problems (PCSPs) are a generalization ofConstraint Satisfaction Problems (CSPs) where each predicate has a strong and aweak form and given a CSP instance, the objective is to distinguish if thestrong form can be satisfied vs. even the weak form cannot be satisfied. Sincetheir formal introduction by Austrin, Guruswami, and H\aa stad, there has beena flurry of works on PCSPs [BBKO19,KO19,WZ20]. The key tool in studying PCSPsis the algebraic framework developed in the context of CSPs where the closureproperties of the satisfying solutions known as the polymorphisms are analyzed. The polymorphisms of PCSPs are much richer than CSPs. In the Boolean case, westill do not know if dichotomy for PCSPs exists analogous to Schaefer'sdichotomy result for CSPs. In this paper, we study a special case of BooleanPCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate$x \leq y$. In the algebraic framework, this is the special case of BooleanPCSPs when the polymorphisms are monotone functions. We prove that BooleanOrdered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1Conjecture [BKM21] which is a perfect completeness surrogate of the UniqueGames Conjecture. Assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP canbe solved in polynomial time if for every $\epsilon>0$, it has polymorphismswhere each coordinate has Shapley value at most $\epsilon$, else it is NP-hard.The algorithmic part of our dichotomy is based on a structural lemma thatBoolean monotone functions with each coordinate having low Shapley value havearbitrarily large threshold functions as minors. The hardness part proceeds byshowing that the Shapley value is consistent under a uniformly random 2-to-1minor. Of independent interest, we show that the Shapley value can beinconsistent under an adversarial 2-to-1 minor.more » « less
-
The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε > 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε > 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices.more » « less
-
We consider three graphs, 𝐺_{7,3}, 𝐺_{7,4}, and 𝐺_{7,6}, related to Keller’s conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size 2^7 = 128. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of ℝ^7 contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of ℝ^8 exists (which we also verify), this completely resolves Keller’s conjecture.more » « less