Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder

  title={Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder},
  author={Per Austrin and Aleksa Stankovic},
Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within \approx 0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within \approx 0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (\approx… 

Figures from this paper

On Regularity of Max-CSPs and Min-CSPs
Simultaneous max-cut is harder to approximate than max-cut
The question whether one can achieve an αgw-minimum approximation algorithm for simultaneous Max-Cut is answered by showing that there exists an absolute constant ε0 ≥ 10-5 such that it is NP-hard to get an (αgw -ε0)-minimum approximation for simultaneousMax-Cut assuming the Unique Games Conjecture.
A characterization of approximability for biased CSPs
This work explores the role played by the bias parameter µ on the approximability of biased CSPs and implies the first tight approximation bounds for the Densest-k-Subgraph problem in the linear bias regime.
Inapproximability of Clustering in Lp Metrics
  • V. Cohen-Addad, S. KarthikC.
  • Computer Science, Mathematics
    2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2019
It is shown that it is hard to approximate the k-means objective in O(log n) dimensional space and an embedding technique is introduced which combines the transcripts of certain communication protocols with the geometric realization of certain graphs that proves standard NP-hardness for the above problems but for smaller approximation factors.
Johnson Coverage Hypothesis: Inapproximability of k-means and k-median in L_p metrics
A new hypothesis called JCH is introduced, which roughly asserts that the well-studied Max k-Coverage problem on set systems is hard to approximate to a factor greater than (1− 1/e), even when the membership graph of the set system is a subgraph of the Johnson graph.
Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems
This work presents a new general and simple method for rounding semi-definite programs, based on Brownian motion, and gives new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion.


Improved Approximation Algorithms for Max-2SAT with Cardinality Constraint
An approximation algorithm with polynomial running time for Max-2SAT-CC and a greedy algorithm with running time O(N log N) and approximation ratio 1/2 that works for clauses of arbitrary length is presented.
An Approximation Algorithm for MAX-2-SAT with Cardinality Constraint
We present a randomized polynomial-time approximation algorithm for the MAX-2-SAT problem in the presence of an extra cardinality constraint which has an asymptotic worst-case ratio of 0.75. This
Balanced max 2-sat might not be the hardest
We show that, assuming the Unique Games Conjecture, it is NP-hard to approximate MAX2SAT within αLLZ-+ε, where 0.9401 < αLLZ- < 0.9402 is the believed approximation ratio of the algorithm of Lewin,
Approximating CSPs with Global Cardinality Constraints Using SDP Hierarchies
The idea of this paper is to find a set of “uncorrelated” vectors using higher order sum of squares (SOS) hierarchy using higher Order Sum of squares hierarchy.
Optimal algorithms and inapproximability results for every CSP?
A generic conversion from SDP integrality gaps to UGC hardness results for every CSP is shown, which achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.
Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems
Improved approximation algorithms for the MAX 2-SAT and MAX DI-CUT problems are obtained, which are essentially the best performance ratios that can be achieved using any combination of prerounding rotations and skewed distributions of hyperplanes, and even using more general families of rounding procedures.
Best Possible Approximation Algorithm for MAX SAT with Cardinality Constraint
A (1-e-1) -approximation algorithm for MAX SAT with cardinality constraint with clauses without negations is obtained, which is the best possible performance guarantee unless P=\NP.
A Proof of CSP Dichotomy Conjecture
  • Dmitriy Zhuk
  • Mathematics, Computer Science
    2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2017
An algorithm is presented that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.
.879-approximation algorithms for MAX CUT and MAX 2SAT
This research presents randomized approximation algorithms for the MAX CUT and MAX 2SAT problems that always deliver solutions of expected value at least .87856 times the optimal value and represents the first use of semidefinite programming in the design of approximation algorithms.
A 7/8-approximation algorithm for MAX 3SAT?
  • H. Karloff, U. Zwick
  • Computer Science
    Proceedings 38th Annual Symposium on Foundations of Computer Science
  • 1997
A randomized approximation algorithm which takes an instance of MAX 3SAT as input that is optimal if the instance-a collection of clauses each of length at most three-is satisfiable, and a method of obtaining direct semidefinite relaxations of any constraint satisfaction problem of the form MAX CSP(F), where F is a finite family of Boolean functions.