Center for Discrete Mathematics and Theoretical Computer Science

About: Center for Discrete Mathematics and Theoretical Computer Science is a facility organization based out in Piscataway, New Jersey, United States. It is known for research contribution in the topics: Local search (optimization) & Optimization problem. The organization has 140 authors who have published 175 publications receiving 2345 citations.

The Majority Rule and Combinatorial Geometry (via the Symmetric Group)

Abstract: The Marquis du Condorcet recognized 200 years ago that majority rule can produce intransitive group preferences if the domain of possible (transitive) individual preference orders is unrestricted. We present results on the cardinality and structure of those maximal sets of permutations for which majority rule produces transitive results (consistent sets). Consistent sets that contain a maximal chain in the Weak Bruhat Order inherit from it an upper semimodular sublattice structure. They are intrinsically related to a special class of hamiltonian graphs called persistent graphs. These graphs in turn have a clean geometric interpretation: they are precisely visibility graphs of staircase polygons. We highlight the main tools used to prove these connections and indicate possible social choice and computational research directions.

On the hardness of learning sparse parities

Abstract: This work investigates the hardness of computing sparse solutions to systems of linear equations over F_2. Consider the k-EvenSet problem: given a homogeneous system of linear equations over F_2 on n variables, decide if there exists a nonzero solution of Hamming weight at most k (i.e. a k-sparse solution). While there is a simple O(n^{k/2})-time algorithm for it, establishing fixed parameter intractability for k-EvenSet has been a notorious open problem. Towards this goal, we show that unless k-Clique can be solved in n^{o(k)} time, k-EvenSet has no poly(n)2^{o(sqrt{k})} time algorithm and no polynomial time algorithm when k = (log n)^{2+eta} for any eta > 0. Our work also shows that the non-homogeneous generalization of the problem -- which we call k-VectorSum -- is W[1]-hard on instances where the number of equations is O(k log n), improving on previous reductions which produced Omega(n) equations. We also show that for any constant eps > 0, given a system of O(exp(O(k))log n) linear equations, it is W[1]-hard to decide if there is a k-sparse linear form satisfying all the equations or if every function on at most k-variables (k-junta) satisfies at most (1/2 + eps)-fraction of the equations. In the setting of computational learning, this shows hardness of approximate non-proper learning of k-parities. In a similar vein, we use the hardness of k-EvenSet to show that that for any constant d, unless k-Clique can be solved in n^{o(k)} time there is no poly(m, n)2^{o(sqrt{k}) time algorithm to decide whether a given set of m points in F_2^n satisfies: (i) there exists a non-trivial k-sparse homogeneous linear form evaluating to 0 on all the points, or (ii) any non-trivial degree d polynomial P supported on at most k variables evaluates to zero on approx. Pr_{F_2^n}[P(z) = 0] fraction of the points i.e., P is fooled by the set of points.

On a local search algorithm for the capacitated max-k-cut problem

Abstract: The local search algorithm for the capacitated max-k -cut problem proposed by Gaur et al. (2008) is not guaranteed to terminate. In this note, we modify the iterative step of their algorithm to make it terminate in a finite number of steps. The modified algorithm is pseudo-polynomial, and in a special case it is strongly polynomial. Moreover, we analyze the worst case bound of the modified algorithm and give some extensions.

Timing-Aware Fill Insertions with Design-Rule and Density Constraints

Abstract: Metal fill insertion has become an essential step to reduce dielectric thickness variation and improve pattern uniformity, which is important in mitigating process variations, thereby achieving better manufacturing yield. However, metal fills could induce coupling capacitance, which is not often considered in existing works that typically focus more on pattern density uniformity, incurring significant problems in timing closure. In this paper, we address the timing-aware fill insertion problem that considers the total capacitance and density constraints simultaneously. First, initial metal fill insertion and design-rule-aware legalization are used to quickly obtain an initial fill insertion solution. Second, from critical conductors to powers/grounds in a circuit, we divide conductors into different equivalent paths and then construct a capacitance graph to globally reduce the capacitance of each equivalent path. Third, we present a density-aware coupling capacitance optimization method and a fast Monte Carlo based fill selection to further reduce the coupling capacitance between any pair of conductors. Finally, we present a density-aware fill deletion method to reduce the fill amounts. We evaluate the performance of our algorithm based on the benchmarks of the 2018 CAD Contest at ICCAD and its official contest evaluator. Compared with the first place team of the contest and the state-of-the-art work, experimental results show that our algorithm achieves the lowest total capacitance and the least fill amount for each benchmark.