Harshil Mittal

Koustav De, Harshil Mittal, Palash Dey et al.

The Kemeny method is one of the popular tools for rank aggregation. However, computing an optimal Kemeny ranking is NP-hard. Consequently, the computational task of finding a Kemeny ranking has been studied under the lens of parameterized complexity with respect to many parameters. We first present a comprehensive relationship, both theoretical and empirical, among these parameters. Further, we study the problem of computing all distinct Kemeny rankings under the lens of parameterized complexity. We consider the target Kemeny score, number of candidates, average distance of input rankings, maximum range of any candidate, and unanimity width as our parameters. For all these parameters, we already have FPT algorithms. We find that any desirable number of Kemeny rankings can also be found without substantial increase in running time. We also present FPT approximation algorithms for Kemeny rank aggregation with respect to these parameters.

Balagopal Komarath, Harshil Mittal, Jayalal Sarma

Arithmetic circuit complexity studies the complexity of computing polynomials using only arithmetic operations such as addition, multiplication, subtraction, and division. Polynomials over rings of integers model counting problems. Similarly, polynomials over semirings such as tropical semirings model optimization problems. Circuits over semirings then model so called pure algorithms, algorithms that only use the operations in the semiring. In this paper, we do a complexity-theoretic study of the power and limitations of circuits (which represent dynamic programs) over semirings: i) We define $\mathsf{VNP}$ over min-plus semirings, which can faithfully represent problems such as computing min-weight perfect matchings and min-weight Hamiltonian cycles where we have efficiently verifiable certificates. Unlike over rings, we complement the values in the certificate for free as complementation is impossible over min-plus semirings. We prove a dichotomy theorem that states that if we only complement logarithmically many values, this class is same as $\mathsf{VP}$ over min-plus semirings. If we complement super-logarithmically many values, then $\mathsf{VNP} \neq \mathsf{VP}$. ii) We consider constant-width ABPs (which are also called incremental dynamic programs that are restricted to use only a constant number of registers) and show that even simple problems like computing the min-weight $2$-edge-matching is impossible with width $2$ (or $2$ registers). However, with width $3$ (or $3$ registers), such programs can compute everything. More generally, we show that constant-depth formulas are efficiently simulated by constant-width ABPs. iii) We show that an exponential hypercube sum (min in the semiring) over even provably weak models such as width-$2$ ABPs and products of linear forms are the same as $\mathsf{VNP}$.

Nagashri Krishnakumar, Harshil Mittal, Jayalal Sarma

The labelled reachability problem for undirected graphs with edges labelled by elements of a monoid $M$ (more generally, groupoids or magmas) captures the classes $\sf{L}$ and $\sf{NL}$. Given a graph $G(V, E)$ labelled by $ϕ~\colon E \to M$, $s,t \in V$ and an accepting subset $F \subseteq M$, the problem asks to test whether there is a walk $P$ from $s$ to $t$ in $G$ where $ϕ(P) \in F$. Ramaswamy et al. (2019) studied the variant where the accepting element is part of the input for aperiodic monoids and groups. Motivated by the success in designing space-bounded algorithms for the undirected graph reachability problem, we study the labelled reachability problem when the accepting set is also fixed. This reveals finer complexity bounds and dichotomies for the problem based on the monoid and the accepting set. Previous results imply that the problem is in $\sf{L}$ for any finite accepting subset when $M$ is a group or belongs to $\sf{DA}$. We prove the following (for finite monoids): 1) For any monoid $M$, the problem is in $\sf{L}$ when the accepting element is the identity of $M$. If the accepting element is an idempotent, under suitable constraints, the problem is $\sf{NL}$-hard. 2) For any commutative monoid $M$, the problem is in $\sf{L}$ for all $F \subseteq M$. 3) For any $\mathcal{L}(\mathcal{R})$-commutative union-of-groups (UoG) monoid $M$, the problem is in $\sf{L}$ for all $F\subseteq M$. We show deterministic logspace algorithms for UoG monoids that are neither $\mathcal{L}$-commutative nor $\mathcal{R}$-commutative, under certain constraints. 4) For the monoids $\sf{BA_2}$ and $\sf{U}$, we show a dichotomy: for all $F \subseteq M$, the problem is either $\sf{NL}$-complete or in $\sf{L}$. Our results exploit the connection between Green's relations in the UoG monoids and the properties of the product graph (a graph introduced by Ramaswamy et al. (2019)).