Aggregating multiple input rankings into a consensus ranking is essential in various fields such as social choice theory, hiring, college admissions, web search, and databases. A major challenge is that the optimal consensus ranking might be biased against individual candidates or groups, especially those from marginalized communities. This concern has led to recent studies focusing on fairness in rank aggregation. The goal is to ensure that candidates from different groups are fairly represented in the top-k positions of the aggregated ranking. We study this fair rank aggregation problem by considering the Kendall tau as the underlying metric. While we know of a polynomial-time approximation scheme (PTAS) for the classical rank aggregation problem, the corresponding fair variant only possesses a quite straightforward 3-approximation algorithm due to Wei et al., SIGMOD'22, and Chakraborty et al., NeurIPS'22, which finds closest fair ranking for each input ranking and then simply outputs the best one. In this paper, we first provide a novel algorithm that achieves (2+ε)-approximation (for any ε > 0), significantly improving over the 3-approximation bound. Next, we provide a 2.881-approximation fair rank aggregation algorithm that works irrespective of the fairness notion, given one can find a closest fair ranking, beating the 3-approximation bound. We complement our theoretical guarantee by performing extensive experiments on various real-world datasets to establish the effectiveness of our algorithm further by comparing it with the performance of state-of-the-art algorithms.

We introduce a new graph-theoretic concept in the area of network monitoring. In this area, one wishes to monitor the vertices and/or the edges of a network (viewed as a graph) in order to detect and prevent failures. Inspired by two notions studied in the literature (edge-geodetic sets and distance-edge-monitoring sets), we define the notion of a monitoring edge-geodetic set (MEG-set for short) of a graph G as an edge-geodetic set S ⊆ V ( G ) of G (that is, every edge of G lies on some shortest path between two vertices of S ) with the additional property that for every edge e of G , there is a vertex pair x , y of S such that e lies on all shortest paths between x and y . The motivation is that, if some edge e is removed from the network (for example if it ceases to function), the monitoring probes x and y will detect the failure since the distance between them will increase. We explore the notion of MEG-sets by deriving the minimum size of a MEG-set for some basic graph classes (trees, cycles, unicyclic graphs, complete graphs, grids, hypercubes, corona products...) and we prove an upper bound using the feedback edge set of the graph. We also show that determining the smallest size of an MEG-set of a graph is NP-hard, even for graphs of maximum degree at most 9.

Several measures exist for string similarity, including notable ones like the edit distance and the indel distance. The former measures the count of insertions, deletions, and substitutions required to transform one string into another, while the latter specifically quantifies the number of insertions and deletions. Many algorithmic solutions explicitly address one of these measures, and frequently techniques applicable to one can also be adapted to work with the other. In this paper, we investigate whether there exists a standardized approach for applying results from one setting to another. Specifically, we demonstrate the capability to reduce questions regarding string similarity over arbitrary alphabets to equivalent questions over a binary alphabet. Furthermore, we illustrate how to transform questions concerning indel distance into equivalent questions based on edit distance. This complements an earlier result of Tiskin (2007) which addresses the inverse direction.

We study geometric variations of the discriminating code problem. In the \emph{discrete version} of the problem, a finite set of points $P$ and a finite set of objects $S$ are given in $\mathbb{R}^d$. The objective is to choose a subset $S^* \subseteq S$ of minimum cardinality such that for each point $p_i \in P$, the subset $S_i^* \subseteq S^*$ covering $p_i$ satisfies $S_i^*\neq \emptyset$, and each pair $p_i,p_j \in P$, $i \neq j$, we have $S_i^* \neq S_j^*$. In the \emph{continuous version} of the problem, the solution set $S^*$ can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case ($d=1$), the points in $P$ are placed on a horizontal line $L$, and the objects in $S$ are finite-length line segments aligned with $L$ (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. Still, for the 1-dimensional discrete version, we design a polynomial-time $2$-approximation algorithm. We also design a PTAS for both discrete and continuous versions in one dimension, for the restriction where the intervals are all required to have the same length. We then study the 2-dimensional case ($d=2$) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-complete, and design polynomial-time approximation algorithms that produce $(16\cdot OPT+1)$-approximate and $(64\cdot OPT+1)$-approximate solutions respectively, using rounding of suitably defined integer linear programming problems. We show that the identifying code problem for axis-parallel unit square intersection graphs (in $d=2$) can be solved in the same manner as for the discrete version of the discriminating code problem for unit square objects.

We study two geometric variations of the discriminating code problem. In the discrete version, a finite set of points P and a finite set of objects S are given in ℝ^d. The objective is to choose a subset S^* ⊆ S of minimum cardinality such that the subsets S_i^* ⊆ S^* covering p_i, satisfy S_i^* ≠ ∅ for each i = 1,2,…, n, and S_i^* ≠ S_j^* for each pair (i,j), i ≠ j. In the continuous version, the solution set S^* can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case (d = 1), the points are placed on some fixed-line L, and the objects in S are finite segments of L (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in 1D. We then design a polynomial-time 2-approximation algorithm for the 1-dimensional discrete case. We also design a PTAS for both discrete and continuous cases when the intervals are all required to have the same length. We then study the 2-dimensional case (d = 2) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-hard, and design polynomial-time approximation algorithms with factors 4+ε and 32+ε, respectively (for every fixed ε > 0).