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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2026.8

Capacitated Partition Vertex Cover and Partition Edge Cover

Authors: Rajni Dabas, Samir Khuller, and Emilie Rivkin

Published in: LIPIcs, Volume 392, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)

Abstract

We study generalizations of the classical Vertex Cover and Edge Cover problems that incorporate group-wise coverage. Our first focus is the Capacitated Partition Vertex Cover (C-PVC) problem in hypergraphs. In C-PVC, we are given a hypergraph with capacities on its vertices and a partition of the hyperedge set into ω distinct groups. The objective is to select a minimum size subset of vertices that satisfies two main conditions: (1) in each group, the total number of covered hyperedges meets a specified threshold, and (2) the number of hyperedges assigned to any vertex respects its capacity constraint. A covered hyperedge is required to be assigned to a selected vertex that belongs to the hyperedge. This formulation generalizes classical Vertex Cover, Partial Vertex Cover, and Partition Vertex Cover. We investigate two primary variants: soft capacitated (multiple copies of a vertex are allowed) and hard capacitated (each vertex can be chosen at most once). Let f denote the rank of the hypergraph (i.e., the maximum number of vertices contained in any single hyperedge). Our main contributions are: (i) an (f+1)-approximation algorithm for the weighted soft-capacitated C-PVC problem, which runs in n^O(ω) time, and (ii) an (f+ε)-approximation algorithm for the unweighted hard-capacitated C-PVC problem, which runs in n^O(ω/ε) time. We also study a natural generalization of the edge cover problem, the Weighted Partition Edge Cover (W-PEC) problem, where each edge has an associated weight, and the vertex set is partitioned into groups. For each group, the goal is to cover at least a specified number of vertices using incident edges, while minimizing the total weight of the selected edges. We present the first exact polynomial-time algorithm for the weighted case, improving runtime from O(ω n³) to O(mn + n²log n) and simplifying the algorithmic structure over prior unweighted approaches (that rely on the tropical matching problem).

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Rajni Dabas, Samir Khuller, and Emilie Rivkin. Capacitated Partition Vertex Cover and Partition Edge Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 8:1-8:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dabas_et_al:LIPIcs.APPROX/RANDOM.2026.8,
  author =	{Dabas, Rajni and Khuller, Samir and Rivkin, Emilie},
  title =	{{Capacitated Partition Vertex Cover and Partition Edge Cover}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
  pages =	{8:1--8:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-449-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{392},
  editor =	{Singh, Mohit and Gur, Tom},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.8},
  URN =		{urn:nbn:de:0030-drops-277257},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2026.8},
  annote =	{Keywords: Approximation algorithms, capacitated vertex cover, iterative rounding}
}

Document

DOI: 10.4230/LIPIcs.FORC.2026.9

Serving Clients Fairly: On Facility Location and k-Median with Fair Outliers

Authors: Rajni Dabas, Samir Khuller, and Emilie Rivkin

Published in: LIPIcs, Volume 368, 7th Symposium on Foundations of Responsible Computing (FORC 2026)

Abstract

Classical clustering problems such as Facility Location and k-Median aim to efficiently serve a set of clients from a subset of facilities - minimizing the total cost of facility openings and client assignments in Facility Location, and minimizing assignment (service) cost under a facility count constraint in k-Median. These problems are highly sensitive to outliers, and therefore researchers have studied variants that allow excluding a small number of clients as outliers to reduce cost. However, in many real-world settings, clients belong to different demographic or functional groups, and unconstrained outlier removal can disproportionately exclude certain groups, raising fairness concerns, especially when the facilities correspond to critically needed facilities for emergencies such as fire stations, hospitals and other emergency services. We study Facility Location with Fair Outliers, where each group is allowed a specified number of outliers, and the objective is to minimize total cost while respecting group-wise fairness constraints. We present a bicriteria approximation with a O(1/ε) approximation factor and (1+ 2ε) factor violation in outliers per group. For k-Median with Fair Outliers, we design a bicriteria approximation with a 4(1+ω/ε) approximation factor and (ω + ε) violation in outliers per group improving on prior work by avoiding dependence on k in outlier violations. We also prove that the problems are W[1]-hard parameterized by ω. We complement our algorithmic contributions with a detailed empirical analysis, demonstrating that fairness can be achieved with negligible increase in cost and that the integrality gap of the standard LP is small in practice.

Cite as

Rajni Dabas, Samir Khuller, and Emilie Rivkin. Serving Clients Fairly: On Facility Location and k-Median with Fair Outliers. In 7th Symposium on Foundations of Responsible Computing (FORC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 368, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dabas_et_al:LIPIcs.FORC.2026.9,
  author =	{Dabas, Rajni and Khuller, Samir and Rivkin, Emilie},
  title =	{{Serving Clients Fairly: On Facility Location and k-Median with Fair Outliers}},
  booktitle =	{7th Symposium on Foundations of Responsible Computing (FORC 2026)},
  pages =	{9:1--9:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-419-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{368},
  editor =	{Lin, Huijia (Rachel)},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2026.9},
  URN =		{urn:nbn:de:0030-drops-259812},
  doi =		{10.4230/LIPIcs.FORC.2026.9},
  annote =	{Keywords: Approximation algorithms, fairness}
}

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Documents authored by Rivkin, Emilie

Capacitated Partition Vertex Cover and Partition Edge Cover

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Serving Clients Fairly: On Facility Location and k-Median with Fair Outliers

Abstract

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