Approximating sparse covering integer programs online

Anupam Gupta, Viswanath Nagarajan

Research output: Chapter in Book/Report/Conference proceedingConference contribution


A covering integer program (CIP) is a mathematical program of the form: min{cT x | Ax ≥ 1, 0 ≤ x ≤ u, x ∈ ℤn}, where A ∈ R≥0mxn, c, u ∈ ℝ ≥0n. In the online setting, the constraints (i.e., the rows of the constraint matrix A) arrive over time, and the algorithm can only increase the coordinates of x to maintain feasibility. As an intermediate step, we consider solving the covering linear program (CLP) online, where the requirement x ∈ ℤn is replaced by x ∈ ℝn. Our main results are (a) an O(logk)-competitive online algorithm for solving the CLP, and (b) an O(logk·logℓ)-competitive randomized online algorithm for solving the CIP. Here k n and ℓ ≤ m respectively denote the maximum number of non-zero entries in any row and column of the constraint matrix A. By a result of Feige and Korman, this is the best possible for polynomial-time online algorithms, even in the special case of set cover (where A ∈ {0,1}mxn and c, u ∈{0,1}n). The novel ingredient of our approach is to allow the dual variables to increase and decrease throughout the course of the algorithm. We show that the previous approaches, which either only raise dual variables, or lower duals only within a guess-and-double framework, cannot give a performance better than O(logn), even when each constraint only has a single variable (i.e., k = 1).

Original languageEnglish (US)
Title of host publicationAutomata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings
Number of pages13
EditionPART 1
StatePublished - 2012
Event39th International Colloquium on Automata, Languages, and Programming, ICALP 2012 - Warwick, United Kingdom
Duration: Jul 9 2012Jul 13 2012

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume7391 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other39th International Colloquium on Automata, Languages, and Programming, ICALP 2012
Country/TerritoryUnited Kingdom

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


Dive into the research topics of 'Approximating sparse covering integer programs online'. Together they form a unique fingerprint.

Cite this