TY - GEN

T1 - Approximating sparse covering integer programs online

AU - Gupta, Anupam

AU - Nagarajan, Viswanath

PY - 2012

Y1 - 2012

N2 - 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).

AB - 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).

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U2 - 10.1007/978-3-642-31594-7_37

DO - 10.1007/978-3-642-31594-7_37

M3 - Conference contribution

AN - SCOPUS:84883783847

SN - 9783642315930

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 436

EP - 448

BT - Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings

T2 - 39th International Colloquium on Automata, Languages, and Programming, ICALP 2012

Y2 - 9 July 2012 through 13 July 2012

ER -