TY - GEN

T1 - On testing convexity and submodularity

AU - Parnas, Michal

AU - Ron, Dana

AU - Rubinfeld, Ronitt

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2002.

PY - 2002

Y1 - 2002

N2 - Submodular and convex functions play an important role in many applications, and in particular in combinatorial optimization. Here we study two special cases: convexity in one dimension and submodularity in two dimensions. The latter type of functions are equivalent to the well known Monge matrices. A matrix V = {vi,j}i=n1,j=n2 i,j=0 is called a Monge matrix if for every 0 ≤ i < r ≤ n1 and 0 ≤ j < s ≤ n2, we have vi,j + vr,s ≤ vi,s + vr,j. If inequality holds in the opposite direction then V is an inverse Monge matrix (supermodular function). Many problems, such as the traveling salesperson problem and various transportation problems, can be solved more efficiently if the input is a Monge matrix. In this work we present a testing algorithm for Monge and inverse Monge matrices, whose running time is O ((log n1 · log n2)/ ε), where ε is the distance parameter for testing. In addition we have an algorithm that tests whether a function f: [n] → R is convex (concave) with running time of O ((log n)/ε).

AB - Submodular and convex functions play an important role in many applications, and in particular in combinatorial optimization. Here we study two special cases: convexity in one dimension and submodularity in two dimensions. The latter type of functions are equivalent to the well known Monge matrices. A matrix V = {vi,j}i=n1,j=n2 i,j=0 is called a Monge matrix if for every 0 ≤ i < r ≤ n1 and 0 ≤ j < s ≤ n2, we have vi,j + vr,s ≤ vi,s + vr,j. If inequality holds in the opposite direction then V is an inverse Monge matrix (supermodular function). Many problems, such as the traveling salesperson problem and various transportation problems, can be solved more efficiently if the input is a Monge matrix. In this work we present a testing algorithm for Monge and inverse Monge matrices, whose running time is O ((log n1 · log n2)/ ε), where ε is the distance parameter for testing. In addition we have an algorithm that tests whether a function f: [n] → R is convex (concave) with running time of O ((log n)/ε).

UR - http://www.scopus.com/inward/record.url?scp=84959040997&partnerID=8YFLogxK

U2 - 10.1007/3-540-45726-7_2

DO - 10.1007/3-540-45726-7_2

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AN - SCOPUS:84959040997

SN - 3540441476

SN - 9783540457268

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

SP - 11

EP - 25

BT - Randomization and Approximation Techniques in Computer Science - 6th International Workshop, RANDOM 2002, Proceedings

A2 - Vadhan, Salil

A2 - Rolim, Jose D. P.

PB - Springer Verlag

T2 - 6th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2002

Y2 - 13 September 2002 through 15 September 2002

ER -