TY - GEN

T1 - Improved approximations for k-Exchange systems (Extended Abstract)

AU - Feldman, Moran

AU - Naor, Joseph

AU - Schwartz, Roy

AU - Ward, Justin

PY - 2011

Y1 - 2011

N2 - Submodular maximization and set systems play a major role in combinatorial optimization. It is long known that the greedy algorithm provides a 1/(k+1)-approximation for maximizing a monotone submodular function over a k-system. For the special case of k-matroid intersection, a local search approach was recently shown to provide an improved approximation of 1 / (k+δ) for arbitrary δ≥0. Unfortunately, many fundamental optimization problems are represented by a k-system which is not a k-intersection. An interesting question is whether the local search approach can be extended to include such problems. We answer this question affirmatively. Motivated by the b-matching and k-set packing problems, as well as the more general matroid k-parity problem, we introduce a new class of set systems called k-exchange systems, that includes k-set packing, b-matching, matroid k-parity in strongly base orderable matroids, and additional combinatorial optimization problems such as: independent set in (k+1)-claw free graphs, asymmetric TSP, job interval selection with identical lengths and frequency allocation on lines. We give a natural local search algorithm which improves upon the current greedy approximation, for this new class of independence systems. Unlike known local search algorithms for similar problems, we use counting arguments to bound the performance of our algorithm. Moreover, we consider additional objective functions and provide improved approximations for them as well. In the case of linear objective functions, we give a non-oblivious local search algorithm, that improves upon existing local search approaches for matroid k-parity.

AB - Submodular maximization and set systems play a major role in combinatorial optimization. It is long known that the greedy algorithm provides a 1/(k+1)-approximation for maximizing a monotone submodular function over a k-system. For the special case of k-matroid intersection, a local search approach was recently shown to provide an improved approximation of 1 / (k+δ) for arbitrary δ≥0. Unfortunately, many fundamental optimization problems are represented by a k-system which is not a k-intersection. An interesting question is whether the local search approach can be extended to include such problems. We answer this question affirmatively. Motivated by the b-matching and k-set packing problems, as well as the more general matroid k-parity problem, we introduce a new class of set systems called k-exchange systems, that includes k-set packing, b-matching, matroid k-parity in strongly base orderable matroids, and additional combinatorial optimization problems such as: independent set in (k+1)-claw free graphs, asymmetric TSP, job interval selection with identical lengths and frequency allocation on lines. We give a natural local search algorithm which improves upon the current greedy approximation, for this new class of independence systems. Unlike known local search algorithms for similar problems, we use counting arguments to bound the performance of our algorithm. Moreover, we consider additional objective functions and provide improved approximations for them as well. In the case of linear objective functions, we give a non-oblivious local search algorithm, that improves upon existing local search approaches for matroid k-parity.

UR - http://www.scopus.com/inward/record.url?scp=80052818288&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-23719-5_66

DO - 10.1007/978-3-642-23719-5_66

M3 - Conference contribution

AN - SCOPUS:80052818288

SN - 9783642237188

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

SP - 784

EP - 798

BT - Algorithms, ESA 2011 - 19th Annual European Symposium, Proceedings

T2 - 19th Annual European Symposium on Algorithms, ESA 2011

Y2 - 5 September 2011 through 9 September 2011

ER -