NDLI: Deterministic Algorithms for Matching and Packing Problems Based on Representative Sets

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Panolan, Fahad Zehavi, Meirav Misra, Neeldhara Goyal, Prachi
Copyright Year	2015
Abstract	In this work, we study the well-known $r$-Dimensional $k$-Matching ($(r,k)$-DM), and $r$-Set $k$-Packing ($(r,k)$-SP) problems. Given a universe $U := U_1 \uplus \cdots \uplus U_r$ and an $r$-uniform family $\mathcal{F} \subseteq U_1 \times \cdots \times U_r$, the $(r,k)$-DM problem asks if $\mathcal{F}$ admits a collection of $k$ mutually disjoint sets. Given a universe $U$ and an $r$-uniform family $\mathcal{F}\subseteq 2^U$, the $(r,k)$-SP problem asks if $\mathcal{F}$ admits a collection of $k$ mutually disjoint sets. We employ techniques based on dynamic programming and representative families. This leads to a deterministic algorithm with running time $\mathcal{O} (2.851^{(r-1)k}\cdot\|\mathcal{F}\|\cdot n\log^2 n\cdot \log W)$ for the weighted version of $(r,k)$-DM, where $W$ is the maximum weight in the input, and a deterministic algorithm with running time $\mathcal{O}(2.851^{(r-0.5501)k}\cdot\|\mathcal{F}\|\cdot n\log^2 n\cdot \log W)$ for the weighted version of $(r,k)$-SP. Thus, we significantly improve the previous best known deterministic running times for $(r,k)$-DM and $(r,k)$-SP and the previous best known running times for their weighted versions. We rely on structural properties of $(r,k)$-DM and $(r,k)$-SP to develop algorithms that are faster than those that can be obtained by a standard use of representative sets. Incorporating the principles of iterative expansion, we obtain a better algorithm for $(3,k)$-DM, running in time $\mathcal{O}(2.004^{3k}\cdot\|\mathcal{F}\| \cdot n\log^2 n)$. We believe that this algorithm demonstrates an interesting application of representative families in conjunction with more traditional techniques. Furthermore, we present kernels of size $\mathcal{O}(e^rr(k-1)^r\log W)$ for the weighted versions of $(r,k)$-DM and $(r,k)$-SP, improving the previous best known kernels of size $\mathcal{O}(r!r(k-1)^r\log W)$ for these problems.
Starting Page	1815
Ending Page	1836
Page Count	22
File Format	PDF
ISSN	08954801
DOI	10.1137/140981290
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	29
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-10-06
Access Restriction	Subscribed
Subject Keyword	3D-matching Analysis of algorithms and problem complexity representative sets set packing $r$-dimensional matching fixed-parameter algorithms Analysis of algorithms Combinatorics iterative expansion
Content Type	Text
Resource Type	Article
Subject	Mathematics

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4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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