A Polynomial-Time Algorithm for Pliable Index Coding

• Published in: IEEE transactions on information theory Vol. 64; no. 2; pp. 979 - 999
• Main Authors: Song, Linqi, Fragouli, Christina
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.02.2018; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">t -requests; Algorithm design and analysis; Algorithms; Coding; Decoding; Encoding; greedy algorithm; index coding; Indexes; Lower bounds; Messages; Pliable index coding; polynomial time algorithm; Polynomials; Probabilistic analysis; random graphs; Servers; Set theory; Silicon; Upper bound
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2017.2752088

In pliable index coding, we consider a server with <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> messages and <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> clients, where each client has as side information a subset of the messages. We seek to minimize the number of broadcast transmissions, so that each client can recover any one unknown message she does not already have. Previous work has shown that the pliable index coding problem is NP-hard and requires at most <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(\log ^{2}(n)) </tex-math></inline-formula> broadcast transmissions, which indicates exponential savings over the conventional index coding that requires in the worst case <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(n) </tex-math></inline-formula> transmissions. In this paper, building on a decoding criterion that we propose, we first design a deterministic polynomial-time algorithm that can realize the exponential benefits, by achieving, in the worst case, a performance upper bounded by <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(\log ^{2}(n)) </tex-math></inline-formula> broadcast transmissions. We extend our algorithm to the <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-requests case, where each client requires <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula> unknown messages that she does not have, and show that our algorithm requires at most <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(t\log (n)+\log ^{2}(n)) </tex-math></inline-formula> broadcast transmissions. We construct lower bound instances that require at least <inline-formula> <tex-math notation="LaTeX">\Omega (\log (n)) </tex-math></inline-formula> transmissions for linear pliable index coding and at least <inline-formula> <tex-math notation="LaTeX">\Omega (t+\log (n)) </tex-math></inline-formula> transmissions for the <inline-formula> <tex-math notation="LaTeX">t </tex-math></inline-formula>-requests case, indicating that both our upper and lower bounds are polynomials of <inline-formula> <tex-math notation="LaTeX">\log (n) </tex-math></inline-formula> and differ within a factor of <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(\log (n)) </tex-math></inline-formula>. We provide a probabilistic analysis over random instances and show that the required number of transmissions is almost surely <inline-formula> <tex-math notation="LaTeX">\Theta (\log (n)) </tex-math></inline-formula>, as compared with the <inline-formula> <tex-math notation="LaTeX">\Theta (n/\log (n)) </tex-math></inline-formula> for index coding. In addition, we show that these upper and lower bounds also hold for vector pliable index coding in the worst case instances and the random graph instances, implying that vector coding does not provide benefits in terms of these bounds. Our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to 50% less transmissions.