Laks V.S. Lakshmanan University of British Columbia Vancouver, Canada laks Joint work with Zeinab...

Laks V.S. Lakshmanan University of British Columbia

Vancouver, Canada http://www.cs.ubc.ca/~laks Joint work with Zeinab Abbassi.

Recommender Systems Revisited – From Items to

Transactions

04/19/23

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PersDB'09, Lyon.

Why Recommendations? 2

The Recommendation Paradigm Suggest content (in most cases, items) to users based on her

profile and past activities. Why Recommendation?

Search queries can be generic: e.g., >90% of Yahoo! Travel queries are general descriptions like family trip.

More so for Social Content Sites ...

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PersDB'09, Lyon, August 2009.

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Why Recommendations?

Social Content Sites Sites where users make friends and share contents

E.g., facebook, del.icio.us, Flickr, etc. Content sites letting you share info with your social buddies

E.g., nytimes.com, indiatimes.com, youtube.com

Recommendation is an indispensible information exploration paradigm on social content sites.

The rich activities and user connections provide lots of opportunities for generating recommendations.

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Overview of RecSys

Item-Based Strategies Estimate the rating of an unrated item (i) by the

user (u) based on its similarity to items already rated and how u rated those items.

Collaborative Filtering Strategies Estimate the rating of i by u based on how u’s

similarity network (either explicit or implicit) rated i.

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Overview of RecSys

X

5

U1

U2

…

Ui

…

I1 I2 … Ij …

Item-based.

Overview of RecSys

X

6

U1

U2

…

Ui

…

I1 I2 … Ij …

User-based: Collaborative Filtering

Overview of RecSys

Fusion Strategies Model/Machine Leaning-based approaches. What’s common among all RecSys algorithms?

Recommend items to users. What if we want to recommend transactions instead?

Motivating Apps User exchanging items

Offline (one-shot) exchanges Asynchronous exchanges

Users buying/selling items for a price

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Talk Outline

MotivationProblem DefinitionRelated WorkOur ApproachExperimental ResultsSummary & Future Work.

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Motivation

Online social networks – emergence and rapid growth.

Users spend more time on online social networks.MySpace and Facebook are among the top 10

websites:

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Motivation – Exchange Markets There are exchange

markets around on the Web today:

OddShoe.org

Peerflix.com (movie exchange)

ReadItSwapIt.co.uk

JoeBarter.com

Intervac PersDB'09, Lyon, August 2009.

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Motivation

Need “Matching” Algorithms Enhance quality of user experience. Let more people be engaged in the system and more

of the time. Monetization.

Lack of comprehensive study of “matching”. Need efficient recommendation algorithms.

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Chinese Postman Problem:

Collect mail from postal station (s). Deliver mail on all streets (edges). Minimize distance covered (fuel consumed).

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Some Related Problems

Cycle Covers:

? ?

Let’s try again!

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Cycle Covers:

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157

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Can we cover all vertices with edge-disjoint cycles? What is the minimum weight of such a cover?

Cycle Covers:

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12

157

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6

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Can we cover all vertices with edge-disjoint cycles? What is the minimum weight of such a cover?

Vertex-disjoint, edge-disjoint, vertex/edge cover, bounded length, etc. – variants.

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Related Work

Graphs – Cycle Cover Problem: Polytime algorithm for Chinese Postman problem on

undirected graphs [Edmonds & Johnson 73]. CPP is NP-hard for mixed graphs [Papadimitriou 76] but

admits a 3/2-approx. [Raghavachari & Veerasamy 99]. Cycle Cover -- cover given set of nodes/edges with set

of min. length cycles. Min. Weight Cycle Covers – a variant of CPP; NP-hard in

general [Thomassen 97]. CC w/ bounds on cycle length (heuristic) [Hochbaum

and Olinick 01]. Approximation algorithms when length is bounded

[Immorlica+ 05].

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Related Work

Recommender Systems: Management science perspective [Murthy & Sarkar

03]. Collaborative filtering [Resnik+ 94, Shani+ 02]. Survey of item-based, collaborative filtering, fusion-

based, and model-based [Adomavicius & Tuzhilin 05].

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Kidney Exchange problem:

4,000 deaths/yr in US. 70,000 waiting for a cadaver kidney.

Related Work 18

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Related Work

Kidney Exchange problem: In kidney transplants frequently the donor’s kidney is not

compatible with the patient’s. Example: A’ is willing to donate her kidney to A and B’

to B but incompatible. However B’ kidney compatible with A and A’ kidney with B.

Motivation: Find feasible exchanges and save more people’s lives.

Medical constraints: no cycle longer than 3!

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Related Work

Bi-cycles in this case: perfect matching, therefore polynomial!

Cycles of length 3 or more: NP-complete.In [Abraham+ 07] solved by Integer Linear

Programming for the problem of United States kidney exchange with real data!

We will look at a more general problem than KE.

Incentivizing exchanges in P2P file-sharing systems [Anagnostakis & Greenwald 04].

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A Model

u v

i

j

Set of users U and a set of items I. Two lists for each user u in U – item list Su,

items u is willing to give away; wish list Wu, items u is looking for.

Network – nodes = users; u v iff there is a feasible transaction from u to v.

Edges labeled with item.

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Example

04/19/23 22PersDB'09, Lyon.

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Different Models

One-shot exchange Markets:Simple exchange markets (swaps). Exchange markets through short

cycles.Probabilistic exchange markets.Wish List as Query List.

Exchange markets over time.

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Simple exchange markets

Only one-by-one transactions.

The problem is to find a set of pairs: [ (u,i) , (v,j)] where i Є Su, j Є

Wu, i Є Wv and j Є Sv. form 2-cycles (swaps).

Typically each user has one instance of any item and also wants one instance of an item in his wish list [ (u,i) , (*,*)] should not appear more than once for each user u, i.e., looking for a set of conflict-free cycles.

04/19/23 PersDB'09, Lyon, August 2009.

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i

ki j

u

w

v

Exchange markets through cycles.

We look for cycles of length more than 2 in the system.

The goal is to find cycles: [ (u_1,i_1) , (u_2,i_2) , (u_3,i_3) , …, (u_k,i_k) ] where i_1 in S_u1, i_1 in W_u2, i_2 in S_u2, i_2 in W_u2, ….

Alice

Amy

BobB7

B4B8

04/19/23 PersDB'09, Lyon, August 2009.

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26Exchange markets through short cycles

Note: A cycle can happen if and only if all the participating edges are realized.

discover short cycles and solve the short cycle cover problem for cycles of length <= k, where k = 3, 4, 5, …

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Probabilistic Exchange Markets

Each edge in the graph has a probability indicating the likelihood of it being realized.

The probability of realizing each edge is independent of the other edges.

Two kinds of probabilities are of interest: Pu(v): what’s the probability u is willing to perform a

transaction with v? Pu(i,j): what’s the probability u is willing to exchange

item i for j?

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Wish list only contains “predicates” instead of items. E.g., horror movie, science fiction, eastern philosophy,

home hardware, …

Item list as before. Users may rate/review items. Matchmaking has to factor in ratings, i.e.,

matching has to use RecSys technology.

-- Will focus on simple exchange, short cycles, and prob. markets in this talk.

Query List as Wish List 28

Goal

Generate recommendations that maximize the (expected) number of items exchanged through the network.

Each user u gets a reco.: Gives = {(give i to v), …} Gets = {(get j from w), …}

Set of reco’s together constitute a set of conflict-free cycles that maximize above metric.

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SimpleMarket Problem

Theorem: Even SimpleMarket problem is NP-complete.

(Contrast with one-by-one kidney exchange.)Reduction from four-cycle partitioning of 4-

partite graphs to our problem.Reduction from three-cycle partitioning of 3-

partite graphs to 4-cycle partitioning of 4-partite graphs.

3-cycle partitionining of 3-partite graphs is NP-complete [Holyer 81, Abraham+ 06].

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ProbMarket

Lemma: The kidney exchange version of ProbMarket can be solved in polynomial time.

Idea: Maximum weighted perfect matching continues to work.

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Algorithms

Maximal set of Cycles

Greedy.

Local Search.

Greedy/Local Search.

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Maximal Algorithm

Initialize the set of cycles CFSC=empty.At each step,

Find an exchange cycle C. Add C to the set of cycles CFSC. Remove all edges in G in conflict with this cycle. Terminate if there is no remaining cycle.

Find an exchange cycle C: Run a DFS or BFS algorithm until you find a backward

edge. BFS tends to find short cycles.

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Greedy Algorithm


Find the best exchange cycle C. Add C to the set of cycles CFSC. Remove all edges in conflict with this cycle. Terminate if there is no remaining cycle.

Find the best exchange cycle C: Try all short cycles and find the cycle with maximum

weight.

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Intermediate Maximal/Greedy

Improve Running Time.

Find the best exchange cycle C: Run BFS from each node v and find a cycle Cv.

Find the cycle Cv with the maximum weight and add it.

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Local search algorithm


Let the current set of cycles be CFSC. For any exchange cycle C that is not already

picked, Try to add C, and remove all cycles in CFSC in

conflict with C If the total weight of CFSC increases, add C to CFSC

and remove all conflicting cycles from CFSC. If no local improvement is possible, output CFSC

and terminate.

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Greedy/Local Search

First, Run the greedy algorithm to find a set of cycles CFSC.

Then, Run the local search algorithm starting from the set CFSC.

How good are these algorithms?

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Set Packing

Our problem is a special case of weighted k-set packing problem:

Given a collection of sets, each of which has an associated real weight and contains at most k elements drawn from a finite base set, find a collection of disjoint sets of maximum total weight.

OutputInput

http://www.cs.sunysb.edu/~algorith/files/set-packing.shtml

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Relation to set packing

Elements (User u gives item i) (User v gets item j)

Sets Cycles of exchanges.Weights of sets:

Short cycle case: weight is 2k for k item exchanges. Probabilistic: weight is [\pi_{e \in C} p(e)]*2k.

Main difference: Sets are not given explicitly.Sets are cycles (given implicitly) and we have

to discover them.

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Quality of Algorithms

Maximal: No guaranteed quality: O((|V| + |E|)|B|) time.

Greedy: 2k-approximation [Chandra & Halldorsson 99]: O(|V|^2k |B|) time.

Local Search: (2K-1)-approximation. [Arkin &Hassin 97]: O(|V|^2k|E|log OPT).

Local Greedy: 2(2k+1)/3-approximation [Arkin & Hassin 97].

More details [Abbassi & L 09].

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Algorithms implemented in MATLAB and run on 2.16 GHz Intel Core 2 Duo CPU and 1 GBof RAM under Windows XP.

Goals: Extent to which allowing cycles of length > 2

increases coverage of users/items. Quality of results of algorithms (Recall: Maximal has

no theoretical guarantees). Scalability.

Synthetic data: Structure as well as user activities follow power law [Newman 03].

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Experiments

%Increase 23 34 45

Maximal 5.56 3.40 2.70

Greedy 7.33 3.81 3.12

Local Search

7.71 3.85 3.35

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Takeaways: Allowing Larger Cycles

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Experimental Results

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Takeaways: Maximal vs Approximation Algorithms

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Skew factor = 1.0, cycle length bound = 4, #users = 25K.

Algorithm #items exchanged

Maximal 60,000

Greedy, Local Search 65,000

Skew factor = 1.5, cycle length bound = 4, #users = 25K.

Algorithm #items exchanged

Maximal 35,000

Greedy, Local Search 41,000

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Summary & Future Work

Market exchanges over online social nets – simple, short cycles, probabilistic.

Related kidney exchange problem – polytime for swaps and NP-complete for k > 2.

Even swaps NP-complete for market exchange. Reduction to weighted k-set packing

approximation algorithms and Maximal (heuristic). Experiments: “Diminishing returns” as k goes up.Maximal – more than one order of magnitude more

efficient and comparable quality! More empirical analysis needed.

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Summary & Future Work 48


Experiments on Real data sets. More efficient approximation algorithms? Randomization? Exchange Markets over time?

Many different objectives: e.g., #items exchanged, fairness, average waiting time, …

Market Price, Buy/Sell. Connection with game theory. Query List as Wish List (think movie in place

of kidney!)

Mining/Analysis of Social Networks (e.g., for viral marketing).

Network Evolution. Diversification in RecSys. Network-aware Search. Social Search – SocialScope. Opportunities for grad students and postdocs. See http://www.cs.ubc.ca/~laks and

UBC CS Grad Programs

Other Projects in Social Networks and A Shameless Ad

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References Cited

N. Immorlica, V. S. Mirrokni, and M. Mahdian, “Cycle cover with short cycles,” in Symposium on Theoretical Aspects of Computer Science (STACS) 2005.

J. D. Hartline, V. S. Mirrokni, and M. Sundararajan. Optimal marketing strategies over social networks. In WWW’08.

J. Edmonds and E. Johnson, “Matching euler tours and the chinese postman problem.” Mathematical programming, vol. 5, pp. 88–124, 1973.

C. H. Papadimitriou, “On the complexity of edge traversing.” Journal of the ACM, vol. 23, no. 3, pp. 544–554, 1976.

B. Rachavachari and J. Veerasamy, “A 3/2-approximation algorithm for the mixed postman problem,” SIAM journal of Discrete Math., vol. 12, pp. 425–433, 1999.

C. Thomassen, “On the complexity of finding a minimum cycle cover of a graph,” SIAM J. Comput., vol. 26, no. 3, pp. 675–677, 1997.

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References Cited

D. Hochbaum and E. Olinick, “The bounded cycle-cover problem,” INFORMS Journal on Computing, vol. 13, no. 2, pp. 104–109, 2001.

B. Murthi and S. Sarkar, “The role of the management sciences in research on personalization,” Management Science, vol. 49, no. 10, pp. 1344–1362, 2003.

P. Resnick, N. Iakovou, M. Sushak, P. Bergstrom, and J. Riedl, “Grouplens: An open architecture for collaborative filtering of netnews,” in Computer Supported Cooperative Work Conference, 1994.

G. Shani, R. Brafman, and D. Heckerman, “An mdp-based recommender system,” in 18th Conference Uncertainty in Artificial Intelligence, August, 2002.

D. J. Abraham, A. Blum, and T. Sandholm, “Clearing algorithms for barter exchange markets: Enabling nationwide kidney exchanges,” inACM Conference on Electronic Commerce, June 13-16 2007, pp. 295–304.

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References Cited

K. G. Anagnostakis and M. B. Greenwald, “Exchange-based incentive mechanisms for peer-to-peer file sharing,” in 24th International Conference on Distributed Computing Systems, 2004, pp. 524–533.

I. Holyer, “The np-completeness of some edge-partition problems.” SIAM Journal of Computing,, vol. 10, no. 4, pp. 713–717, Nov. 1981.

D. J. Abraham, N. Chen, V. Kumar, and V. Mirrokni, “Assignment problems in rental markets,” in WINE 2006, 2006, pp. 198–213.

B. Chandra and M. Halldorsson, “Greedy local improvement and weighted set packing approximation,” in SODA 1999, 1999, pp. 169–176.

E. M. Arkin and R. Hassin, “On local search for weighted k-set packing.” in ESA 1997, 1997, pp. 13–22.

Zeinab Abbassi, Laks V. S. Lakshmanan: On Efficient Recommendations for Online Exchange Markets. ICDE 2009: 712-723.

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