₍b₎ { retweeted favorited @ mentioned 6:00PM 6:05PM 9:00PM ₍a₎ { INFORMATION FLOW FOLLOW RELATIONSHIP POST BY USER p POST BY OTHER USER Broadcast-driven Communication on Twitter
each of the author’s tweets receives less attention. (Counts, ICWSM ’11) Interviews revealed that users were unfollowed for their bursts of tweets, regardless of the tweet content. (Kwak, CHI ’11) Survey showed that high posting frequencies tend to irritate users, especially when originating from weaker ties. (Koroleva, Cognition or Effect, ’11) Diversity enforcement on a social networking system newsfeed. (U.S. Patent 13/716,002, Facebook, Dec. ’12) Naive Strategy: Flood.
author tweets frequently each of the author’s tweets receives less attention. (Counts, ICWSM ’11) Interviews revealed that users were unfollowed for their bursts of tweets, regardless of the tweet content. (Kwak, CHI ’11) Survey showed that high posting frequencies tend to irritate users, especially when originating from weaker ties. (Koroleva, Cognition or Effect, ’11) Diversity enforcement on a social networking system newsfeed. (U.S. Patent 13/716,002, Facebook, Dec. ’12) The Frequency Problem.
Experiments on Twitter data a. Discuss information overload (Rodriguez, ICWSM ’14). b. Validate the existence of bursty circadian rhythms. c. Introduce and quantify monotony aversion.
Experiments on Twitter data a. Formulate the process of timeline information exchange — links schedules, attention and behavioural phenomena. b. Formulate the attention potential objective function. c. Formulate the problem as a non- linear integer program, propose an algorithm to solve it.
timeline. (Counts, ICWSM ’11; Rodriguez, ICWSM ’14) e chances of a user forwarding an incoming mation may depend on the in-ﬂow rate itself, o which the user is overloaded. Here, we elu- tion by investigating the relationship between w rate and the retweet out-ﬂow rate. hows how the retweet out-ﬂow rate r varies et in-ﬂow rate . Interestingly, as the in-ﬂow the retweeting rate increases, but at a de- other words, the retweet rate seems to follow shing returns with respect to the in-ﬂow rate; , r( 1 + ) r( 1) r( 2 + ) 2 . Figure 2(b) shows how the probability n incoming tweet r varies against the in- erestingly, we ﬁnd two different in-ﬂow rate ⇠30 tweets/hour, the retweeting probability 10-3 10-2 10-1 0 20 40 60 80 100 P(q) Queue length, q 1-30 30-1000 (a) Queue position distribution 0 100 200 300 400 500 10-1 100 101 10 Queue position, q Tweet In-flow [twe Median Mean (90%) (b) Average queue Figure 3: Queue position. Panel (a) shows the emp tribution and panel (b) shows average/median p a tweet on the user’s queue (feed) at the time w retweeted for different in-ﬂow rates, where q = 0 tweet was at the top of the user’s queue at retweet ATTENTION (a) POST BY USER p 0% 100% POST BY OTHER USER q’s TIMELINE
10−6 10−5 10−4 10−3 10−2 10−1 100 P(τ) ONE DAY ONE WEEK ONE MONTH α = −1.33 Dataset Weiboscope (Fu, Internet Computing, ’13) 14M users, each has >1000 followers. 13M tweets, entire 2012. Experiment Distribution of the time between tweets. Observations Power-law curve due to bursts. Systematic deviations due to circadian rhythms. Bursty Circadian Rhythms
Cognition or Effect, ’11. U.S. Patent 13/716,002, Facebook, Dec. ’12. Validation None. POST BY USER p IRRITATION (b) 0% 100% POST BY OTHER USER q’s TIMELINE
each of the author’s tweets receives less attention. (Counts, ICWSM ’11) Interviews revealed that users were unfollowed for their bursts of tweets, regardless of the tweet content. (Kwak, CHI ’11) Survey showed that high posting frequencies tend to irritate users, especially when originating from weaker ties. (Koroleva, Cognition or Effect, ’11) Diversity enforcement on a social networking system newsfeed. (U.S. Patent 13/716,002, Facebook, Dec. ’12) Evidence
Cluster size. Tweet Cluster position. Was it reacted to? CLUSTER A CLUSTER B CLUSTER C CLUSTER D 1 1 2 1 1 (a) POST BY USER p POST BY OTHER USER Monotony Aversion
timeline. For any tweet on u’s timeline We can calculate the size of its cluster, c ∈ N. We can calculate its cluster position, p ∈ N. 1 1 2 1 1 (a) 1 2 2 1 1 1 1 2 1 1
timeline. For any tweet on u’s timeline We can calculate the size of its cluster, c ∈ N. We can calculate its cluster position, p ∈ N. We can see whether it was reacted to, r ∈ {T, F}. 1 1 2 1 1 (a) 1 2 2 1 1 1 1 2 1 1
2 1 1 1 1 2 1 1 Notation c ∈ N (size of the cluster) p ∈ N (cluster position) r ∈ {T, F} (whether tweet was reacted to) Experiment We can empirically estimate, for every cluster size c: Abstract Scheduling Broadcasts in a Network of Timelines Emaad Ahmed Manzoor Pr(R = 1|C = c) = #[(R = 1) \ (C = c)] #[(R = 0) \ (C = c)] + #[(R = 1) \ (C = c)] Broadcasts and timelines are the primary mechanism of information exchange in online social platforms today. Ser- vices like Facebook, Twitter and Instagram have enabled P(R|C=1) = 2/3 P(R|C=2) = 1/2
Cognition or Effect, ’11. U.S. Patent 13/716,002, Facebook, Dec. ’12. Validation Larger (user-clustered) tweet clusters cause more irritation, resulting in fewer retweets. Follow-up Are tweets independent of each other in their contribution to irritation? POST BY USER p IRRITATION (b) 0% 100% POST BY OTHER USER q’s TIMELINE
chunked (Counts, ICWSM ’11). Validation None. POST BY USER p POST BY OTHER USER (b) INDEPENDENT CONSUMPTION HYPOTHESIS CHUNKED CONSUMPTION HYPOTHESIS IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% Monotony Aversion
p) POST BY USER p POST BY OTHER USER (b) INDEPENDENT CONSUMPTION HYPOTHESIS CHUNKED CONSUMPTION HYPOTHESIS IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% Monotony Aversion
p) POST BY USER p POST BY OTHER USER (b) INDEPENDENT CONSUMPTION HYPOTHESIS CHUNKED CONSUMPTION HYPOTHESIS IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% Monotony Aversion Fixed
p) POST BY USER p POST BY OTHER USER (b) INDEPENDENT CONSUMPTION HYPOTHESIS CHUNKED CONSUMPTION HYPOTHESIS IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% Monotony Aversion Fixed Increase
independent (Leong, WWW, ’14) Validation Tweet consumption is chunked. Tweets lower on the timeline affect the attention/irritation obtained/caused by the ones above. POST BY USER p POST BY OTHER USER (b) INDEPENDENT CONSUMPTION HYPOTHESIS CHUNKED CONSUMPTION HYPOTHESIS IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% IRRITATION 0% 100% Monotony Aversion
c p Competitor activity c 0 c 1 c 2 c Producer/Competitor Schedules (a) p u 1 u 0 c Timeline Construction (c) Network (b) Social Network c c 0 x 0 c 1 x 1 c 2 x 2 u 0 u 1 c c 2 x 2 c 0 x 0 c 1 x 1 Follower Timelines Goal of this section Formally link schedules, timelines, behavioural phenomena and attention.
c p Competitor activity c 0 c 1 c 2 c Producer/Competitor Schedules (a) p u 1 u 0 c Timeline Construction (c) Network (b) c c 0 x 0 c 1 x 1 c 2 x 2 u 0 u 1 c c 2 x 2 c 0 x 0 c 1 x 1 Follower Timelines Social Network Timeline Information Exchange Goal of this section Formally link schedules, timelines, behavioural phenomena and attention.
c p Competitor activity c 0 c 1 c 2 c Producer/Competitor Schedules (a) p u 1 u 0 c Timeline Construction (c) Network (b) c c 0 x 0 c 1 x 1 c 2 x 2 u 0 u 1 c c 2 x 2 c 0 x 0 c 1 x 1 Follower Timelines Social Network Timeline Information Exchange Attention Potential Goal of this section Formally link schedules, timelines, behavioural phenomena and attention.
of time slots) N = x0 + x1 + x2 (Total number of intended posts) p u 1 u 0 c Timeline Construction (c) Network (b) Two followers. One competitor. p u0 logs in at time slot 2. u1 logs in at time slot 1. (ui logs in at time slot σi.)
of time slots) N = x0 + x1 + x2 (Total number of intended posts) p u 1 u 0 c Timeline Construction (c) Network (b) Two followers. One competitor. p σ0 = 2, σ1 = 1
schedule for u1. x0 x1 x2 c00 c10 c20 c c01 c11 c21 c p u 1 u 0 c Timeline Construction (c) Network (b) Two followers. One competitor. p σ0 = 2, σ1 = 1
schedule for u1. Drop subscript. c00 c10 c20 x0 x1 x2 p c c01 c11 c21 c c0 c1 c2 c = p u 1 u 0 c Timeline Construction (c) Network (b) Two followers. One competitor. σ0 = 2, σ1 = 1
1 u 0 c Timeline Construction (c) Network (b) v00 v01 x00 x01 v10 v11 x10 x11 v20 v21 x20 x21 zij = Â m=0 xmj + Â n=0 vnj ( What remains is to derive the number of producer posts xij and competitor posts vij in the cluster from the pro- ducer schedule X and aggregate competitor schedules C respectively. The following relations hold: xij = x ((sj i) mod S) ( vij = c ((sj i) mod S)j ( The attention potential of the timeline constructed for What remains is to derive the number of producer posts xij and competitor posts vij in the cluster from the pro- ducer schedule X and aggregate competitor schedules C respectively. The following relations hold: xij = x ((sj i) mod S) ( vij = c ((sj i) mod S)j ( The attention potential of the timeline constructed for a follower j is the sum of attention potentials of all the producer’s clusters, ÂS 1 i=0 fij( X , C , s, r, d). Each timeline Linking schedules, circadian rhythms and timelines: u0 c1 x1 c0 x0 c2 x2 c2 x2 c1 x1 c0 x0 Constructing Timelines u1 σ0 = 2, σ1 = 1
R(d;ρj), the user survival function, is a decreasing function of d, parameterised by ρj. Eg. R(d;ρj) = (1 - ρj)d 0 < ρj < 1 is the “quitting probability” of user j. ρj = 0.2 ρj = 0.5 ρj = 0.8 d + k d = z10 x
R(d;ρj), the user survival function, is a decreasing function of d, parameterised by ρj. Eg. R(d;ρj) = (1 - ρj)d 0 < ρj < 1 is the “quitting probability” of user j. ρj = 0.2 ρj = 0.5 ρj = 0.8 C(x;δj) = Pr(uj views a cluster of size x) C(x;δj), the cluster survival function, is a decreasing function of x, parameterised by δj. d + k d = z10 x
C(x;δj) = Pr(uj views a cluster of size x) Pr(a post at depth d+k in a cluster of size x is viewed by user uj;ρj, δj) = C(x;δj)R(d+k;ρj) d + k d = z10 x
C(x;δj) = Pr(uj views a cluster of size x) Pr(a post at depth d+k in a cluster of size x is viewed by user uj;ρj, δj) = C(x;δj)R(d+k;ρj) Attention potential of the cluster d + k d = z10 x until that post and the cluster containing that post survives for that follower. Since these events are independent, the probability of this happening is given by R(d; rj)C(x; dj). This is the attention potential of a post at depth d for follower j. The attention potential of a cluster is the sum of attention potentials of the posts within it. Given a cluster containing xij posts, let there be a total of zij posts on the timeline above the ﬁrst post in this clus- ter. The attention potential of this cluster is given by the following function of the producer schedule: fij(x 0, x 1, . . . , xS 1) = C(xij; dj) xij Â k=1 R(zij + k; rj) ( 3 . 1 ) Attention po the broadcas We can calculate the number of posts above the ﬁrst post in a given cluster by the following formula: zij = i 1 Â m=0 xmj + i Â n=0 vnj ( 3 . 2 ) Nu ﬁrs What remains is to derive the number of producer posts
Scoring Timelines z10 happening is given by R(d; rj)C(x; dj). This is the attention potential of a post at depth d for follower j. The attention potential of a cluster is the sum of attention potentials of the posts within it. Given a cluster containing xij posts, let there be a total of zij posts on the timeline above the ﬁrst post in this clus- ter. The attention potential of this cluster is given by the following function of the producer schedule: fij(x 0, x 1, . . . , xS 1) = C(xij; dj) xij Â k=1 R(zij + k; rj) ( 3 . 1 ) t We can calculate the number of posts above the ﬁrst a given cluster by the following formula: zij = i 1 Â m=0 xmj + i Â n=0 vnj What remains is to derive the number of producer po xij and competitor posts vij in the cluster from the pr ducer schedule X and aggregate competitor schedule respectively. The following relations hold: xij = x ((sj i) mod S) vij = c ((sj i) mod S)j d = z10 xij = x ((sj i) mod S) ( 3 . 3 vij = c ((sj i) mod S)j ( 3 . 4 The attention potential of the timeline constructed for a follower j is the sum of attention potentials of all the producer’s clusters, ÂS 1 i=0 fij( X , C , s, r, d). Each timeline may be weighted by a factor gj encoding a preference for speciﬁc characteristics, such as inﬂuence or tendency to retweet. If there are U followers in total, the attention potential of a schedule is the sum of attention potentials of all followers’ timelines: F( X , C , s, r, d) = S 1 Â i=0 U 1 Â j=0 fij( X , C , s, r, d) ( 3 . 5 Total attention potential = the broadcast sche 3.3 An Optimisation Problem An optimal schedule maximises attention potential by ﬁnding the perfect post timing and frequency conﬁguration satisfying both these balances. We can succinctly write this optimisation problem as the following nonlinear integer program: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, ( 3 . 6 ) The broadcast nonlinear int
timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over An Optimisation Problem ptimal schedule maximises attention potential by ng the perfect post timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over
timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over An Optimisation Problem ptimal schedule maximises attention potential by ng the perfect post timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over Balance Post volume & Visibility Irritation & Overload
timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over Nonlinear Knapsack Problem (Bretthauer, European Journal of Operational Research, ’02) An Optimisation Problem ptimal schedule maximises attention potential by ng the perfect post timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over
timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over Nonlinear Knapsack Problem (Bretthauer, European Journal of Operational Research, ’02) The objective is non-separable, non-concave, non- quasiconcave. An Optimisation Problem ptimal schedule maximises attention potential by ng the perfect post timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over
timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over Nonlinear Knapsack Problem (Bretthauer, European Journal of Operational Research, ’02) The objective is non-separable, non-concave, non- quasiconcave. Our Approach An Optimisation Problem ptimal schedule maximises attention potential by ng the perfect post timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over Essentially greedy hill-climbing, can be combined with random restarts. The form of the program reveals that it is an instance of the nonlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and The nonlinear knapsack algorithms and applicati Journal of Operational Res function is non-separable: it cannot be decomposed into a linear combination of functions gi(xi), separate for each dimension. Additionally, it is nonconcave in general over the reals. This places the problem outside the realm for which efﬁcient general algorithms have been devised. Greedily exploring the discrete state space is a common local optimisation strategy. For constrained integer pro- grams, greedy algorithms are typically applications of the method of marginal allocation. When adapted to our prob- lem, this corresponds to the following algorithm: Algorithm 1 : Marginal Allocation Input : Constraint N, initial solution X 0 2 ZS + begin Y X 0 D 0 repeat i argmaxk F( Y + e k) F( Y ) Y prev Y Y Y + e i D Y Y prev until D 0 or Ây2 Y y > N return Y prev end e k 2 ZS + is the kth unit v In each iteration, the al a single post to the slot w the maximum increase in function value. It termin adding a post to any slo the total intended posts does not improve the ob value. Essentially, this is a hil algorithm starting from solution and is guarante at an optimum, which m
all days. Start hour recorded after an inactive gap of 8 hours. Micro-level Experiment (c) 12AM 12AM 12PM 6AM 6PM SUN SAT FRI THU WED TUE MON TWEET SENT
or Effect, 2011) Measured as the probability of this follower reacting to the producer. Assume geometric cluster survival. Micro-level Experiment POST BY USER p IRRITATION (b) 0% 100% POST BY OTHER USER q’s TIMELINE
posts consumed per login = µ. Assume geometric user survival. µ = (1 - ρ) /ρ ⇒ ρ = 1 / (1 + µ) Micro-level Experiment ATTENTION (a) POST BY USER p 0% 100% POST BY OTHER USER q’s TIMELINE
PEAK Increased post frequency during times of peak follower activity. GRAVEYARD Post during the “graveyard shift”. (“The Late Night Infomercial Effect”, https://archive.is/ykNb6) SMART Marginal allocation with 20 random restarts. Number of posts in a slot constrained to < 10. Baseline Scheduling Strategies Micro-level Experiment Our Approach
Experiments on Twitter data a. Timeline information exchange. b. Attention potential objective function. c. Nonlinear knapsack problem formulation. Summary
Haewoon Kwak and Panos Kalnis. KDD 2015 (awaiting decision). Method and Apparatus for Scheduling Broadcasts in Social Networks. Emaad Ahmed Manzoor and Panos Kalnis. US Patent Application 62/118,570 (filed February 20, 2015). Publications
timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over Nonlinear Knapsack Problem (Bretthauer, European Journal of Operational Research, ’02) Objective function characteristics Non-separable, non-concave, non-quasiconcave. Notes Ignoring the depth factor (set ρj = 1) makes it separable. The resulting cluster attention potential is sigmoidal. The problem is then that of maximising a sum of sigmoids (Udell, unpublished, ’14) Approximation guarantees = ? An Optimisation Problem ptimal schedule maximises attention potential by ng the perfect post timing and frequency conﬁguration ying both these balances. We can succinctly write this misation problem as the following nonlinear integer am: maximise X F( X , C , s, r, d) subject to Â xi2 X xi N, X ✓ ZS + . ( 3 . 6 ) The broadcast scheduling problem as a nonlinear integer program. orm of the program reveals that it is an instance of onlinear knapsack problem1. Our speciﬁc objective 1 Kurt M Bretthauer and Bala Shetty. The nonlinear knapsack problem– algorithms and applications. European Journal of Operational Research, 2002 ion is non-separable: it cannot be decomposed into a r combination of functions gi(xi), separate for each nsion. Additionally, it is nonconcave in general over