Scheduling Meetings

I came across this problem in real life and thought it could be made into an interesting puzzle. I will enjoy seeing how my eventual solution could be improved!

Here's the situation.

• There are 290 themes that need to be discussed.
• Each theme requires discussion by some subset of 22 people.
• We need to schedule an efficient number of meetings so that:
• All themes get discussed once
• All the people who are impacted by the theme are in the discussion.

It would, of course, be possible to have 290 meetings with just the subset of people who are needed.

On the other extreme, we could get all 22 people together and have one giant meeting.

The themes will be quite quick to discuss (say 10 minutes), but we don't want to waste people's time sitting in meetings where many of the themes do not impact them.

So the plan will be to have a number of meetings with different subsets of the people and cover all the themes.

Part of the puzzle is to think about what efficient means here in real life terms. So if you have some ideas to improve the scoring then you can include that and I may update the scoring system in response. There's a qualitative aspect to this where neither of the two extremes above is satisfactory.

But for the sake of scoring, here's how we will do it:

1. Convening a meeting wastes 15 minutes of time per participant
2. Being in a meeting wastes 10 minutes per theme for each participant who is not involved with the theme
3. A meeting wastes 5 minutes per participant every 10 themes (coffee break)

So we want to minimize the waste.

By my reckoning:

• One big meeting of everyone wastes 55,180 minutes
• 290 separate meetings wastes 13,620 minutes
• The solution I found wastes 9,530 minutes

I calculated this with a python script. With one big meeting we have:

• 21 participants are required (I should have realized that person R doesn't meet anyone) and we are going to go through all 290 themes.
• The meeting wastes 15 mins * 21 people = 315 mins to convene (scoring rule 1)
• The first theme (1516) only engages 4 people (B, M, P, and T). So the other 17 people are wasting ten minutes each. So add 10 mins * 17 people = 170 mins to the waste (scoring rule 2)
• This will have to be repeated for the other 289 ideas. This is easy to calculate. There are 908 people themes (sum of the 1s in the below database), so it's (21 people * 290 themes)-908 involved people = 5182 uninvolved people. So total waste from scoring rule 2 is 51,820
• After 10 themes, 5 mins * 21 people are wasted, so over the course of the whole meeting there will be 29 such breaks which will waste 5 min * 21 people * 29 breaks = 3045 mins (scoring rule 3)
• So the total waste is 55,180mins

If we have 290 meetings with just the right people:

• It wastes 908 * 15 mins = 13,620 mins (scoring rule 1) to convene the meetings
• No time is wasted in the meetings (scoring rule 2)
• No meetings have more than 10 themes (scoring rule 3)
• Total waste: 13,620 mins

The themes are listed below along with the participants.

And in a .CSV format:

Theme,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V
1516,,1,,,,,,,,,,,1,,,1,,,,1,,
2339,,1,,,,1,,,,,,1,,,,,,,1,,1,
4134,1,1,,,,1,,1,,,,,,1,,,,,,,,
1567,,,,,,,,,,,1,1,,,,,,,1,1,1,
1526,1,,,,,,1,,,,,,1,,,1,,,,,,1
3718,1,,,,1,,1,,,,,,,,,,,,1,,,1
2791,1,,,,1,1,1,,,,,1,,,,,,,,,,
574,,,,,,1,1,1,,1,,1,,,,,,,,,,
1627,1,1,,1,1,,,,,,1,,,,,,,,,1,,
3246,,,,,,,,,,,1,1,1,,1,1,,,,1,,
3120,1,1,1,,,,,,,,1,,,,,,,,1,,1,
3893,,,1,1,1,,,1,,,1,1,,,,1,,,,,,
3265,,,1,,,,,1,,,1,1,1,,,1,,,,1,,
363,1,1,1,,,1,,,,,1,,,,,,,,1,,1,
1709,1,,,,,,1,1,,,,,1,,,,,,1,,1,1
3500,,,1,1,1,,1,1,,,1,,,,,1,,,,,,
3125,,,1,,,,1,1,,,1,1,1,,1,,,,,1,,
3428,1,1,1,,,,1,1,,,,,1,,1,,,,1,1,,
244,1,1,,,1,,,,,,1,1,1,,,1,,,1,1,,
2448,1,1,1,,,,,1,,,1,1,1,,,1,,,,1,,
3434,,,,,,,1,,,,,,,,,,,,,,1,1
1240,,1,,1,,,,,,,,,,,1,,,,,,,
476,,,,,,,,,,,,1,,,,,,,,1,1,1
3025,1,,,,,,,,,,1,,1,,,,,,,,1,
442,,1,,,,1,,,,,,,1,,,,,,1,,,
3400,1,1,,,,,,,,,1,1,,,,,,,,,1,
371,1,,,,,,1,1,,,,,,,1,,,,,,,1
1939,1,,,,,,1,1,,,,,1,,,,,,,,,1
4241,,,,,,,,,,,,,,,,,1,,1,,,
971,,,,,,,,,,,,,,,,,1,,,,1,
1117,,,,,,,,,,1,,,1,,,,,,,,,
258,,,,,,,,,,1,,,,1,,,,,,,,
611,,,,,,,1,,,,,1,,,,,,,,,,1
1703,,,,1,,,1,,,,1,,,,,,,,,,,
2136,,,,1,1,,,1,,,,,,,,,,,,,,
3307,,,,,,,,,,,1,,,,1,1,,,,,,
879,,1,,,,1,,,,,,,,,,,,,1,,1,
3404,,,,,,,1,,,,1,1,,,,,,,,,,1
1737,1,,1,,1,,,,1,,,,,,,,,,,,,
1736,1,,1,,1,,,,1,,,,,,,,,,,,,
1768,1,,,,1,,1,,,,,,1,,,,,,,,,
1447,,,,,,,,,,1,1,1,,,1,,,,,,,
3843,1,,,1,1,,1,,,,,,1,,,,,,,,,
3890,,,,1,1,,,,,,,,1,,,,,,,,,
3435,,,,1,1,,,,,,,,,,,1,,,,,,
956,,,,1,,,,,,,,1,1,1,,,,,,,,
1491,,,,1,1,1,1,,,,,1,1,1,,1,,,,,,
2216,,1,,,1,,,,,,,,,,,,,,,1,,
500,1,,,,1,,,,,,,,1,,,,,,,,,
962,1,,1,,1,,,1,,,1,,,,,,,,,,,
3308,1,1,1,,1,,,1,,,1,1,,,1,,,,,1,,
3218,1,1,1,,1,,,1,,,1,1,1,,1,,,,,1,,
4245,,,1,,,,,1,,,,,,,,,,,,,,
1759,,1,,,,,,,,,,,1,,,,,,,,1,
3999,1,,,,,,,,,,,,,,1,,,,,,1,
3934,,,,,1,,,,,,,,,,,,,,1,,1,
3624,,,,,,,,,,,1,1,,,1,,,,,1,,
2376,1,1,,,,,,,,,,,,,1,,,,,,1,
2866,,,,,1,,,,,,,,,,,,,,1,1,1,
33,,,,1,1,,,,,,1,1,,,,,,,,,,
3432,1,1,,,,,,,,,,1,1,,,,,,,1,,
985,1,,1,1,1,,,,,,1,,,,,,,,,,,
3176,1,,,1,1,,,,,,1,,,,,1,,,,,,
647,,,1,1,1,,,,,,1,1,,,,1,,,,,,
497,,,1,1,1,,,,,,1,1,,,,1,,,,,,
818,1,,1,1,1,,,,,,1,1,,,,1,,,,,,
2731,1,1,1,,,,,,,,1,1,1,,1,,,,,1,,
3374,1,1,1,,,,,1,,,1,1,1,,1,,,,,1,,
3429,1,1,,,1,,,,,,,,1,,1,,,,1,1,1,1
2218,1,1,1,,,,,1,,,1,1,1,,1,,,,1,1,,
3841,,,,,,,,,,,,,1,,,,,,,,,
1694,,,,,,,1,1,,,,,,,,,,,,,,
3342,,,,,,,1,1,,,,,,,,,,,,,,
3966,,,,,,,1,1,,,,,,,,,,,,,,
2670,,,,,,,1,1,,,,,,,,,,,,,,
3655,,,,,,,1,1,,,,,,,,,,,,,,
3967,,,,,,,1,1,,,,,,,,,,,,,,
92,1,,,,,,,1,,,,,,,,,,,,,,
3192,1,,,,,,,1,,,,,,,,,,,,,,
2514,,,,1,,,,,,,,,,,,1,,,,,,
137,,,,1,,,,,,,,,,,,1,,,,,,
138,,,,1,,,,,,,,,,,,1,,,,,,
147,,,,1,,,,,,,,,,,,1,,,,,,
3223,,,1,,,,,,,,,,,,,1,,,,,,
3443,,,1,,,,,,,,,,,,,1,,,,,,
996,,,1,1,,,,,,,,,,,,,,,,,,
3469,,,1,1,,,,,,,,,,,,,,,,,,
1087,1,,,,,,,,,,,,,,,,,,1,,,
915,1,,,,,,,,,,,,,,,,,,1,,,
2798,1,,,,,,,,,,,,,,,,,,1,,,
2369,1,1,,,,,,,,,,,,,,,,,,,,
2805,1,,,,,1,,,,,,,,,,,,,,,,
1661,1,,,,,1,,,,,,,,,,,,,,,,
2785,1,,,,,1,,,,,,,,,,,,,,,,
3001,1,,,,,,,,,,,,1,,,,,,,,,
3727,1,,,,,,,,,,,,1,,,,,,,,,
2103,,1,,,,1,,,,,,,,,,,,,,,,
2368,,1,,,,1,,,,,,,,,,,,,,,,
3831,,,,,,1,,,,,,,,,,,,,,,1,
2533,,,,1,1,,,,,,,,,,,,,,,,,
2449,,,,1,1,,,,,,,,,,,,,,,,,
2202,,,,,1,,,,,,1,,,,,,,,,,,
1730,,,,,1,,,,1,,,,,,,,,,,,,
1731,,,,1,,,,,1,,,,,,,,,,,,,
1738,,,,1,,,,,1,,,,,,,,,,,,,
1125,,,1,,1,,,,,,,,,,,,,,,,,
2236,,,,1,,,,1,,,,,,,,,,,,,,
2889,,,,1,,,,1,,,,,,,,,,,,,,
3659,,,,1,,,,1,,,,,,,,,,,,,,
3636,,,,,,,,1,,,,,,,,1,,,,,,
3657,,,,,,,,1,,,,,,,,1,,,,,,
718,,,,,,,,,,,1,,,,,1,,,,,,
1701,,,,,1,,1,,,,,,,,,,,,,,,
1697,,,,,1,,1,,,,,,,,,,,,,,,
1691,,,,,,,1,,,,,,1,,,,,,,,,
3163,,,,,,,1,,,,,,1,,,,,,,,,
3224,,,,,,,1,,,,,,,1,,,,,,,,
3272,,,,,,,1,,,,,,,1,,,,,,,,
3319,,,,,,,,,,,,,1,1,,,,,,,,
576,,,,,,,,1,,,,,1,,,,,,,,,
1900,,,,,,,,,,,,,,1,,,1,,,,,
2863,,,,,,,,,,,,,,1,,,1,,,,,
3851,,,,,,,,,,,,,,1,,,1,,,,,
3961,,,,,,,,,,,,,,,,,,,1,,1,
2446,,,,,,,1,,,,,,,,,,,,,,1,
3204,,,,,,,1,,,,,,,,,,,,,,1,
3302,,,,,,,,,,,,,,1,,1,,,,,,
3376,1,1,,,,,,,,,,,,,,,,,1,,,
285,1,1,,,1,,,,,,,,,,,,,,,,,
116,1,,,,1,,,,,,,,,,,,,,1,,,
2433,1,1,,,,,,,,,,,,,,,,,,,1,
307,1,1,,,,,,,,,,,,,,,,,,,1,
2447,1,1,,,,,,,,,,,,,,,,,,,1,
2462,1,1,,,,,,,,,,,,,,,,,,,1,
2480,1,1,,,,,,,,,,,,,,,,,,,1,
2491,1,1,,,,,,,,,,,,,,,,,,,1,
1922,,1,,,,1,,,,,,,,,,,,,,,1,
1298,1,,,,1,,,,,,1,,,,,,,,,,,
1640,,,,1,1,,,,,,1,,,,,,,,,,,
2773,,,,1,1,,,,,,1,,,,,,,,,,,
3041,,,,1,1,,,,,,1,,,,,,,,,,,
2993,1,,,1,1,,,,,,,,,,,,,,,,,
1192,,,1,1,1,,,,,,,,,,,,,,,,,
1734,,,1,1,,,,,1,,,,,,,,,,,,,
1741,,,1,1,,,,,1,,,,,,,,,,,,,
105,,,1,1,,,,,1,,,,,,,,,,,,,
216,,,,1,,,,1,,,1,,,,,,,,,,,
2638,,,,1,,,,1,,,1,,,,,,,,,,,
1880,,,,,,,1,,,,,,1,1,,,,,,,,
1903,,,,,,,1,1,,,,,,,,,1,,,,,
2390,,,,,,,1,1,,,,,,,,,1,,,,,
2913,,,,,,,,,,,,,1,1,,,1,,,,,
3119,1,,,,,,,,,,,1,,,,,,,,,1,
836,,,,,,,,,,,,1,,,,,,,1,,1,
2378,,,,,1,,,,,,1,1,,,,,,,,,,
2199,1,1,,,1,,,,,,,,,,,,,,1,,,
4242,,,,,,,1,1,,,,,,1,,,1,,,,,
2767,1,1,,,,,,,,,,1,,,,,,,,,1,
3014,1,1,,,,,1,,,,,,,,,,,,,,1,
477,1,,,,1,,,,,,1,,,,,1,,,,,,
1350,,,,,,,,,,,1,1,,1,,1,,,,,,
2047,,,,,,,,,,,1,1,,1,,1,,,,,,
3503,,,,,,,,,,,1,1,,1,,1,,,,,,
4207,,,,,,,,,,,1,1,,1,,1,,,,,,
303,1,1,,,,,,,,,,1,,,,,,,1,,1,
318,,,,,1,,,,,,1,1,,1,,1,,,,,,
475,,,,,,,,,,,,1,,,,,,,,,,
2600,,,,,,,,,,,,1,,,,,,,,,,
4159,,,,,,,,,,,,1,,,,,,,,,,
4195,,,,,,,,,,,,1,,,,,,,,,,
2201,,,,,1,,,,,,,1,,,,,,,,,,
454,,,,,1,,,,,,,1,,,,,,,,,,
1025,,,,,1,,,,,,,1,,,,,,,,,,
1302,,,,,1,,,,,,,1,,,,,,,,,,
4346,,,,,1,,,,,,,1,,,,,,,,,,
485,,,,,,,,,,,1,1,,,,,,,,,,
589,,,,,,,,,,,1,1,,,,,,,,,,
681,,,,,,,,,,,1,1,,,,,,,,,,
2513,,,,,,,,,,,1,1,,,,,,,,,,
316,,,,,,,,,,,1,1,,,,,,,,,,
3066,,,,,,,,,,,1,1,,,,,,,,,,
2792,1,,,,,,,,,,,1,,,,,,,,,,
3191,1,,,,,,,,,,,1,,,,,,,,,,
3506,1,,,,,,,,,,1,,,,,,,,,,,
1245,,,,,,,,,,1,1,,,,,,,,,,,
1121,,,1,,,,,,,,,1,,,,,,,,,,
3219,,1,,,,,,,,,,1,,,,,,,,,1,
284,,,,,,,,,,1,1,1,,,,,,,,,,
295,,,,,,,,,,1,1,1,,,,,,,,,,
1242,,,,,,,,,,1,1,1,,,,,,,,,,
1439,,,,,,,,,,1,1,1,,,,,,,,,,
721,,,,,,,,,,,1,1,,,,1,,,,,,
1160,,,1,,,,,,,,1,,,,,1,,,,,,
289,,,1,,,,,,,,1,,,,,1,,,,,,
3466,,,1,,,,,,,,1,,,,,1,,,,,,
1128,,,,1,,,,,,,1,1,,,,,,,,,,
2763,,1,,,,,,,,,1,1,,,,,,,,,1,
3095,,1,,,,,,,,,1,1,,,,,,,,,1,
3096,,1,,,,,,,,,1,1,,,,,,,,,1,
2946,,,1,1,,,,,,,1,,,,,1,,,,,,
2947,,,1,1,,,,,,,1,,,,,1,,,,,,
2949,,,1,1,,,,,,,1,,,,,1,,,,,,
1693,,,,,,,1,,,,,1,,,,,,,,,,
1878,,,,,,,1,,,,,1,,,,,,,,,,
3704,,,,,,,1,,,,,1,,,,,,,,,,
3830,,,,,,,1,,,,,1,,,,,,,,,,
4243,,,,,,,1,,,,,1,,,,,,,,,,
1294,,,,,,,,,,1,,1,,,,,,,,,,
2741,,,,,,,,,,1,,1,,,,,,,,,,
2855,,,,,,,,,,1,,1,,,,,,,,,,
3142,,,,,,,,,,1,,1,,,,,,,,,,
3809,,,,,,,,,,1,,1,,,,,,,,,,
359,1,,,,,,,,,,,,,,,,,,,,,1
1030,1,,,,,,,,,,,,,,,,,,1,,,1
1717,1,,,,,,,,,,,,,,,,,,1,,,1
3000,1,,,,,,1,,,,,,,,,,,,,,,1
2062,1,,,,,,1,,,,,,,,,,,,,,,1
3359,1,,,,,,1,,,,,,,,,,,,1,,,1
746,1,,,,,,1,,,,,,,,,,,,1,,,1
1379,1,,,,,,1,,,,,,,,,,,,1,,,1
3360,1,,,,,,1,,,,,,,,,,,,1,,,1
4036,1,,,,,,1,,,,,,,,,,,,1,,,1
1474,,,,,1,,,,,,,,,,,,,,,,,
730,1,,,,,,,,,,,,,,,,,,,,,
141,1,,,,,,,,,,,,,,,,,,,,,
1542,1,,,,,,,,,,,,,,,,,,,,,
1662,1,,,,,,,,,,,,,,,,,,,,,
1747,1,,,,,,,,,,,,,,,,,,,,,
2171,1,,,,,,,,,,,,,,,,,,,,,
115,1,,,,1,,,,,,,,,,,,,,,,,
3129,1,,,,1,,,,,,,,,,,,,,,,,
3274,1,,,,1,,,,,,,,,,,,,,,,,
1368,1,,,,1,,,,,,,,,,,,,,,,,
58,,,,1,,,1,,,,,,,,,,,,,,,
35,,,,1,,,1,,,,,,,,,,,,,,,
38,,,,1,,,1,,,,,,,,,,,,,,,
39,,,,1,,,1,,,,,,,,,,,,,,,
47,,,,1,,,1,,,,,,,,,,,,,,,
49,,,,1,,,1,,,,,,,,,,,,,,,
62,,,,1,,,1,,,,,,,,,,,,,,,
68,,,,1,,,1,,,,,,,,,,,,,,,
73,,,,1,,,1,,,,,,,,,,,,,,,
91,,,,1,,,1,,,,,,,,,,,,,,,
93,,,,1,,,1,,,,,,,,,,,,,,,
3169,,,,1,,,1,,,,,,,,,,,,,,,
631,,,,,,,1,,,,,,,,,,,,,,,1
1710,,,,,,,,1,,,,,,,,,,,,,,1
2312,,,,,,,1,,,,,,,,,,,,,,,1
2313,,,,,,,1,,,,,,,,,,,,,,,1
4052,,,,,,,1,,,,,,,,,,,,,,,1
2980,,,,,,,,1,,,,,,,,,,,,,,1
2982,,,,,,,,1,,,,,,,,,,,,,,1
4305,,,,,,,,1,,,,,,,,,,,,,,1
761,,,,,,,1,1,,,,,,,,,,,,,,1
824,,,,,,,1,1,,,,,,,,,,,,,,1
1532,,,,,,,1,1,,,,,,,,,,,,,,1
1942,,,,,,,1,1,,,,,,,,,,,,,,1
3239,,,,,,,,1,,,,,,,,,,,1,,,1
3877,1,,,,,,1,1,,,,,,,,,,,,,,1
791,1,,,,,,,1,,,,,,,,,,,1,,,1
832,1,,,,,,,1,,,,,,,,,,,1,,,1
1976,1,,,,,,1,1,,,,,,,,,,,,,,1
2067,1,,,,,,,1,,,,,,,,,,,1,,,1
3231,1,,,,,,1,1,,,,,,,,,,,,,,1
3338,1,,,,,,1,1,,,,,,,,,,,,,,1
770,1,,,,,,1,1,,,,,,,,,,,1,,,1
792,1,,,,,,1,1,,,,,,,,,,,1,,,1
71,1,,,,,,1,1,,,,,,,,,,,1,,,1
844,1,,,,,,1,1,,,,,,,,,,,1,,,1
1530,1,,,,,,1,1,,,,,,,,,,,1,,,1
2060,1,,,,,,1,1,,,,,,,,,,,1,,,1
3394,1,,,,,,1,1,,,,,,,,,,,1,,,1
1050,,,,,,,,,,,1,,,,,,,,,,,
3591,,,,,,,,,,,1,,,,,,,,,,,
2536,,,1,,,,,,,,1,,,,,,,,,,,
3020,,,1,,,,,,,,1,,,,,,,,,,,
3182,,,1,,,,,,,,1,,,,,,,,,,,
3183,,,1,,,,,,,,1,,,,,,,,,,,
3553,,,1,,,,,,,,1,,,,,,,,,,,
3564,,,1,,,,,,,,1,,,,,,,,,,,
3629,,,1,,,,,,,,1,,,,,,,,,,,
4082,,,1,,,,,,,,1,,,,,,,,,,,
3,,,1,,,,,,,,1,,,,,,,,,,,
676,,,1,,,,,,,,1,,,,,,,,,,,
698,,,1,,,,,,,,1,,,,,,,,,,,
909,,,1,,,,,,,,1,,,,,,,,,,,
1926,,,1,,,,,,,,1,,,,,,,,,,,
1499,,,1,,,,,,,,1,,,,,,,,,,,
2912,,,1,,,,,,,,1,,,,,,,,,,,
3453,,,1,,,,,,,,1,,,,,,,,,,,

• @Oray, I've added a detailed calculation. Hopefully, it is a little clearer. The essence of the problem is not complicated, but it's a little difficult to explain clearly. Let me know if it's clear or not. This problem would suit you! Feb 27, 2018 at 18:13
• I think it is clear to me, i just want to understand why there is theme with only one person involved.. Feb 28, 2018 at 9:02
• @Untitpoi Example: Theme 1050 and 3591. Right? Feb 28, 2018 at 11:59
• There are a few with just one person. They just need to be scheduled with at least that one person. It doesn't make much sense in real life, but the puzzle still works. Feb 28, 2018 at 20:06
• This almost sounds like something you could throw on code-golf as well. It'd be interesting to see what those folks come up with. Mar 9, 2018 at 15:25

2 Answers

Via mixed integer linear programming, I found a solution that uses $$76$$ meetings and has a total waste of $$4230 + 1900 + 45 = 6175$$:

1 : 1516 3432
2 : 303 363 2339 3120 3400
3 : 4134
4 : 476 836 1567
5 : 371 1526 1939
6 : 1709 3428 3429 3718
7 : 574 2791
8 : 1627 2216
9 : 244 2218 2448 2731 3218 3246 3265 3308 3374
10 : 477 497 647 818 962 985 3176 3893
11 : 3500
12 : 3125
13 : 2446 3204 3434
14 : 1240
15 : 3025
16 : 442 879 1759
17 : 971 3961 4241
18 : 1117
19 : 258
20 : 611 3404
21 : 1703
22 : 2136 3435
23 : 718 3307
24 : 1730 1736 1737
25 : 500 1768 2993 3843 3890
26 : 1447 3624
27 : 956 1491
28 : 4245
29 : 2376 3999
30 : 2866 3934
31 : 33 1128 1640 2378 2773 3041
32 : 3001 3727 3841
33 : 1694 2670 3342 3655 3966 3967
34 : 92 3192
35 : 137 138 147 2514
36 : 3223 3443
37 : 105 996 1731 1734 1738 1741 3469
38 : 915 1087 2798
39 : 307 2369 2433 2447 2462 2480 2491
40 : 1661 2785 2805
41 : 1922 2103 2368 3831
42 : 1125 1192 2449 2533
43 : 1298 2202 3506
44 : 2236 2889 3659
45 : 3636 3657
46 : 1697 1701
47 : 1691 1880 3163 3224 3272
48 : 1900 2863 2913 3319 3851
49 : 576
50 : 721 1350 2047 3302 3503 4207
51 : 116 285 2199 3376
52 : 216 2638
53 : 1903 2390 4242
54 : 2767 3014 3119
55 : 318
56 : 475 2600 4159 4195
57 : 454 1025 1302 2201 4346
58 : 316 485 589 681 2513 3066
59 : 2792 3191
60 : 284 295 1242 1245 1439
61 : 1121
62 : 2763 3095 3096 3219
63 : 289 1160 2946 2947 2949 3466
64 : 1693 1878 3704 3830 4243
65 : 1294 2741 2855 3142 3809
66 : 359
67 : 746 1030 1379 1717 2062 3000 3359 3360 4036
68 : 115 1368 1474 3129 3274
69 : 141 730 1542 1662 1747 2171
70 : 35 38 39 47 49 58 62 68 73 91 93 3169
71 : 631 2312 2313 4052
72 : 1710 2980 2982 4305
73 : 761 824 1532 1942
74 : 71 770 791 792 832 844 1530 1976 2060 2067 3231 3239 3338 3394 3877
75 : 1050 3591
76 : 3 676 698 909 1499 1926 2536 2912 3020 3182 3183 3453 3553 3564 3629 4082


There are many sets of themes that require the exact same sets of participants. For example, the last 16 themes listed all have the same set of participants. Obviously, it would be more efficient to combine each set into a single meeting.

What I would recommend would be to enumerate all the distinct sets of participants required, and count how many themes correspond to each set. Then work with the sets with the smallest numbers of related themes. For those sets, find the ones with the highest overlaps between them. It should be obvious where meetings could be combined that would have a few employees wasting time because they are not interested int theme, but a larger number would benefit from having fewer meetings to attend.