Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets...
Transcript of Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets...
![Page 1: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/1.jpg)
Covering Small Independent Sets and Separators(with Applications)
Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Roohani Sharma, Meirav Zehavi
Recent Advances in AlgorithmsNISER BhubaneswarFebruary 10th, 2019
![Page 2: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/2.jpg)
Table of Contents
1. Introduction/Literature2. Tool 13. Tool 24. Applications5. Concluding Remarks
![Page 3: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/3.jpg)
Table of Contents
1. A Combinatorial Question - Tool 12. Applications of Tool 13. Stumble upon Literature4. Resolve some open questions using Tool 15. Need for the design of Tool 26. Design of Tool 27. Concluding Remarks
Behind the
scenes
![Page 4: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/4.jpg)
n edges
F (G,1) 2 F (G,2) n + 2
F (G,3)
F (G,k)
Can the dependence of k be removed from the exponent on n? ?
2
✓n
2
◆+ 2
2k✓n
k
◆+ 2
⇡ nO(k)
Given a graph G and an integer k, an independent set covering family (ISCF) for (G,k) is a family of independent sets of G, say F (G,k), such that for any independent set X of G of size at most k, there exists Y ⊆ F (G,k), such that X ⊆ Y.
![Page 5: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/5.jpg)
Independent Set Covering Lemma (ISCL)If G is d-degenerate, then for any k, there is an ISCF for (G,k) of size
2O(k log kd) log n.
In fact, such a family can be found in 2O(k log kd) (n+m). log n time.
Tool 1:
![Page 6: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/6.jpg)
Towards Randomized Independent Set Covering Lemma
Given: A d-degenerate graph G, an integer kOutput: An independent set Y such that for any independent set X of size at most k, the Pr( ) �
<latexit sha1_base64="Ko0Gjx7n2SdQ6Ni9VKlPO/VDYMg=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cKthbaUDbbSbt0s4m7G6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvSATXxnW/ndLK6tr6RnmzsrW9s7tX3T9o6zhVDFssFrHqBFSj4BJbhhuBnUQhjQKBD8H4JvcfnlBpHst7M0nQj+hQ8pAzanKpN8THfrXm1t0ZyDLxClKDAs1+9as3iFkaoTRMUK27npsYP6PKcCZwWumlGhPKxnSIXUsljVD72ezWKTmxyoCEsbIlDZmpvycyGmk9iQLbGVEz0oteLv7ndVMTXvkZl0lqULL5ojAVxMQkf5wMuEJmxMQSyhS3txI2oooyY+Op2BC8xZeXSfus7p3X3buLWuO6iKMMR3AMp+DBJTTgFprQAgYjeIZXeHMi58V5dz7mrSWnmDmEP3A+fwAODY49</latexit>
X ✓ Y<latexit sha1_base64="D/2I2LpEzfh36C0kV3Lt63nfKNQ=">AAAB9HicdVDJSgNBEO2JW4xb1KOXxiB4Cj2J2W5BLx4jGBNJhtDTqUma9Czp7gmEId/hxYMiXv0Yb/6NnUVQ0QcFj/eqqKrnRoIrTciHlVpb39jcSm9ndnb39g+yh0d3KowlgyYLRSjbLlUgeABNzbWAdiSB+q6Alju6mvutCUjFw+BWTyNwfDoIuMcZ1UZy2rirYleBhjG+72VzJE/KpVqRYJIvEbtSqxlCSLlaLGDbkDlyaIVGL/ve7Ycs9iHQTFClOjaJtJNQqTkTMMt0YwURZSM6gI6hAfVBOcni6Bk+M0ofe6E0FWi8UL9PJNRXauq7ptOneqh+e3PxL68Ta6/qJDyIYg0BWy7yYoF1iOcJ4D6XwLSYGkKZ5OZWzIZUUqZNThkTwten+H9yV8jbxTy5ucjVL1dxpNEJOkXnyEYVVEfXqIGaiKExekBP6NmaWI/Wi/W6bE1Zq5lj9APW2yea75IA</latexit>
1
2k(d+1)<latexit sha1_base64="W5aNk24gFMkkx6MtpUXh3us0i7w=">AAAB/XicdVDJSgNBEO2JW4xbXG5eGoMQEUJPYrZb0IvHCGaBZAw9PT1Jk56F7h4hDoO/4sWDIl79D2/+jZ1FUNEHBY/3qqiqZ4ecSYXQh5FaWl5ZXUuvZzY2t7Z3srt7bRlEgtAWCXggujaWlDOfthRTnHZDQbFnc9qxxxdTv3NLhWSBf60mIbU8PPSZywhWWhpkD/quwCQ2k7h4E4/zzql5kiSDbA4VUKVcLyGICmVkVut1TRCq1EpFaGoyRQ4s0Bxk3/tOQCKP+opwLGXPRKGyYiwUI5wmmX4kaYjJGA9pT1Mfe1Ra8ez6BB5rxYFuIHT5Cs7U7xMx9qSceLbu9LAayd/eVPzL60XKrVkx88NIUZ/MF7kRhyqA0yigwwQlik80wUQwfSskI6zjUDqwjA7h61P4P2kXC2apgK7Oco3zRRxpcAiOQB6YoAoa4BI0QQsQcAcewBN4Nu6NR+PFeJ23pozFzD74AePtE+DHlNs=</latexit>
Goal
For each vertex v ∈ V(G), colour it either red or blue, uniformly at random.Experiment
Graph G
![Page 7: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/7.jpg)
Towards Randomized Independent Set Covering Lemma
Given: A d-degenerate graph G, an integer kOutput: An independent set Y such that for any independent set X of size at most k, the Pr( ) �
<latexit sha1_base64="Ko0Gjx7n2SdQ6Ni9VKlPO/VDYMg=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cKthbaUDbbSbt0s4m7G6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvSATXxnW/ndLK6tr6RnmzsrW9s7tX3T9o6zhVDFssFrHqBFSj4BJbhhuBnUQhjQKBD8H4JvcfnlBpHst7M0nQj+hQ8pAzanKpN8THfrXm1t0ZyDLxClKDAs1+9as3iFkaoTRMUK27npsYP6PKcCZwWumlGhPKxnSIXUsljVD72ezWKTmxyoCEsbIlDZmpvycyGmk9iQLbGVEz0oteLv7ndVMTXvkZl0lqULL5ojAVxMQkf5wMuEJmxMQSyhS3txI2oooyY+Op2BC8xZeXSfus7p3X3buLWuO6iKMMR3AMp+DBJTTgFprQAgYjeIZXeHMi58V5dz7mrSWnmDmEP3A+fwAODY49</latexit>
X ✓ Y<latexit sha1_base64="D/2I2LpEzfh36C0kV3Lt63nfKNQ=">AAAB9HicdVDJSgNBEO2JW4xb1KOXxiB4Cj2J2W5BLx4jGBNJhtDTqUma9Czp7gmEId/hxYMiXv0Yb/6NnUVQ0QcFj/eqqKrnRoIrTciHlVpb39jcSm9ndnb39g+yh0d3KowlgyYLRSjbLlUgeABNzbWAdiSB+q6Alju6mvutCUjFw+BWTyNwfDoIuMcZ1UZy2rirYleBhjG+72VzJE/KpVqRYJIvEbtSqxlCSLlaLGDbkDlyaIVGL/ve7Ycs9iHQTFClOjaJtJNQqTkTMMt0YwURZSM6gI6hAfVBOcni6Bk+M0ofe6E0FWi8UL9PJNRXauq7ptOneqh+e3PxL68Ta6/qJDyIYg0BWy7yYoF1iOcJ4D6XwLSYGkKZ5OZWzIZUUqZNThkTwten+H9yV8jbxTy5ucjVL1dxpNEJOkXnyEYVVEfXqIGaiKExekBP6NmaWI/Wi/W6bE1Zq5lj9APW2yea75IA</latexit>
1
2k(d+1)<latexit sha1_base64="W5aNk24gFMkkx6MtpUXh3us0i7w=">AAAB/XicdVDJSgNBEO2JW4xbXG5eGoMQEUJPYrZb0IvHCGaBZAw9PT1Jk56F7h4hDoO/4sWDIl79D2/+jZ1FUNEHBY/3qqiqZ4ecSYXQh5FaWl5ZXUuvZzY2t7Z3srt7bRlEgtAWCXggujaWlDOfthRTnHZDQbFnc9qxxxdTv3NLhWSBf60mIbU8PPSZywhWWhpkD/quwCQ2k7h4E4/zzql5kiSDbA4VUKVcLyGICmVkVut1TRCq1EpFaGoyRQ4s0Bxk3/tOQCKP+opwLGXPRKGyYiwUI5wmmX4kaYjJGA9pT1Mfe1Ra8ez6BB5rxYFuIHT5Cs7U7xMx9qSceLbu9LAayd/eVPzL60XKrVkx88NIUZ/MF7kRhyqA0yigwwQlik80wUQwfSskI6zjUDqwjA7h61P4P2kXC2apgK7Oco3zRRxpcAiOQB6YoAoa4BI0QQsQcAcewBN4Nu6NR+PFeJ23pozFzD74AePtE+DHlNs=</latexit>
Goal
For each vertex v ∈ V(G), colour it either red or blue, uniformly at random.Experiment
Graph G
![Page 8: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/8.jpg)
Graph G
RED = set of all vertices that are coloured redBLUE = set of all vertices that are coloured blue
GOOD EVENT = RED contains all vertices of X and none of its forward neighbours (i.e. all the forward neighbours of X are in BLUE)
Claim : If GOOD EVENT happens, then X ⊆ IND_RED
IND_RED= {v: v ∈ RED and all its forward neighbours in BLUE}
Graph G
Pr(GOOD EVENT) � 1
2|X|1
2|Nf (X)| �1
2k(d+1)
![Page 9: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/9.jpg)
Towards Randomized Independent Set Covering Lemma
Given: A d-degenerate graph G, an integer kOutput: An independent set Y such that for any independent set X of size at most k, the Pr( ) �
<latexit sha1_base64="Ko0Gjx7n2SdQ6Ni9VKlPO/VDYMg=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cKthbaUDbbSbt0s4m7G6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvSATXxnW/ndLK6tr6RnmzsrW9s7tX3T9o6zhVDFssFrHqBFSj4BJbhhuBnUQhjQKBD8H4JvcfnlBpHst7M0nQj+hQ8pAzanKpN8THfrXm1t0ZyDLxClKDAs1+9as3iFkaoTRMUK27npsYP6PKcCZwWumlGhPKxnSIXUsljVD72ezWKTmxyoCEsbIlDZmpvycyGmk9iQLbGVEz0oteLv7ndVMTXvkZl0lqULL5ojAVxMQkf5wMuEJmxMQSyhS3txI2oooyY+Op2BC8xZeXSfus7p3X3buLWuO6iKMMR3AMp+DBJTTgFprQAgYjeIZXeHMi58V5dz7mrSWnmDmEP3A+fwAODY49</latexit>
X ✓ Y<latexit sha1_base64="D/2I2LpEzfh36C0kV3Lt63nfKNQ=">AAAB9HicdVDJSgNBEO2JW4xb1KOXxiB4Cj2J2W5BLx4jGBNJhtDTqUma9Czp7gmEId/hxYMiXv0Yb/6NnUVQ0QcFj/eqqKrnRoIrTciHlVpb39jcSm9ndnb39g+yh0d3KowlgyYLRSjbLlUgeABNzbWAdiSB+q6Alju6mvutCUjFw+BWTyNwfDoIuMcZ1UZy2rirYleBhjG+72VzJE/KpVqRYJIvEbtSqxlCSLlaLGDbkDlyaIVGL/ve7Ycs9iHQTFClOjaJtJNQqTkTMMt0YwURZSM6gI6hAfVBOcni6Bk+M0ofe6E0FWi8UL9PJNRXauq7ptOneqh+e3PxL68Ta6/qJDyIYg0BWy7yYoF1iOcJ4D6XwLSYGkKZ5OZWzIZUUqZNThkTwten+H9yV8jbxTy5ucjVL1dxpNEJOkXnyEYVVEfXqIGaiKExekBP6NmaWI/Wi/W6bE1Zq5lj9APW2yea75IA</latexit>
1
2k(d+1)<latexit sha1_base64="W5aNk24gFMkkx6MtpUXh3us0i7w=">AAAB/XicdVDJSgNBEO2JW4xbXG5eGoMQEUJPYrZb0IvHCGaBZAw9PT1Jk56F7h4hDoO/4sWDIl79D2/+jZ1FUNEHBY/3qqiqZ4ecSYXQh5FaWl5ZXUuvZzY2t7Z3srt7bRlEgtAWCXggujaWlDOfthRTnHZDQbFnc9qxxxdTv3NLhWSBf60mIbU8PPSZywhWWhpkD/quwCQ2k7h4E4/zzql5kiSDbA4VUKVcLyGICmVkVut1TRCq1EpFaGoyRQ4s0Bxk3/tOQCKP+opwLGXPRKGyYiwUI5wmmX4kaYjJGA9pT1Mfe1Ra8ez6BB5rxYFuIHT5Cs7U7xMx9qSceLbu9LAayd/eVPzL60XKrVkx88NIUZ/MF7kRhyqA0yigwwQlik80wUQwfSskI6zjUDqwjA7h61P4P2kXC2apgK7Oco3zRRxpcAiOQB6YoAoa4BI0QQsQcAcewBN4Nu6NR+PFeJ23pozFzD74AePtE+DHlNs=</latexit>
Goal
For each vertex v ∈ V(G), colour it either red or blue, uniformly at random.Experiment
![Page 10: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/10.jpg)
Towards Randomized Independent Set Covering Lemma
Given: A d-degenerate graph G, an integer kOutput: An independent set Y such that for any independent set X of size at most k, the Pr( ) �
<latexit sha1_base64="Ko0Gjx7n2SdQ6Ni9VKlPO/VDYMg=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cKthbaUDbbSbt0s4m7G6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvSATXxnW/ndLK6tr6RnmzsrW9s7tX3T9o6zhVDFssFrHqBFSj4BJbhhuBnUQhjQKBD8H4JvcfnlBpHst7M0nQj+hQ8pAzanKpN8THfrXm1t0ZyDLxClKDAs1+9as3iFkaoTRMUK27npsYP6PKcCZwWumlGhPKxnSIXUsljVD72ezWKTmxyoCEsbIlDZmpvycyGmk9iQLbGVEz0oteLv7ndVMTXvkZl0lqULL5ojAVxMQkf5wMuEJmxMQSyhS3txI2oooyY+Op2BC8xZeXSfus7p3X3buLWuO6iKMMR3AMp+DBJTTgFprQAgYjeIZXeHMi58V5dz7mrSWnmDmEP3A+fwAODY49</latexit>
X ✓ Y<latexit sha1_base64="D/2I2LpEzfh36C0kV3Lt63nfKNQ=">AAAB9HicdVDJSgNBEO2JW4xb1KOXxiB4Cj2J2W5BLx4jGBNJhtDTqUma9Czp7gmEId/hxYMiXv0Yb/6NnUVQ0QcFj/eqqKrnRoIrTciHlVpb39jcSm9ndnb39g+yh0d3KowlgyYLRSjbLlUgeABNzbWAdiSB+q6Alju6mvutCUjFw+BWTyNwfDoIuMcZ1UZy2rirYleBhjG+72VzJE/KpVqRYJIvEbtSqxlCSLlaLGDbkDlyaIVGL/ve7Ycs9iHQTFClOjaJtJNQqTkTMMt0YwURZSM6gI6hAfVBOcni6Bk+M0ofe6E0FWi8UL9PJNRXauq7ptOneqh+e3PxL68Ta6/qJDyIYg0BWy7yYoF1iOcJ4D6XwLSYGkKZ5OZWzIZUUqZNThkTwten+H9yV8jbxTy5ucjVL1dxpNEJOkXnyEYVVEfXqIGaiKExekBP6NmaWI/Wi/W6bE1Zq5lj9APW2yea75IA</latexit>
1
2k(d+1)<latexit sha1_base64="W5aNk24gFMkkx6MtpUXh3us0i7w=">AAAB/XicdVDJSgNBEO2JW4xbXG5eGoMQEUJPYrZb0IvHCGaBZAw9PT1Jk56F7h4hDoO/4sWDIl79D2/+jZ1FUNEHBY/3qqiqZ4ecSYXQh5FaWl5ZXUuvZzY2t7Z3srt7bRlEgtAWCXggujaWlDOfthRTnHZDQbFnc9qxxxdTv3NLhWSBf60mIbU8PPSZywhWWhpkD/quwCQ2k7h4E4/zzql5kiSDbA4VUKVcLyGICmVkVut1TRCq1EpFaGoyRQ4s0Bxk3/tOQCKP+opwLGXPRKGyYiwUI5wmmX4kaYjJGA9pT1Mfe1Ra8ez6BB5rxYFuIHT5Cs7U7xMx9qSceLbu9LAayd/eVPzL60XKrVkx88NIUZ/MF7kRhyqA0yigwwQlik80wUQwfSskI6zjUDqwjA7h61P4P2kXC2apgK7Oco3zRRxpcAiOQB6YoAoa4BI0QQsQcAcewBN4Nu6NR+PFeJ23pozFzD74AePtE+DHlNs=</latexit>
Goal
For each vertex v ∈ V(G), colour it either red or blue, uniformly at random.Experiment
color v red with probability
color v blue with probability
1
d+ 1
d
d+ 1
1
2O(k log kd)
![Page 11: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/11.jpg)
Randomized Independent Set Covering Lemma
Randomized Independent Set Covering Lemma (ISCL)
There is an algorithm that given a d-degenerate graph G and an integer k, outputs a family F (G,k) such that:• F (G,k) is an ISCF for (G,k) with probability at least 1- 1/n,
• |F (G,k)| ≤ 2O(k log kd) log n• Running time of the algorithm is O(|F (G,k)| . (n+m)).
Given: A d-degenerate graph G, an integer kOutput: An independent set Y such that for any independent set X of size at most k, the Pr( ) �
<latexit sha1_base64="Ko0Gjx7n2SdQ6Ni9VKlPO/VDYMg=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cKthbaUDbbSbt0s4m7G6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvSATXxnW/ndLK6tr6RnmzsrW9s7tX3T9o6zhVDFssFrHqBFSj4BJbhhuBnUQhjQKBD8H4JvcfnlBpHst7M0nQj+hQ8pAzanKpN8THfrXm1t0ZyDLxClKDAs1+9as3iFkaoTRMUK27npsYP6PKcCZwWumlGhPKxnSIXUsljVD72ezWKTmxyoCEsbIlDZmpvycyGmk9iQLbGVEz0oteLv7ndVMTXvkZl0lqULL5ojAVxMQkf5wMuEJmxMQSyhS3txI2oooyY+Op2BC8xZeXSfus7p3X3buLWuO6iKMMR3AMp+DBJTTgFprQAgYjeIZXeHMi58V5dz7mrSWnmDmEP3A+fwAODY49</latexit>
X ✓ Y<latexit sha1_base64="D/2I2LpEzfh36C0kV3Lt63nfKNQ=">AAAB9HicdVDJSgNBEO2JW4xb1KOXxiB4Cj2J2W5BLx4jGBNJhtDTqUma9Czp7gmEId/hxYMiXv0Yb/6NnUVQ0QcFj/eqqKrnRoIrTciHlVpb39jcSm9ndnb39g+yh0d3KowlgyYLRSjbLlUgeABNzbWAdiSB+q6Alju6mvutCUjFw+BWTyNwfDoIuMcZ1UZy2rirYleBhjG+72VzJE/KpVqRYJIvEbtSqxlCSLlaLGDbkDlyaIVGL/ve7Ycs9iHQTFClOjaJtJNQqTkTMMt0YwURZSM6gI6hAfVBOcni6Bk+M0ofe6E0FWi8UL9PJNRXauq7ptOneqh+e3PxL68Ta6/qJDyIYg0BWy7yYoF1iOcJ4D6XwLSYGkKZ5OZWzIZUUqZNThkTwten+H9yV8jbxTy5ucjVL1dxpNEJOkXnyEYVVEfXqIGaiKExekBP6NmaWI/Wi/W6bE1Zq5lj9APW2yea75IA</latexit>
1
2O(k log kd)
![Page 12: Covering Small Independent Sets and Separators (with ...aritra/...Bhubaneshwar10022019.pdf · Sets and Separators (with Applications) Daniel Lokshtanov, Fahad Panolan, Saket Saurabh,](https://reader034.fdocuments.us/reader034/viewer/2022051915/60064cb64571c11479122eb2/html5/thumbnails/12.jpg)
Deterministic Independent Set Covering Lemma
|U| = nFamily of functions {f1,…,ft}
fi : U -> [q]For each S ⊆ U, |S| ≤ l,
there exists some fi such that fi is injective on S
U
fi
1
2
q
(n,l,q)-perfect hash family, (q ≥ l)
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Deterministic Independent Set Covering Lemma (n,l,q)-perfect hash family, (q ≥ l)
|U| = nFamily of functions {f1,…,ft}
fi : U -> [q]For each S ⊆ U, |S| ≤ l,
there exists some fi such that fi is injective on S
U
fi
1
2
q
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Deterministic Independent Set Covering Lemma (n,l,q)-perfect hash family
|U| = nFamily of functions {f1,…,ft}
fi : U -> [q]
For each S ⊆ U, |S| ≤ l,
there exists some fi such that fi is injective on S
U
fi
1
2
q
Fredman, Komlos, Szemeredi [J. ACM ‘84]
For any n,l, a (n,l, lO(1))-perfect hash family of size lO(1) log n can be computed in time
lO(1) n log n.
Compute for l = k+kd
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Deterministic Independent Set Covering Lemma (ISCL)
There is an algorithm that given a d-degenerate graph G and an integer k, runs in time 2O(k log kd) (n+m) log n, and outputs an ISCF for (G,k) of size 2O(k log kd) log n.
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Applications: Design of Fixed-Parameter
Tractable Algorithms
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Vertex Deletion Problems
Input: A graph G , an integer kQuestion: Does there exist a set of at most k vertices, say S, such that G-S has a property 𝝥?
• Feedback Vertex Set (FVS): 𝝥 is a forest.
• Odd Cycle Transversal (OCT): 𝝥 is a bipartite graph.
• Planar Vertex Deletion (PVD): 𝝥 is a planar graph.
• s-t Separator: 𝝥 is no path from s to t.• …
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Conflict-free Vertex Deletion Problems
Input: A graph G , an integer kQuestion: Does there exist a set of at most k vertices, say S, such that G-S has a property 𝝥 and S is conflict-free (independent set)?
• Conflict-free Feedback Vertex Set (FVS)• Conflict-free Odd Cycle Transversal (OCT)• Conflict-free Planar Vertex Deletion (PVD)• Conflict-free s-t Separator• …
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“Reusing” algorithms of vertex deletion problems to design algorithms for
Conflict-free Vertex Deletion Problems
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FPT Algorithms Conflict-free Vertex Deletion Problems on d-degenerate
graphs (using ISCL)
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Deterministic Independent Set Covering Lemma (ISCL)There is an algorithm that given a d-degenerate graph G and an integer k,
runs in time 2O(k log d) (n+m) log n, and outputs an ISCF for (G,k) of size
2O(k log d) log n.
Conflict-free s-t Separator on d-degenerate graphs
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Conflict-free s-t Separator on d-degenerate graphs
Input: A graph G, an integer k, vertices s and tQuestion: Does there exist a set S, such that |S| ≤ k, S is an independent set in G and G-S has no path from s to t.
• Compute ISCF for (G,k), say F = {Y1, … , Yt}, where t = 2O(k log kd) log n (from
ISCL). Time Taken: 2O(k log kd) (n+m) log n
Input: A graph G, an integer k, vertices s and t, Y ⊆ V(G)
Question: Does there exist a set S, such that |S| ≤ k, S ⊆ Y and G-S has no path
from s to t.
Annotated s-t Separator
(G,k,s,t)
(G,k,s,t,Y1) (G,k,s,t,Y2) (G,k,s,t,Yt)
is a YES instance
if and only if
one of them is a YES instance
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Annotated s-t Separator Input: A graph G, an integer k, vertices s and t, Y ⊆ V(G)
Question: Does there exist a set S, such that |S| ≤ k, S ⊆ Y and G-S has no path
from s to t.
Input: A graph G, an integer k, vertices s and t, w : V(G) -> N Question: Does there exist a set S, such that |S| ≤ k, w(S) ≤ k and G-S has no path from s to t.
Weighted s-t Separator
Assign weights to vertices, w(v) =1 if v ∈ Y, otherwise w(v) = k+1.
Annotated s-t Separator can be solved in O(k . (n+m)) time.
Conflict-free s-t Separator on d-degenerate graphs can be solved in 2O(k log kd) (n+m) log n time.
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FPT Algorithms Conflict-free Vertex Deletion Problems on general
graphs
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Conflict-free Feedback Vertex Set on general graphs
Approximate Feedback Vertex set, |X| ≤ c k
Forest,R = V(G) \ X
Compute ISCF for (G[X],k), say F2
|F2| = 2O(k)
(Using Brute force)
Compute ISCF for (G[R],k), say F1
|F1| = 2O(k log k) log n
(using ISCL)
F = {Y∪Z : Y∈F1, Z∈F2} is an ISCF for G.
|F | = 2kO(1)
log n
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Open Problem at Dagstuhl Seminar Structure Theory and FPT Algorithms for Graphs, Digraphs and Hypergraphs
2007
of P-sequences, determine whether there is a common supersequence of length at most n+ k.This problem is at least as hard as directed feedback vertex set [FHS03], which was recentlysolved (and presented at this workshop) [CLL07, RS07].
References
[CLL07] Jianer Chen, Yang Liu, and Songjian Lu. Directed feedback vertex set problemis FPT. Preprint, 2007. http://kathrin.dagstuhl.de/files/Materials/07/07281/07281.ChenJianer.Paper.pdf
[FHS03] Michael Fellows, Michael Hallett, and Ulrike Stege. Analogs & duals of theMAST problem for sequences & trees. Journal of Algorithms 49:192–216, 2003.doi:10.1016/S0196-6774(03)00081-6
[RS07] Igor Razgon and Barry O’Sullivan. Directed feedback vertex set is fixed-parametertractable. Preprint, arXiv:0707.0282 [cs.DS], 2007.
Almost 2-coloringHenning FernauU. [email protected]
Is the following problem fixed-parameter tractable? Given a graph G and a parameter k,determine whether G has a vertex 3-coloring such that one color class has at most k vertices.In other words, the goal is to remove an independent set of k vertices such that the remaininggraph is bipartite.
Even Set / Minimum Distance in Linear CodesMichael FellowsU. [email protected]
Is the following problem fixed-parameter tractable? Given a graph G and a parameter k,determine whether there is a set S of between 1 and k vertices such that, for every vertex vin G, |N [v] \ S| is even. (Here N [v] denotes the set of neighbors of v in G.)
Almost 2-SATMichael FellowsU. [email protected]
Delete k Clauses 2-SAT is the following problem: given a 2-SAT formula, can you delete kclauses to make it satisfiable? This problem is equivalent in the fixed-parameter-tractabilitysense to the following problem: given a graph G having a perfect matching (so its minimumvertex cover has size at least n/2), does G have a vertex cover of size at most n/2 + k? Arethese problems fixed-parameter tractable?
3
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Open Problem at Dagstuhl Seminar Structure Theory and FPT Algorithms for Graphs, Digraphs and Hypergraphs
2007
of P-sequences, determine whether there is a common supersequence of length at most n+ k.This problem is at least as hard as directed feedback vertex set [FHS03], which was recentlysolved (and presented at this workshop) [CLL07, RS07].
References
[CLL07] Jianer Chen, Yang Liu, and Songjian Lu. Directed feedback vertex set problemis FPT. Preprint, 2007. http://kathrin.dagstuhl.de/files/Materials/07/07281/07281.ChenJianer.Paper.pdf
[FHS03] Michael Fellows, Michael Hallett, and Ulrike Stege. Analogs & duals of theMAST problem for sequences & trees. Journal of Algorithms 49:192–216, 2003.doi:10.1016/S0196-6774(03)00081-6
[RS07] Igor Razgon and Barry O’Sullivan. Directed feedback vertex set is fixed-parametertractable. Preprint, arXiv:0707.0282 [cs.DS], 2007.
Almost 2-coloringHenning FernauU. [email protected]
Is the following problem fixed-parameter tractable? Given a graph G and a parameter k,determine whether G has a vertex 3-coloring such that one color class has at most k vertices.In other words, the goal is to remove an independent set of k vertices such that the remaininggraph is bipartite.
Even Set / Minimum Distance in Linear CodesMichael FellowsU. [email protected]
Is the following problem fixed-parameter tractable? Given a graph G and a parameter k,determine whether there is a set S of between 1 and k vertices such that, for every vertex vin G, |N [v] \ S| is even. (Here N [v] denotes the set of neighbors of v in G.)
Almost 2-SATMichael FellowsU. [email protected]
Delete k Clauses 2-SAT is the following problem: given a 2-SAT formula, can you delete kclauses to make it satisfiable? This problem is equivalent in the fixed-parameter-tractabilitysense to the following problem: given a graph G having a perfect matching (so its minimumvertex cover has size at least n/2), does G have a vertex cover of size at most n/2 + k? Arethese problems fixed-parameter tractable?
3
Is Conflict-free Odd Cycle Transversal FPT?Reed, Smith, Vetta : Finding odd cycle transversals.
[Operations Research Letters] (2004)❖
Is Conflict-free s-t Separator FPT?
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Is Conflict-free s-t Separator FPT?Marx, O’Sullivan, Razgon : Finding small separators in linear time via
treewidth reduction. [ACM Trans. Algorithms] (2013)❖
Yes! 22kO(1)
(n+m)
Open Problems from Marx et al.1.Is it possible to improve the dependence of k to 2k
O(1) ?
2.Is Conflict-free Multicut FPT?
1.Yes! 2kO(1)
(n+m) log n
2.Yes! 2O(k3) n3 (n+m)
Lokshtanov, Panolan, Saurabh, S., Zehavi: Covering small independent sets and separators with applications to parameterized algorithms [SODA] (2018)
❖
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Overview of Marx et el [TALG 2013] approach
Is Conflict-free s-t Separator FPT?
1. Treewidth Reduction Step2. Dynamic Programming on bounded treewidth graph
(G,k,s,t)
Treewidth Reduction Step
(G,k,s,t) (G’,k,s,t)f(k). (n+m) time
preserves all minimal s-t separators of size
at most k
treewidth(G’) = 2kO(1)
G’ is an “induced subgraph” of G
Dynamic Programming on bounded treewidth graph (G’)
Time taken: exponential in treewidth ⇒ 22kO(1)
(n+m)
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Our approach [SODA 2018]1. Treewidth Reduction Step2. Dynamic Programming on bounded treewidth graph
Treewidth Reduction Step
(G,k,s,t) (G’,k,s,t)f(k). (n+m) time
preserves all minimal s-t separators of size
at most k
treewidth(G’) = 2kO(1)
G’ is an “induced subgraph” of G
Dynamic Programming on bounded treewidth graph (G’)
ISCL
ISCL on (G’,k)degeneracy(G’) ≤ treewidth(G’) = 2kO(1)
Conflict-free s-t Separator on d-degenerate graphs can be solved in 2O(k log kd) (n+m) log n time.
Conflict-free s-t Separator on general graphs can be solved in
2kO(1)
(n+m) log n time.
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Conflict-free MulticutInput: A graph G, an integer k, terminal pairs T={(s1,t1), … , (sp,tp)}
Question: Does there exist a set S, such that |S| ≤ k, S is an independent set in G and, for all i∈{1,…,p}, there is no path from si to ti in G-s.
s1
s2
s3
t1
t2
T3
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Tool 2: Degeneracy Reduction Preserving Minimal Multicuts
There exists a polynomial time algorithm that given a graph G, a set of terminal pairs T={(s1,t1), … , (sp,tp)} and an integer k, returns an
induced subgraph G’ of G and T’ ⊆ T such that:• Every minimal multicut of T in G of size at most k is a minimal multicut of T’ in G’,
• Every minimal multicut of T’ in G’ of size at most k is a minimal multicut of T in G.
• Degeneracy of G’ is 2O(k).
(G,T,k) (G’,T’,k)
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Concluding Remarks
Extensions:ISCL for nowhere dense graphs
Barriers:• ISCL for general graphs• Induced Matching Covering on 1-degenerate graphs• Acyclic Subgraphs Covering on 2-degenerate graphs• r-scattered sets covering on 1-degenerate graphs
• ISCL beyond nowhere dense graphs?• Covering other families
Open:?
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Thank you!