Covering Small Independent Sets and Separators (with Applications) - PowerPoint PPT Presentation

Covering Small Independent Sets and Separators (with Applications) Recent Advances in Algorithms NISER Bhubaneswar February 10th, 2019 Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Roohani Sharma, Meirav Zehavi

Table of Contents 1. Introduction/Literature 2. Tool 1 3. Tool 2 4. Applications 5. Concluding Remarks

Table of Contents Behind the scenes 1. A Combinatorial Question - Tool 1 2. Applications of Tool 1 3. Stumble upon Literature 4. Resolve some open questions using Tool 1 5. Need for the design of Tool 2 6. Design of Tool 2 7 . Concluding Remarks

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 . F (G,1) 2 F (G,2) n + 2 ✓ n ◆ F (G,3) 2 + 2 2 ✓ n ◆ 2 k F (G,k) + 2 k ≈ n O ( k ) ? Can the dependence of k be removed from the exponent on n? n edges

Tool 1: 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.

<latexit sha1_base64="Ko0Gjx7n2SdQ6Ni9VKlPO/VDYMg=">AB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cKthbaUDbSbt0s4m7G6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvSATXxnW/ndLK6tr6RnmzsrW9s7tX3T9o6zhVDFsFrHqBFSj4BJbhuBnUQhjQKBD8H4JvcfnlBpHst7M0nQj+hQ8pAzanKpN8THfrXm1t0ZyDLxClKDAs1+9as3iFkaoTRMUK27npsYP6PKcCZwWumlGhPKxnSIXUsljVD72ezWKTmxyoCEsbIlDZmpvycyGmk9iQLbGVEz0oteLv7ndVMTXvkZl0lqUL5ojAVxMQkf5wMuEJmxMQSyhS3txI2oyY+Op2BC8xZeXSfus7p3X3buLWuO6iKMR3AMp+DBJTgFprQAgYjeIZXeHMi58V5dz7mrSWnmDmEP3A+fwAODY49</latexit> <latexit sha1_base64="D/2I2LpEzfh36C0kV3Lt63nfKNQ=">AB9HicdVDJSgNBEO2JW4xb1KOXxiB4Cj2J2W5BLx4jGBNJhtDTqUma9Czp7gmEId/hxYMiXv0Yb/6NnUVQ0QcFj/eqKrnRoIrTciHlVpb39jcSm9ndnb39g+yh0d3KowlgyYLRSjbLlUgeABNzbWAdiSB+q6Alju6mvutCUjFw+BWTyNwfDoIuMcZ1UZy2rirYleBhjG+72VzJE/KpVqRYJIvEbtSqxlCSLlaLGDbkDlyaIVGL/ve7Ycs9iHQTFClOjaJtJNQqTkTMt0YwURZSM6gI6hAfVBOcni6Bk+M0ofe6E0FWi8UL9PJNRXauq7ptOneqh+e3PxL68Ta6/qJDyIYg0BWy7yYoF1iOcJ4D6XwLSYGkKZ5OZWzIZUqZNThkTwten+H9yV8jbxTy5ucjVL1dxpNEJOkXnyEYVEfXqIGaiKExekBP6NmaWI/Wi/W6bE1Zq5lj9APW2yea75IA</latexit> <latexit sha1_base64="W5aNk24gFMkx6MtpUXh3us0i7w=">AB/XicdVDJSgNBEO2JW4xbXG5eGoMQEUJPYrZb0IvHCGaBZAw9PT1Jk56F7h4hDoO/4sWDIl79D2/+jZ1FUNEHBY/3qiqZ4ecSYXQh5FaWl5ZXUuvZzY2t7Z3srt7bRlEgtAWCXgujaWlDOfthRTnHZDQbFnc9qxdTv3NLhWSBf60mIbU8PSZywhWhpkD/quwCQ2k7h4E4/zql5kiSDbA4VUKVcLyGICmVkVut1TRCq1EpFaGoyRQ4s0Bxk3/tOQCKP+opwLGXPRKGyYiwUI5wmX4kaYjJGA9pT1Mfe1Ra8ez6B5rxYFuIHT5Cs7U7xMx9qSceLbu9LAayd/eVPzL60XKrVkx8NIUZ/MF7kRhyqA0yigwQlik80wUQwfSskI6zjUDqwjA7h61P4P2kXC2apgK7Oco3zRxpcAiOQB6YoAoa4BI0QsQcAcewBN4Nu6NR+PFeJ23pozFzD74AePtE+DHlNs=</latexit> Towards Randomized Independent Set Covering Lemma Given: A d-degenerate graph G, an integer k Goal Output: An independent set Y such that for any independent set X of size at most k, the Pr( ) 1 X ⊆ Y ≥ 2 k ( d +1) For each vertex v ∈ V(G), colour it either red or blue, uniformly Experiment at random. Graph G

<latexit sha1_base64="Ko0Gjx7n2SdQ6Ni9VKlPO/VDYMg=">AB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lU0GPRi8cKthbaUDbSbt0s4m7G6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvSATXxnW/ndLK6tr6RnmzsrW9s7tX3T9o6zhVDFsFrHqBFSj4BJbhuBnUQhjQKBD8H4JvcfnlBpHst7M0nQj+hQ8pAzanKpN8THfrXm1t0ZyDLxClKDAs1+9as3iFkaoTRMUK27npsYP6PKcCZwWumlGhPKxnSIXUsljVD72ezWKTmxyoCEsbIlDZmpvycyGmk9iQLbGVEz0oteLv7ndVMTXvkZl0lqUL5ojAVxMQkf5wMuEJmxMQSyhS3txI2oyY+Op2BC8xZeXSfus7p3X3buLWuO6iKMR3AMp+DBJTgFprQAgYjeIZXeHMi58V5dz7mrSWnmDmEP3A+fwAODY49</latexit> <latexit sha1_base64="D/2I2LpEzfh36C0kV3Lt63nfKNQ=">AB9HicdVDJSgNBEO2JW4xb1KOXxiB4Cj2J2W5BLx4jGBNJhtDTqUma9Czp7gmEId/hxYMiXv0Yb/6NnUVQ0QcFj/eqKrnRoIrTciHlVpb39jcSm9ndnb39g+yh0d3KowlgyYLRSjbLlUgeABNzbWAdiSB+q6Alju6mvutCUjFw+BWTyNwfDoIuMcZ1UZy2rirYleBhjG+72VzJE/KpVqRYJIvEbtSqxlCSLlaLGDbkDlyaIVGL/ve7Ycs9iHQTFClOjaJtJNQqTkTMt0YwURZSM6gI6hAfVBOcni6Bk+M0ofe6E0FWi8UL9PJNRXauq7ptOneqh+e3PxL68Ta6/qJDyIYg0BWy7yYoF1iOcJ4D6XwLSYGkKZ5OZWzIZUqZNThkTwten+H9yV8jbxTy5ucjVL1dxpNEJOkXnyEYVEfXqIGaiKExekBP6NmaWI/Wi/W6bE1Zq5lj9APW2yea75IA</latexit> <latexit sha1_base64="W5aNk24gFMkx6MtpUXh3us0i7w=">AB/XicdVDJSgNBEO2JW4xbXG5eGoMQEUJPYrZb0IvHCGaBZAw9PT1Jk56F7h4hDoO/4sWDIl79D2/+jZ1FUNEHBY/3qiqZ4ecSYXQh5FaWl5ZXUuvZzY2t7Z3srt7bRlEgtAWCXgujaWlDOfthRTnHZDQbFnc9qxdTv3NLhWSBf60mIbU8PSZywhWhpkD/quwCQ2k7h4E4/zql5kiSDbA4VUKVcLyGICmVkVut1TRCq1EpFaGoyRQ4s0Bxk3/tOQCKP+opwLGXPRKGyYiwUI5wmX4kaYjJGA9pT1Mfe1Ra8ez6B5rxYFuIHT5Cs7U7xMx9qSceLbu9LAayd/eVPzL60XKrVkx8NIUZ/MF7kRhyqA0yigwQlik80wUQwfSskI6zjUDqwjA7h61P4P2kXC2apgK7Oco3zRxpcAiOQB6YoAoa4BI0QsQcAcewBN4Nu6NR+PFeJ23pozFzD74AePtE+DHlNs=</latexit> Towards Randomized Independent Set Covering Lemma Given: A d-degenerate graph G, an integer k Goal Output: An independent set Y such that for any independent set X of size at most k, the Pr( ) 1 X ⊆ Y ≥ 2 k ( d +1) For each vertex v ∈ V(G), colour it either red or blue, uniformly Experiment at random. Graph G

Graph G RED = set of all vertices that are coloured red BLUE = 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) Graph G IND_RED= {v: v ∈ RED and all its forward neighbours in BLUE} Claim : If GOOD EVENT happens, then X ⊆ IND_RED 1 1 1 Pr(GOOD EVENT) ≥ 2 | N f ( X ) | ≥ 2 | X | 2 k ( d + 1 )

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