Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834686
M. F. Bollauf, Hsuan-Yin Lin, Øyvind Ytrehus
Lattice coding for the Gaussian wiretap channel is considered, where the goal is to ensure reliable communication between two authorized parties while preventing an eavesdropper from learning the transmitted messages. Recently, a measure called secrecy gain was proposed as a design criterion to quantify the secrecy-goodness of the applied lattice code. In this paper, the theta series of the so-called formally unimodular lattices obtained by Construction A4 from codes over ${{mathbb{Z}}_4}$ is derived, and we provide a universal approach to determine their secrecy gains. Initial results indicate that Construction A4 lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Furthermore, a new code construction of formally self-dual ${{mathbb{Z}}_4}$-linear codes is presented.
{"title":"On the Secrecy Gain of Formally Unimodular Construction A4 Lattices","authors":"M. F. Bollauf, Hsuan-Yin Lin, Øyvind Ytrehus","doi":"10.1109/ISIT50566.2022.9834686","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834686","url":null,"abstract":"Lattice coding for the Gaussian wiretap channel is considered, where the goal is to ensure reliable communication between two authorized parties while preventing an eavesdropper from learning the transmitted messages. Recently, a measure called secrecy gain was proposed as a design criterion to quantify the secrecy-goodness of the applied lattice code. In this paper, the theta series of the so-called formally unimodular lattices obtained by Construction A4 from codes over ${{mathbb{Z}}_4}$ is derived, and we provide a universal approach to determine their secrecy gains. Initial results indicate that Construction A4 lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Furthermore, a new code construction of formally self-dual ${{mathbb{Z}}_4}$-linear codes is presented.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125586251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834806
Daewon Seo, Yongjune Kim
In practical simultaneous information and energy transmission (SIET), the exact energy harvesting function is usually unavailable because a harvesting circuit is nonlinear and nonideal. In this work, we consider a SIET problem where the harvesting function is accessible only at sample points that are experimentally taken in the presence of noise. Assuming that the harvesting function is of bounded variation that may have discontinuities, we propose to design SIET based on the wavelet-reconstructed harvesting function. The main focus is its asymptotic performance of expected energy and information rate. Specifically, we propose to design a SIET system based on the wavelet-reconstructed harvesting function with soft-thresholding estimation. Then, the expected loss in energy transmission asymptotically vanishes as the number of samples grows, which turns out to be optimal up to a logarithmic factor. The expected loss in information transmission also vanishes if the target energy delivery is in the interior of the deliverable energy range.
{"title":"Information and Energy Transmission with Wavelet-Reconstructed Harvesting Functions","authors":"Daewon Seo, Yongjune Kim","doi":"10.1109/ISIT50566.2022.9834806","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834806","url":null,"abstract":"In practical simultaneous information and energy transmission (SIET), the exact energy harvesting function is usually unavailable because a harvesting circuit is nonlinear and nonideal. In this work, we consider a SIET problem where the harvesting function is accessible only at sample points that are experimentally taken in the presence of noise. Assuming that the harvesting function is of bounded variation that may have discontinuities, we propose to design SIET based on the wavelet-reconstructed harvesting function. The main focus is its asymptotic performance of expected energy and information rate. Specifically, we propose to design a SIET system based on the wavelet-reconstructed harvesting function with soft-thresholding estimation. Then, the expected loss in energy transmission asymptotically vanishes as the number of samples grows, which turns out to be optimal up to a logarithmic factor. The expected loss in information transmission also vanishes if the target energy delivery is in the interior of the deliverable energy range.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125705565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834436
M. Baniasadi, Yu Chen, M. Safari
In this paper, we investigate the capacity of the continuous spectrum of nonlinear frequency division multiplexing (NFDM) systems when data is encoded using b-modulation. Recently, a tractable channel model is proposed for such optical fiber communication systems, describing the received b-modulated signal with an input-dependent complex Gaussian distributed noise. Considering this channel model, we prove that the capacity-achieving distribution is unique and discrete. A search algorithm is proposed to determine the optimal discrete input distribution that maximizes the mutual information. The numerical results show that the optimal input distributions form ring-based constellations. We compare these optimal designs with some conventional ring-based amplitude and phase shift keying (APSK) modulation schemes including 16 APSK, 32 APSK and 64 APSK, to show the achievable performance gains.
{"title":"On the Capacity of b-Modulated Nonlinear Frequency Division Multiplexing","authors":"M. Baniasadi, Yu Chen, M. Safari","doi":"10.1109/ISIT50566.2022.9834436","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834436","url":null,"abstract":"In this paper, we investigate the capacity of the continuous spectrum of nonlinear frequency division multiplexing (NFDM) systems when data is encoded using b-modulation. Recently, a tractable channel model is proposed for such optical fiber communication systems, describing the received b-modulated signal with an input-dependent complex Gaussian distributed noise. Considering this channel model, we prove that the capacity-achieving distribution is unique and discrete. A search algorithm is proposed to determine the optimal discrete input distribution that maximizes the mutual information. The numerical results show that the optimal input distributions form ring-based constellations. We compare these optimal designs with some conventional ring-based amplitude and phase shift keying (APSK) modulation schemes including 16 APSK, 32 APSK and 64 APSK, to show the achievable performance gains.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129874773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834825
Yaniv Sadeh, Ori Rottenstreich, Haim Kaplan
Traffic splitting is a required functionality in networks, for example for load balancing over paths or servers, or by the source’s access restrictions. The capacities of the servers (or the number of users with particular access restrictions) determine the sizes of the parts into which traffic should be split. A recent approach implements traffic splitting within the ternary content addressable memory (TCAM), which is often available in switches. It is important to reduce the amount of memory allocated for this task since TCAMs are power consuming. We analyze the expected size of a representation, for uniformly random ordered partitions. We show that the expected representation size of a random partition is at least half the size for the worst-case partition, and is linear in the number of parts and in the logarithm of the size of the address space.
{"title":"Coding Size of Traffic Partition in Switch Memories","authors":"Yaniv Sadeh, Ori Rottenstreich, Haim Kaplan","doi":"10.1109/ISIT50566.2022.9834825","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834825","url":null,"abstract":"Traffic splitting is a required functionality in networks, for example for load balancing over paths or servers, or by the source’s access restrictions. The capacities of the servers (or the number of users with particular access restrictions) determine the sizes of the parts into which traffic should be split. A recent approach implements traffic splitting within the ternary content addressable memory (TCAM), which is often available in switches. It is important to reduce the amount of memory allocated for this task since TCAMs are power consuming. We analyze the expected size of a representation, for uniformly random ordered partitions. We show that the expected representation size of a random partition is at least half the size for the worst-case partition, and is linear in the number of parts and in the logarithm of the size of the address space.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130181467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834697
Chih-Chun Wang
One canonical example of Age-Of-Information (AoI) minimization is the update-through-queues models. Existing results fall into two categories: The open-loop setting for which the sender is oblivious of the actual packet departure time, versus the closed-loop setting for which the decision is based on instantaneous Acknowledgement (ACK). Neither setting perfectly reflects modern networked systems, which almost always rely on feedback that experiences some delay. Motivated by this observation, this work subjects the ACK traffic to an independent queue so that the closed-loop decision is made based on delayed feedback. Near-optimal schedulers have been devised, which smoothly transition from the instantaneous-ACK to the openloop schemes depending on how long the feedback delay is. The results thus quantify the benefits of delayed feedback for AoI minimization in the update-through-queues systems.
{"title":"How Useful Is Delayed Feedback in AoI Minimization — A Study on Systems With Queues in Both Forward and Backward Directions","authors":"Chih-Chun Wang","doi":"10.1109/ISIT50566.2022.9834697","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834697","url":null,"abstract":"One canonical example of Age-Of-Information (AoI) minimization is the update-through-queues models. Existing results fall into two categories: The open-loop setting for which the sender is oblivious of the actual packet departure time, versus the closed-loop setting for which the decision is based on instantaneous Acknowledgement (ACK). Neither setting perfectly reflects modern networked systems, which almost always rely on feedback that experiences some delay. Motivated by this observation, this work subjects the ACK traffic to an independent queue so that the closed-loop decision is made based on delayed feedback. Near-optimal schedulers have been devised, which smoothly transition from the instantaneous-ACK to the openloop schemes depending on how long the feedback delay is. The results thus quantify the benefits of delayed feedback for AoI minimization in the update-through-queues systems.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129464250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834746
N. Weinberger, M. Yemini
We introduce a multi-armed bandit problem with information-based rewards. At each round, a player chooses an arm, observes a symbol, and receives an unobserved reward in the form of the symbol’s self-information. The player aims to maximize the expected total reward associated with the entropy values of the arms played. We propose two algorithms based on upper confidence bounds (UCB) for this model. The first algorithm optimistically corrects the bias term in the entropy estimation. The second algorithm relies on data-dependent UCBs that adapt to sources with small entropy values. We provide performance guarantees by upper bounding the expected regret of each of the algorithms, and compare their asymptotic behavior to the Lai-Robbins lower bound. Finally, we provide numerical results illustrating the regret of the algorithms presented.
{"title":"Upper Confidence Interval Strategies for Multi-Armed Bandits with Entropy Rewards","authors":"N. Weinberger, M. Yemini","doi":"10.1109/ISIT50566.2022.9834746","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834746","url":null,"abstract":"We introduce a multi-armed bandit problem with information-based rewards. At each round, a player chooses an arm, observes a symbol, and receives an unobserved reward in the form of the symbol’s self-information. The player aims to maximize the expected total reward associated with the entropy values of the arms played. We propose two algorithms based on upper confidence bounds (UCB) for this model. The first algorithm optimistically corrects the bias term in the entropy estimation. The second algorithm relies on data-dependent UCBs that adapt to sources with small entropy values. We provide performance guarantees by upper bounding the expected regret of each of the algorithms, and compare their asymptotic behavior to the Lai-Robbins lower bound. Finally, we provide numerical results illustrating the regret of the algorithms presented.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129536420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834882
Xuan Guang, Y. Bai, R. Yeung
The problem of secure network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a target function f, of which the inputs are generated at multiple source nodes, while a wiretapper, who can access any one but not more than one wiretap set in a given collection of wiretap sets, obtains no information about the source inputs. The secure computing rate of a secure function-computing network code is the average number of times the target function can be securely computed for one use of the network. In the paper, we are interested in securely computing linear functions with the wiretapper who can eavesdrop any subset of edges up to a certain size r, referred to as the security level. We obtain an improved upper bound on the secure computing capacity, which is applicable to arbitrary network topologies and arbitrary security levels. When the security level r is equal to 0, our improved upper bound reduces to the computing capacity without security consideration. Furthermore, by applying the improved upper bound, we obtain a non-trivial upper bound on the maximum security level such that the function can be securely computed with a positive rate. We also present a lower bound on the secure computing capacity and give some sufficient conditions in terms of the network topology on the tightness of the lower bound. Together with the improved upper bound, the secure computing capacity for a class of security models can be fully characterized.
{"title":"An Improved Capacity Bound for Secure Network Function Computation","authors":"Xuan Guang, Y. Bai, R. Yeung","doi":"10.1109/ISIT50566.2022.9834882","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834882","url":null,"abstract":"The problem of secure network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a target function f, of which the inputs are generated at multiple source nodes, while a wiretapper, who can access any one but not more than one wiretap set in a given collection of wiretap sets, obtains no information about the source inputs. The secure computing rate of a secure function-computing network code is the average number of times the target function can be securely computed for one use of the network. In the paper, we are interested in securely computing linear functions with the wiretapper who can eavesdrop any subset of edges up to a certain size r, referred to as the security level. We obtain an improved upper bound on the secure computing capacity, which is applicable to arbitrary network topologies and arbitrary security levels. When the security level r is equal to 0, our improved upper bound reduces to the computing capacity without security consideration. Furthermore, by applying the improved upper bound, we obtain a non-trivial upper bound on the maximum security level such that the function can be securely computed with a positive rate. We also present a lower bound on the secure computing capacity and give some sufficient conditions in terms of the network topology on the tightness of the lower bound. Together with the improved upper bound, the secure computing capacity for a class of security models can be fully characterized.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129595330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834810
Shota Saito
For the problem of parameter estimation in a Bayesian setting, information-theoretic lower bounds of the Bayes risk have been investigated. Previous studies have proven the lower bound of the Bayes risk in a different manner and characterized the lower bound via different quantities such as the mutual information, Sibson’s α-mutual information, and Csiszár’s f-informativity. In this paper, we introduce an inequality called a "meta-bound for lower bounds of the Bayes risk" and show that the previous results can be derived from this bound.
{"title":"On Meta-Bound for Lower Bounds of Bayes Risk","authors":"Shota Saito","doi":"10.1109/ISIT50566.2022.9834810","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834810","url":null,"abstract":"For the problem of parameter estimation in a Bayesian setting, information-theoretic lower bounds of the Bayes risk have been investigated. Previous studies have proven the lower bound of the Bayes risk in a different manner and characterized the lower bound via different quantities such as the mutual information, Sibson’s α-mutual information, and Csiszár’s f-informativity. In this paper, we introduce an inequality called a \"meta-bound for lower bounds of the Bayes risk\" and show that the previous results can be derived from this bound.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"2016 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128069360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834869
Hanxu Hou, Y. Han, Bo Bai, Gong Zhang
Large-scale high code-rate maximum distance separable (MDS) codes are critical and important in distributed storage systems that can provide high fault tolerance with extremely small storage redundancy. Repair access (defined as the total amount of symbols accessed in repairing one single-node failure) is a key metric of designing MDS codes. In large-scale MDS codes, one single-node failure can be recovered by connecting a large number of helper nodes. However, one or more helper nodes may be busy and can not send symbols during the repair process. In this paper, we define the total amount of symbols accessed in repairing one single-node failure with one or more busy nodes as the repair access with busy-node. We then propose a class of MDS array codes over a well-designed binary cyclic ring that is with small sub-packetization, small repair access, small repair access with busy-node, and small encoding complexity.
{"title":"Towards Efficient Repair and Coding of Binary MDS Array Codes with Small Sub-packetization","authors":"Hanxu Hou, Y. Han, Bo Bai, Gong Zhang","doi":"10.1109/ISIT50566.2022.9834869","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834869","url":null,"abstract":"Large-scale high code-rate maximum distance separable (MDS) codes are critical and important in distributed storage systems that can provide high fault tolerance with extremely small storage redundancy. Repair access (defined as the total amount of symbols accessed in repairing one single-node failure) is a key metric of designing MDS codes. In large-scale MDS codes, one single-node failure can be recovered by connecting a large number of helper nodes. However, one or more helper nodes may be busy and can not send symbols during the repair process. In this paper, we define the total amount of symbols accessed in repairing one single-node failure with one or more busy nodes as the repair access with busy-node. We then propose a class of MDS array codes over a well-designed binary cyclic ring that is with small sub-packetization, small repair access, small repair access with busy-node, and small encoding complexity.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128581984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-26DOI: 10.1109/ISIT50566.2022.9834647
D. Gamarnik, Eren C. Kizildag
We focus on the problem of algorithmically finding a near-optimal solution for the (random) number partitioning problem (NPP), a problem that is of great practical and theoretical significance. The NPP possesses a striking gap between the existential and best algorithmic guarantee: when its input has i.i.d. standard Gaussian entries, the optimal value of NPP is $Theta left( {sqrt n {2^{ - n}}} right)$ (w.h.p.); whereas the best polynomial-time algorithm achieves an exponentially worse value of only ${2^{ - Theta left( {{{log }^2}n} right)}}$ (w.h.p.). In this paper, we inquire into the origin of this gap by studying the landscape of the NPP through the lens of statistical physics and establish the presence of the Overlap Gap Property (OGP), a topological barrier for large classes of algorithms. We then leverage the OGP to establish that (a) sufficiently stable algorithms fail to find a near-optimal solution with value below $left. {{2^{ - omega (nlog - 1/5}}n} right)$; and (b) a very natural Monte Carlo Markov Chain dynamics mixes slowly. A technical innovation of our paper is that we consider the overlap structure of m–tuples of near- optimal solutions where m itself grows in n. Our hardness result for stable algorithms is based on a Ramsey-theoretic argument from extremal combinatorics. To the best of our knowledge, this is the first usage of Ramsey Theory to show algorithmic hardness.
{"title":"The Random Number Partitioning Problem: Overlap Gap Property and Algorithmic Barriers","authors":"D. Gamarnik, Eren C. Kizildag","doi":"10.1109/ISIT50566.2022.9834647","DOIUrl":"https://doi.org/10.1109/ISIT50566.2022.9834647","url":null,"abstract":"We focus on the problem of algorithmically finding a near-optimal solution for the (random) number partitioning problem (NPP), a problem that is of great practical and theoretical significance. The NPP possesses a striking gap between the existential and best algorithmic guarantee: when its input has i.i.d. standard Gaussian entries, the optimal value of NPP is $Theta left( {sqrt n {2^{ - n}}} right)$ (w.h.p.); whereas the best polynomial-time algorithm achieves an exponentially worse value of only ${2^{ - Theta left( {{{log }^2}n} right)}}$ (w.h.p.). In this paper, we inquire into the origin of this gap by studying the landscape of the NPP through the lens of statistical physics and establish the presence of the Overlap Gap Property (OGP), a topological barrier for large classes of algorithms. We then leverage the OGP to establish that (a) sufficiently stable algorithms fail to find a near-optimal solution with value below $left. {{2^{ - omega (nlog - 1/5}}n} right)$; and (b) a very natural Monte Carlo Markov Chain dynamics mixes slowly. A technical innovation of our paper is that we consider the overlap structure of m–tuples of near- optimal solutions where m itself grows in n. Our hardness result for stable algorithms is based on a Ramsey-theoretic argument from extremal combinatorics. To the best of our knowledge, this is the first usage of Ramsey Theory to show algorithmic hardness.","PeriodicalId":348168,"journal":{"name":"2022 IEEE International Symposium on Information Theory (ISIT)","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124517121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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