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Consensus Halving is PPA-Complete [article]

Aris Filos-Ratsikas, Paul W. Goldberg
<span title="2017-11-13">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that the computational problem CONSENSUS-HALVING is PPA-complete, the first PPA-completeness result for a problem whose definition does not involve an explicit circuit.  ...  We also show that an approximate version of this problem is polynomial-time equivalent to NECKLACE SPLITTING, which establishes PPAD-hardness for NECKLACE SPLITTING, and suggests that it is also PPA-complete  ...  The reason for believing that Necklace-splitting is PPA-complete, is that it would be surprising if, in relaxing the approximation parameter ǫ from inverse-exponential to inverse-polynomial, we should  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1711.04503v1">arXiv:1711.04503v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fjtv7mmyg5gwzhl7ugofmilvtu">fatcat:fjtv7mmyg5gwzhl7ugofmilvtu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200826064528/https://arxiv.org/pdf/1711.04503v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/12/99/129978fbca3233c72b042246a5139705ef36436f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1711.04503v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Consensus halving is PPA-complete

Aris Filos-Ratsikas, Paul W. Goldberg
<span title="">2018</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jlc5kugafjg4dl7ozagimqfcbm" style="color: black;">Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2018</a> </i> &nbsp;
We show that the computational problem CONSENSUS-HALVING is PPA-complete, the first PPA-completeness result for a problem whose definition does not involve an explicit circuit.  ...  the technical clutter from the main event, which is the reduction from there to Consensus-halving (Section 4).  ...  The reason for believing that Necklace-splitting is PPA-complete, is that it would be surprising if, in relaxing the approximation parameter from inverse-exponential to inverse-polynomial, we should lose  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3188745.3188880">doi:10.1145/3188745.3188880</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/stoc/Filos-RatsikasG18.html">dblp:conf/stoc/Filos-RatsikasG18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3fhsxwrncbhpxkylfjxvvbthpu">fatcat:3fhsxwrncbhpxkylfjxvvbthpu</a> </span>
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Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem

Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos, Paul G. Spirakis, Michael Wagner
<span title="2019-07-08">2019</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/pf6v2q6ji5h7birkbrtpdc2uhy" style="color: black;">International Colloquium on Automata, Languages and Programming</a> </i> &nbsp;
We study the problem of finding an exact solution to the consensus halving problem.  ...  While recent work has shown that the approximate version of this problem is PPA-complete [28, 29] , we show that the exact version is much harder.  ...  The PPA-hardness results for consensus halving show that piecewise-linearconsensus halving is PPA-hard, which completes the containment [28, 29] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.icalp.2019.138">doi:10.4230/lipics.icalp.2019.138</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/icalp/DeligkasFMS19.html">dblp:conf/icalp/DeligkasFMS19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/s3idzyyotjg4bf6zhrnllcodhq">fatcat:s3idzyyotjg4bf6zhrnllcodhq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200529125429/https://drops.dagstuhl.de/opus/volltexte/2019/10714/pdf/LIPIcs-ICALP-2019-138.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ab/c6/abc6938a28338fb154d94d6b68a734259000841f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.icalp.2019.138"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

The complexity of splitting necklaces and bisecting ham sandwiches

Aris Filos-Ratsikas, Paul W. Goldberg
<span title="">2019</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jlc5kugafjg4dl7ozagimqfcbm" style="color: black;">Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing - STOC 2019</a> </i> &nbsp;
We do this via a PPA-completeness result for an approximate version of the Consensus-halving problem, strengthening our recent result that the problem is PPA-complete for inverseexponential precision.  ...  We resolve the computational complexity of two problems known as Necklace-splitting and Discrete Ham Sandwich showing that they are PPA-complete.  ...  In this paper, we prove that ε-Consensus-halving is PPA-complete for ε inversely polynomial, thus obtaining the desired PPA-completeness of Necklace-splitting.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/3313276.3316334">doi:10.1145/3313276.3316334</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/stoc/Filos-RatsikasG19.html">dblp:conf/stoc/Filos-RatsikasG19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/25oaczverzdx3fvnz7wsfa7c3i">fatcat:25oaczverzdx3fvnz7wsfa7c3i</a> </span>
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Hardness Results for Consensus-Halving

Aris Filos-Ratsikas, Søren Kristoffer Stiil Frederiksen, Paul W. Goldberg, Jie Zhang, Michael Wagner
<span title="2018-08-20">2018</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5b35w7jagjckfngb6pj76nbam4" style="color: black;">International Symposium on Mathematical Foundations of Computer Science</a> </i> &nbsp;
It was recently proven in [13] that the problem of computing an -approximate Consensus-halving solution (for n agents and n cuts) is PPA-complete when is inverse-exponential.  ...  The Consensus-halving problem is the problem of dividing an object into two portions, such that each of n agents has equal valuation for the two portions.  ...  Consensus-Halving with n cuts is in PPA In this section, we prove that (n, n, )-ConHalving is in PPA.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.mfcs.2018.24">doi:10.4230/lipics.mfcs.2018.24</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/mfcs/Filos-RatsikasF18.html">dblp:conf/mfcs/Filos-RatsikasF18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xahdsvaltzdedo6vw7irnllj3m">fatcat:xahdsvaltzdedo6vw7irnllj3m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220126053327/https://drops.dagstuhl.de/opus/volltexte/2018/9606/pdf/LIPIcs-MFCS-2018-24.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/84/14/841404ab2b5420bd07984d2d5b7fb723f1c956a1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.mfcs.2018.24"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Computing Exact Solutions of Consensus Halving and the Borsuk-Ulam Theorem [article]

Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos, Paul G. Spirakis
<span title="2021-02-09">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study the problem of finding an exact solution to the consensus halving problem.  ...  While recent work has shown that the approximate version of this problem is PPA-complete, we show that the exact version is much harder.  ...  The PPA-hardness results for consensus halving show that piecewise-linear-consensus halving is PPA-hard, which completes the containment [28, 27] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1903.03101v2">arXiv:1903.03101v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/c2lfmqtr2vhxdhcovcli5qaeom">fatcat:c2lfmqtr2vhxdhcovcli5qaeom</a> </span>
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Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents [article]

Argyrios Deligkas, Aris Filos-Ratsikas, Alexandros Hollender
<span title="2020-07-29">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents.  ...  For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete.  ...  Acknowledgements Alexandros Hollender is supported by an EPSRC doctoral studentship (Reference 1892947).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2007.15125v1">arXiv:2007.15125v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t7n37vel35fhbffhgyesdkco4y">fatcat:t7n37vel35fhbffhgyesdkco4y</a> </span>
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Hardness Results for Consensus-Halving [article]

Aris Filos-Ratsikas, Soren Kristoffer Stiil Frederiksen, Paul W. Goldberg, Jie Zhang
<span title="2018-08-08">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that computing ϵ-approximate consensus-halving solution using n cuts is in PPA, and is PPAD-hard, where ϵ is some positive constant; the problem remains PPAD-hard when we allow a constant number  ...  The ϵ-approximate consensus-halving problem allows each agent to have an ϵ discrepancy on the values of the portions.  ...  From Consensus-halving to Tucker First, we establish the reduction from Consensus-Halving to Tucker. Lemma 4.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1609.05136v2">arXiv:1609.05136v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xfjpay6zdrbdfl6kvy7jqlpqzu">fatcat:xfjpay6zdrbdfl6kvy7jqlpqzu</a> </span>
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Consensus Division in an Arbitrary Ratio [article]

Paul W. Goldberg, Jiawei Li
<span title="2022-02-13">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Letting α∈[0,1] denote the desired ratio, this generalises the PPA-complete consensus-halving problem, in which α=1/2.  ...  , the problem is in the Turing closure of PPA-k; 3. when 2n cuts are allowed, the problem belongs to PPA for any α; furthermore, the problem belong to PPA-p for any prime p if 2(p-1)·⌈ p/2 ⌉/⌊ p/2 ⌋· n  ...  Jiawei Li wants to thank Xiaotie Deng for introducing the consensus-halving problem to him.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2202.06949v1">arXiv:2202.06949v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ns32vj6nifaj5b7iztwcmzmise">fatcat:ns32vj6nifaj5b7iztwcmzmise</a> </span>
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The Complexity of Finding Fair Independent Sets in Cycles [article]

Ishay Haviv
<span title="2020-11-04">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that the problem of finding such an independent set is 𝖯𝖯𝖠-complete.  ...  𝖠-complete as well.  ...  Consensus Halving Consider the following variant of the Consensus Halving problem, denoted CON-HALVING. (I + ) − µ i (I − )| ≤ ε.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2011.01770v2">arXiv:2011.01770v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qndff4jjofcbxdx57ubskfvymq">fatcat:qndff4jjofcbxdx57ubskfvymq</a> </span>
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Pizza Sharing is PPA-hard [article]

Argyrios Deligkas, John Fearnley, Themistoklis Melissourgos
<span title="2021-07-14">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that finding an approximate solution is PPA-hard for the straight-cut problem, and PPA-complete for the square-cut problem, while finding an exact solution for the square-cut problem is FIXP-hard  ...  We also show that decision variants of the square-cut problem are hard: we show that the approximate problem is NP-complete, and the exact problem is ETR-complete.  ...  However, we remark that BU-completeness of either exact consensus halving or any of the pizza sharing problems is yet to be proven.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2012.14236v2">arXiv:2012.14236v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6rfteqavqzeyfob3vc7pbqzmba">fatcat:6rfteqavqzeyfob3vc7pbqzmba</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210717075728/https://arxiv.org/pdf/2012.14236v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/31/fb/31fb5fecc62a1742a3ecde265d525586a0f7efa6.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2012.14236v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Consensus-Halving: Does It Ever Get Easier? [article]

Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, Manolis Zampetakis
<span title="2020-07-31">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-1/k-Division problem.  ...  In particular, we prove that ε-Consensus-1/3-Division is PPAD-hard.  ...  Acknowledgments Alexandros Hollender is supported by an EPSRC doctoral studentship (Reference 1892947).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.11437v3">arXiv:2002.11437v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xtxndhl7p5hmrjqxusbz4faf6y">fatcat:xtxndhl7p5hmrjqxusbz4faf6y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200913155612/https://arxiv.org/pdf/2002.11437v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/01/da/01da473461d540a6660e53c83f66e2794174e343.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.11437v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches [article]

Aris Filos-Ratsikas, Paul W. Goldberg
<span title="2018-11-05">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We do this via a PPA-completeness result for an approximate version of the CONSENSUS-HALVING problem, strengthening our recent result that the problem is PPA-complete for inverse-exponential precision.  ...  At the heart of our construction is a smooth embedding of the high-dimensional M\"obius strip in the CONSENSUS-HALVING problem.  ...  Most of the present paper is devoted to proving the following theorem. Theorem 2.1 ε-Consensus-halving is PPA-complete for some inverse-polynomial ε.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.12559v2">arXiv:1805.12559v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/efniwwot2beenjmix3eiouu6eq">fatcat:efniwwot2beenjmix3eiouu6eq</a> </span>
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Consensus Halving for Sets of Items [article]

Paul W. Goldberg, Alexandros Hollender, Ayumi Igarashi, Pasin Manurangsi, Warut Suksompong
<span title="2020-07-14">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Prior work has shown that when the resource is represented by an interval, a consensus halving with at most n cuts always exists, but is hard to compute even for agents with simple valuation functions.  ...  Consensus halving refers to the problem of dividing a resource into two parts so that every agent values both parts equally.  ...  This is in fact not a coincidence: recently showed that ε-approximate consensus halving is PPA-complete, implying that the problem is unlikely to admit a polynomial-time algorithm.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2007.06754v1">arXiv:2007.06754v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/p7vmyijotvajpari4xupui3jsy">fatcat:p7vmyijotvajpari4xupui3jsy</a> </span>
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Efficient Splitting of Measures and Necklaces [article]

Noga Alon, Andrei Graur
<span title="2020-06-30">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
It is known that this is possible even for ϵ = 0. NECKLACE SPLITTING is a discrete version of ϵ-CONSENSUS SPLITTING.  ...  We provide approximation algorithms for two problems, known as NECKLACE SPLITTING and ϵ-CONSENSUS SPLITTING.  ...  A problem is PPA-complete if and only if it is polynomially equivalent to the canonical problem LEAF, described in [23] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2006.16613v1">arXiv:2006.16613v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zmjna7flnvez7nu5pdoovs2l3a">fatcat:zmjna7flnvez7nu5pdoovs2l3a</a> </span>
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