In this paper, we initiate the study of the protein folding problem from an integer linear programming perspective. The particular variant of protein folding that we examine is known as the hydrophobic-hydrophilic (HP) model of protein folding on the integer lattice. This problem is known to be NP-hard and also maxSNP-hard. We examine various alternate formulations for the planar version of this problem and present some preliminary computational results. Hopefully, this sets the stage for a polyhedral combinatorics assault on this important problem.
{"title":"Protein Folding on Lattices: An Integer Programming Approach","authors":"V. Chandru, M. Rao, G. Swaminathan","doi":"10.2139/SSRN.2154543","DOIUrl":"https://doi.org/10.2139/SSRN.2154543","url":null,"abstract":"In this paper, we initiate the study of the protein folding problem from an integer linear programming perspective. The particular variant of protein folding that we examine is known as the hydrophobic-hydrophilic (HP) model of protein folding on the integer lattice. This problem is known to be NP-hard and also maxSNP-hard. We examine various alternate formulations for the planar version of this problem and present some preliminary computational results. Hopefully, this sets the stage for a polyhedral combinatorics assault on this important problem.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128664800","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 : 2002-01-22DOI: 10.1137/1.9780898718805.ch7
Annegret K. Wagler
For all perfect graphs, the stable set polytope STAB$(G)$ coincides with the fractional stable set polytope QSTAB$(G)$, whereas STAB$(G) subset$ QSTAB$(G)$ holds iff $G$ is imperfect. Padberg asked in the early seventies for ``almost'' perfect graphs. He characterized those graphs for which the difference between STAB$(G)$ and QSTAB$(G)$ is smallest possible. We develop this idea further and define three polytopes between STAB$(G)$ and QSTAB$(G)$ by allowing certain sets of cutting planes only to cut off all the fractional vertices of QSTAB$(G)$. The difference between QSTAB$(G)$ and the largest of the three polytopes coinciding with STAB$(G)$ gives some information on the stage of imperfectness of the graph~$G$. We obtain a nested collection of three superclasses of perfect graphs and survey which graphs are known to belong to one of those three superclasses. This answers the question: which graphs are ``almost'' perfect?
{"title":"Relaxing Perfectness: Which Graphs Are \"Almost\" Perfect?","authors":"Annegret K. Wagler","doi":"10.1137/1.9780898718805.ch7","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch7","url":null,"abstract":"For all perfect graphs, the stable set polytope STAB$(G)$ coincides with the fractional stable set polytope QSTAB$(G)$, whereas STAB$(G) subset$ QSTAB$(G)$ holds iff $G$ is imperfect. Padberg asked in the early seventies for ``almost'' perfect graphs. He characterized those graphs for which the difference between STAB$(G)$ and QSTAB$(G)$ is smallest possible. We develop this idea further and define three polytopes between STAB$(G)$ and QSTAB$(G)$ by allowing certain sets of cutting planes only to cut off all the fractional vertices of QSTAB$(G)$. The difference between QSTAB$(G)$ and the largest of the three polytopes coinciding with STAB$(G)$ gives some information on the stage of imperfectness of the graph~$G$. We obtain a nested collection of three superclasses of perfect graphs and survey which graphs are known to belong to one of those three superclasses. This answers the question: which graphs are ``almost'' perfect?","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"08 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116697049","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 : 2001-12-14DOI: 10.1137/1.9780898718805.ch13
V. Kaibel
The edge expansion of a graph is the minimum quotient of the number of edges in a cut and the size of the smaller one among the two node sets separated by the cut. Bounding the edge expansion from below is important for bounding the ``mixing time'' of a random walk on the graph from above. It has been conjectured by Mihail and Vazirani that the graph of every 0/1-polytope has edge expansion at least one. A proof of this (or even a weaker) conjecture would imply solutions of several long-standing open problems in the theory of randomized approximate counting. We present different techniques for bounding the edge expansion of a 0/1-polytope from below. By means of these tools we show that several classes of 0/1-polytopes indeed have graphs with edge expansion at least one. These classes include all 0/1-polytopes of dimension at most five, all simple 0/1-polytopes, all hypersimplices, all stable set polytopes, and all (perfect) matching polytopes.
{"title":"On the Expansion of Graphs of 0/1-Polytopes","authors":"V. Kaibel","doi":"10.1137/1.9780898718805.ch13","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch13","url":null,"abstract":"The edge expansion of a graph is the minimum quotient of the number of edges in a cut and the size of the smaller one among the two node sets separated by the cut. Bounding the edge expansion from below is important for bounding the ``mixing time'' of a random walk on the graph from above. It has been conjectured by Mihail and Vazirani that the graph of every 0/1-polytope has edge expansion at least one. A proof of this (or even a weaker) conjecture would imply solutions of several long-standing open problems in the theory of randomized approximate counting. We present different techniques for bounding the edge expansion of a 0/1-polytope from below. By means of these tools we show that several classes of 0/1-polytopes indeed have graphs with edge expansion at least one. These classes include all 0/1-polytopes of dimension at most five, all simple 0/1-polytopes, all hypersimplices, all stable set polytopes, and all (perfect) matching polytopes.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2001-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114694922","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 : 1900-01-01DOI: 10.1137/1.9780898718805.ch9
H. Kerivin, A. Mahjoub, Charles Nocq
{"title":"(1, 2)-Survivable Networks: Facets and Branch-and-Cut","authors":"H. Kerivin, A. Mahjoub, Charles Nocq","doi":"10.1137/1.9780898718805.ch9","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch9","url":null,"abstract":"","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"13 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120918153","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 : 1900-01-01DOI: 10.1137/1.9780898718805.ch14
G. Ziegler
Viewed geometrically, the simplex algorithm on a (primally and dually non-degenerate) linear program traces a monotone edge path from the starting vertex to the (unique) optimum. Which path it takes depends on the pivot rule. In this paper we survey geometric and combinatorial aspects of the situation: How do “real” linear programs and their polyhedra look like? How long can simplex paths be in the worst case? Do short paths always exist? Can we expect randomized pivot rules (such as Random Edge) or deterministic rules (such as Zadeh’s rule) to find short paths? MSC 2000. 90C05, 52B11
{"title":"Typical and Extremal Linear Programs","authors":"G. Ziegler","doi":"10.1137/1.9780898718805.ch14","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch14","url":null,"abstract":"Viewed geometrically, the simplex algorithm on a (primally and dually non-degenerate) linear program traces a monotone edge path from the starting vertex to the (unique) optimum. Which path it takes depends on the pivot rule. In this paper we survey geometric and combinatorial aspects of the situation: How do “real” linear programs and their polyhedra look like? How long can simplex paths be in the worst case? Do short paths always exist? Can we expect randomized pivot rules (such as Random Edge) or deterministic rules (such as Zadeh’s rule) to find short paths? MSC 2000. 90C05, 52B11","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"187 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115207059","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 : 1900-01-01DOI: 10.1137/1.9780898718805.ch4
M. Conforti, G. Cornuéjols, G. Zambelli
Two classical theorems of Ghouila-Houri and Berge characterize total unimodularity and balancedness in terms of equitable bicolorings and bicolorings, respectively. In this paper, we prove a bicoloring result that provides a common generalization of these two theorems.
{"title":"Bicolorings and Equitable Bicolorings of Matrices","authors":"M. Conforti, G. Cornuéjols, G. Zambelli","doi":"10.1137/1.9780898718805.ch4","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch4","url":null,"abstract":"Two classical theorems of Ghouila-Houri and Berge characterize total unimodularity and balancedness in terms of equitable bicolorings and bicolorings, respectively. In this paper, we prove a bicoloring result that provides a common generalization of these two theorems.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131502956","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 : 1900-01-01DOI: 10.1137/1.9780898718805.ch3
R. Borndörfer
This article investigates a certain class of combinatorial packing problems and some polyhedral relations between such problems and the set packing problem.
研究了一类组合布局问题及其与集合布局问题的多面体关系。
{"title":"Combinatorial Packing Problems","authors":"R. Borndörfer","doi":"10.1137/1.9780898718805.ch3","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch3","url":null,"abstract":"This article investigates a certain class of combinatorial packing problems and some polyhedral relations between such problems and the set packing problem.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128914912","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 : 1900-01-01DOI: 10.1137/1.9780898718805.ch8
M. Grötschel
A subset ${cal C}$ of the power set of a finite set $E$ is called cardinality homogeneous if, whenever ${cal C}$ contains some set $F$, ${cal C}$ contains all subsets of $E$ of cardinality $|F|$. Examples of such set systems ${cal C}$ are the sets of circuits and the sets of cycles of uniform matroids and the sets of all even or of all odd cardinality subsets of $E$. With each cardinality homogeneous set system ${cal C}$, we associate the polytope $P({cal C})$, the convex hull of the incidence vectors of all sets in ${cal C}$, and provide a complete and nonredundant linear description of $P({cal C})$. We show that a greedy algorithm optimizes any linear function over $P({cal C})$, give an explicit optimum solution of the dual linear program, and provide a polynomial time separation algorithm for the class of polytopes of type $P({cal C})$.
{"title":"Cardinality Homogeneous Set Systems, Cycles in Matroids, and Associated Polytopes","authors":"M. Grötschel","doi":"10.1137/1.9780898718805.ch8","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch8","url":null,"abstract":"A subset ${cal C}$ of the power set of a finite set $E$ is called cardinality homogeneous if, whenever ${cal C}$ contains some set $F$, ${cal C}$ contains all subsets of $E$ of cardinality $|F|$. Examples of such set systems ${cal C}$ are the sets of circuits and the sets of cycles of uniform matroids and the sets of all even or of all odd cardinality subsets of $E$. With each cardinality homogeneous set system ${cal C}$, we associate the polytope $P({cal C})$, the convex hull of the incidence vectors of all sets in ${cal C}$, and provide a complete and nonredundant linear description of $P({cal C})$. We show that a greedy algorithm optimizes any linear function over $P({cal C})$, give an explicit optimum solution of the dual linear program, and provide a polynomial time separation algorithm for the class of polytopes of type $P({cal C})$.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126649274","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 : 1900-01-01DOI: 10.1137/1.9780898718805.ch17
Nathan W. Brixius, K. Anstreicher
In a paper Leon Steinberg described a backboard wiring problem that has resisted solution for years The problem concerns the placement of computer components so as to minimize the total amount of wiring required to connect them In the particular instance considered by Steinberg components with a total of interconnections are to be placed on a backboard with open positions The geometry of the backboard is illustrated in Figure To formulate the wiring problem mathematically it is convenient to add dummy components with no connections to any others so that the numbers of components and locations are both n Letting aik be the number of wires that connect components i and k bjl be the distance between locations j and l on the backboard and doubling the objective the problem can be written in the form
{"title":"The Steinberg Wiring Problem","authors":"Nathan W. Brixius, K. Anstreicher","doi":"10.1137/1.9780898718805.ch17","DOIUrl":"https://doi.org/10.1137/1.9780898718805.ch17","url":null,"abstract":"In a paper Leon Steinberg described a backboard wiring problem that has resisted solution for years The problem concerns the placement of computer components so as to minimize the total amount of wiring required to connect them In the particular instance considered by Steinberg components with a total of interconnections are to be placed on a backboard with open positions The geometry of the backboard is illustrated in Figure To formulate the wiring problem mathematically it is convenient to add dummy components with no connections to any others so that the numbers of components and locations are both n Letting aik be the number of wires that connect components i and k bjl be the distance between locations j and l on the backboard and doubling the objective the problem can be written in the form","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128599716","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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