A minimum edge coloring of a bipartite graph is a partition of the edges into &Dgr; matchings, where &Dgr; is the maximum degree in the graph. Coloring algorithms are presented that use time O(min(¦E¦ &Dgr; log n, ¦E¦ @@@@n log n, n2log &Dgr;)) and space O(n&Dgr;). This compares favorably to the previous O(¦E¦ [equation] log &Dgr;) time bound. The coloring algorithms also find maximum matchings on regular (or semi-regular) bipartite graphs. The time bounds compare favorably to the O(¦E¦ @@@@n) matching algorithm, expect when [equation] ≤ &Dgr; ≤ @@@@n log n.
二部图的最小边着色是将边划分为&Dgr;匹配，其中&Dgr;是图中的最大度。提出了使用时间为O(min(…)&Dgr;log n，……@@@@n log n, n2log &Dgr;))和空间0 (n&Dgr;)这比之前的O(…[等式]log &Dgr;)时间限制有利。着色算法也能在正则(或半正则)二部图上找到最大匹配。除了当[方程]≤&Dgr;≤@@@@n log n。
{"title":"Algorithms for edge coloring bipartite graphs","authors":"H. Gabow, O. Kariv","doi":"10.1145/800133.804346","DOIUrl":"https://doi.org/10.1145/800133.804346","url":null,"abstract":"A minimum edge coloring of a bipartite graph is a partition of the edges into &Dgr; matchings, where &Dgr; is the maximum degree in the graph. Coloring algorithms are presented that use time O(min(¦E¦ &Dgr; log n, ¦E¦ @@@@n log n, n2log &Dgr;)) and space O(n&Dgr;). This compares favorably to the previous O(¦E¦ [equation] log &Dgr;) time bound. The coloring algorithms also find maximum matchings on regular (or semi-regular) bipartite graphs. The time bounds compare favorably to the O(¦E¦ @@@@n) matching algorithm, expect when [equation] ≤ &Dgr; ≤ @@@@n log n.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123102445","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}
Let A be the problem of minimizing c1x1+...+cnxn subject to certain constraints on x=(x1,...,xn), and let B be the problem of minimizing (a0+a1x1+...+anxn)/(b0+b1x1+...+bnxn) subject to the same constraints, assuming the denominator is always positive. It is shown that if A is solvable within O[p(n)] comparisons and O[q(n)] additions, then B is solvable in time O[p(n)(q(n)+p(n))]. This applies to most of the “network” algorithms. Consequently, minimum ratio cycles, minimum ratio spanning trees, minimum ratio (simple) paths, maximum ratio weighted matchings, etc., can be computed within polynomial-time in the number of variables. This improves a result of E. L. Lawler, namely, that a minimum ratio cycle can be computed within a time bound which is polynomial in the number of bits required to specify an instance of the problem. A recent result on minimum ratio spanning trees by R. Chandrasekaran is also improved by the general arguments presented in this paper. Algorithms of time-complexity O(¦E¦ • ¦V¦2•log¦V¦) for a minimum ratio cycle and O(¦E¦ • log2¦V¦ • log log ¦V¦) for a minimum ratio spanning tree are developed.
设A是最小化c1x1+…的问题+cnxn在x=(x1，…，xn)上有一定约束，设B为在相同约束下最小化(a0+a1x1+…+anxn)/(b0+b1x1+…+bnxn)的问题，假设分母总是正的。证明了如果A在O[p(n)]个比较和O[q(n)]个加法内可解，那么B在O[p(n))(q(n)+p(n)))时间内可解。这适用于大多数“网络”算法。因此，最小比值循环、最小比值生成树、最小比值(简单)路径、最大比值加权匹配等，都可以在变量数量的多项式时间内计算出来。这改进了E. L. Lawler的一个结果，即可以在指定问题实例所需的比特数的多项式的时间范围内计算最小比率周期。本文提出的一般论点也改进了R. Chandrasekaran关于最小比值生成树的最新结果。给出了最小比值生成树的时间复杂度为O(……V…2·log…)和O(……log2·V…·log log…)的算法。
{"title":"Combinatorial optimization with rational objective functions","authors":"N. Megiddo","doi":"10.1145/800133.804326","DOIUrl":"https://doi.org/10.1145/800133.804326","url":null,"abstract":"Let A be the problem of minimizing c1x1+...+cnxn subject to certain constraints on x=(x1,...,xn), and let B be the problem of minimizing (a0+a1x1+...+anxn)/(b0+b1x1+...+bnxn) subject to the same constraints, assuming the denominator is always positive. It is shown that if A is solvable within O[p(n)] comparisons and O[q(n)] additions, then B is solvable in time O[p(n)(q(n)+p(n))]. This applies to most of the “network” algorithms. Consequently, minimum ratio cycles, minimum ratio spanning trees, minimum ratio (simple) paths, maximum ratio weighted matchings, etc., can be computed within polynomial-time in the number of variables. This improves a result of E. L. Lawler, namely, that a minimum ratio cycle can be computed within a time bound which is polynomial in the number of bits required to specify an instance of the problem. A recent result on minimum ratio spanning trees by R. Chandrasekaran is also improved by the general arguments presented in this paper. Algorithms of time-complexity O(¦E¦ • ¦V¦2•log¦V¦) for a minimum ratio cycle and O(¦E¦ • log2¦V¦ • log log ¦V¦) for a minimum ratio spanning tree are developed.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115547691","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}
A perfect matching in a graph H may be viewed as a collection of subgraphs of H, each of which is isomorphic to K2, whose vertex sets partition the vertex set of H. This is naturally generalized by replacing K2 by an arbitrary graph G. We show that if G contains a component with at least three vertices then this generalized matching problem is NP-complete. These generalized matchings have numerous applications including the minimization of second-order conflicts in examination scheduling.
{"title":"On the completeness of a generalized matching problem","authors":"D. Kirkpatrick, P. Hell","doi":"10.1145/800133.804353","DOIUrl":"https://doi.org/10.1145/800133.804353","url":null,"abstract":"A perfect matching in a graph H may be viewed as a collection of subgraphs of H, each of which is isomorphic to K2, whose vertex sets partition the vertex set of H. This is naturally generalized by replacing K2 by an arbitrary graph G. We show that if G contains a component with at least three vertices then this generalized matching problem is NP-complete. These generalized matchings have numerous applications including the minimization of second-order conflicts in examination scheduling.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131550925","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}
We prove that any multivariate polynomial P of degree d that can be computed with C(P) multiplications-divisions can be computed in O(log d.log C(P)) parallel steps and O(log d) parallel multiplicative steps.
{"title":"On the parallel evaluation of multivariate polynomials","authors":"L. Hyafil","doi":"10.1145/800133.804347","DOIUrl":"https://doi.org/10.1145/800133.804347","url":null,"abstract":"We prove that any multivariate polynomial P of degree d that can be computed with C(P) multiplications-divisions can be computed in O(log d.log C(P)) parallel steps and O(log d) parallel multiplicative steps.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113979099","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}
Let R @@@@ NP be the collection of languages L such that for some polynomial time computable predicate P(x,y) and constant k, L = {x|@@@@y, |y|=|x|k,P(x,y)} = {x|@@@@ at least 2|x|k−1 values of y, |y|=|x|k,P(x,y)}. Let U @@@@ NP be the collection of languages L such that for some polynomial time computable predicate P(x,y) and constant k, L = {x|@@@@y,|y|=|x|k,P(x,y)}= {x|@@@@ unique y, |y|=|x|k,p(x,y)}. Let RA,UA, PA,NPA,CO-NPA be the relativization of these classes with respect to an oracle A as in [ 5 ]. Then for some oracle E (NPE @@@@ CO-NPE) = UE = RE = PE @@@@ NPE while for some other oracle D CO-NPD = NPD = UD = RD @@@@ PD.
设R @@@@ NP是语言L的集合，使得对于某个多项式时间可计算谓词P(x,y)和常数k, L = {x|@@@@y， |y|=|x|k,P(x,y)} = {x|@@@@至少有2个|x|k−1个y的值，|y|=|x|k,P(x,y)}。设U @@@@ NP是语言L的集合，使得对于某个多项式时间可计算谓词P(x,y)和常数k, L = {x|@@@@y，|y|=|x|k,P(x,y)}= {x|@@@@唯一y，|y|=|x|k,P(x,y)}。设RA,UA, PA,NPA,CO-NPA为这些类相对于oracle A的相对化，如[5]所示。然后，对于某些oracle E (NPE @@@@ CO-NPE) = UE = RE = PE @@@@ NPE，而对于另一些oracle D CO-NPD = NPD = UD = RD @@@@ PD。
{"title":"Relativized questions involving probabilistic algorithms","authors":"C. Rackoff","doi":"10.1145/800133.804363","DOIUrl":"https://doi.org/10.1145/800133.804363","url":null,"abstract":"Let R @@@@ NP be the collection of languages L such that for some polynomial time computable predicate P(x,y) and constant k, L = {x|@@@@y, |y|=|x|k,P(x,y)} = {x|@@@@ at least 2|x|k−1 values of y, |y|=|x|k,P(x,y)}. Let U @@@@ NP be the collection of languages L such that for some polynomial time computable predicate P(x,y) and constant k, L = {x|@@@@y,|y|=|x|k,P(x,y)}= {x|@@@@ unique y, |y|=|x|k,p(x,y)}. Let RA,UA, PA,NPA,CO-NPA be the relativization of these classes with respect to an oracle A as in [ 5 ]. Then for some oracle E (NPE @@@@ CO-NPE) = UE = RE = PE @@@@ NPE while for some other oracle D CO-NPD = NPD = UD = RD @@@@ PD.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114539818","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}
L. Carter, R. W. Floyd, John Gill, G. Markowsky, M. Wegman
In this paper we consider the question of how much space is needed to represent a set. Given a finite universe U and some subset V (called the vocabulary), an exact membership tester is a procedure that for each element s in U determines if s is in V. An approximate membership tester is allowed to make mistakes: we require that the membership tester correctly accepts every element of V, but we allow it to also accept a small fraction of the elements of U - V.
{"title":"Exact and approximate membership testers","authors":"L. Carter, R. W. Floyd, John Gill, G. Markowsky, M. Wegman","doi":"10.1145/800133.804332","DOIUrl":"https://doi.org/10.1145/800133.804332","url":null,"abstract":"In this paper we consider the question of how much space is needed to represent a set. Given a finite universe U and some subset V (called the vocabulary), an exact membership tester is a procedure that for each element s in U determines if s is in V. An approximate membership tester is allowed to make mistakes: we require that the membership tester correctly accepts every element of V, but we allow it to also accept a small fraction of the elements of U - V.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124033367","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}
If &pgr; is a graph property, the general node(edge) deletion problem can be stated as follows: Find the minimum number of nodes(edges), whose deletion results in a subgraph satisfying property &pgr;. In this paper we show that if &pgr; belongs to a rather broad class of properties (the class of properties that are hereditary on induced subgraphs) then the node-deletion problem is NP-complete, and the same is true for several restrictions of it. For the same class of properties, requiring the remaining graph to be connected does not change the NP-complete status of the problem; moreover for a certain subclass, finding any "reasonable" approximation is also NP-complete. Edge-deletion problems seem to be less amenable to such generalizations. We show however that for several common properties (e.g. planar, outer-planar, line-graph, transitive digraph) the edge-deletion problem is NP-complete.
{"title":"Node-and edge-deletion NP-complete problems","authors":"M. Yannakakis","doi":"10.1145/800133.804355","DOIUrl":"https://doi.org/10.1145/800133.804355","url":null,"abstract":"If &pgr; is a graph property, the general node(edge) deletion problem can be stated as follows: Find the minimum number of nodes(edges), whose deletion results in a subgraph satisfying property &pgr;. In this paper we show that if &pgr; belongs to a rather broad class of properties (the class of properties that are hereditary on induced subgraphs) then the node-deletion problem is NP-complete, and the same is true for several restrictions of it. For the same class of properties, requiring the remaining graph to be connected does not change the NP-complete status of the problem; moreover for a certain subclass, finding any \"reasonable\" approximation is also NP-complete. Edge-deletion problems seem to be less amenable to such generalizations. We show however that for several common properties (e.g. planar, outer-planar, line-graph, transitive digraph) the edge-deletion problem is NP-complete.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126297989","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}
Systems of nonlinear equations of the form D: Ay = σ.(x), where A is an m×n matrix of rational constants and y = (Y1,...,yn), σ(x) = (σ1(x),..., σm (x)) are column vectors are considered. Each σi(x) is of the form ri(x) or @@@@ri(x)@@@@, where ri(x) is a rational function of x with rational coefficients. It is shown that the problem of determining for a given system D whether there exists a nonnegative integral solution (y1,...,yn,X) satisfying D is decidable. In fact, the problem is NP-complete when restricted to systems D in which the maximum degree of the polynomials defining the σi(x)'s is bounded by some fixed polynomial in the length of the representation of D. Some recent results connecting Diophantine equations and counter machines are briefly mentioned.
{"title":"An NP-complete number-theoretic problem","authors":"E. Gurari, O. Ibarra","doi":"10.1145/800133.804349","DOIUrl":"https://doi.org/10.1145/800133.804349","url":null,"abstract":"Systems of nonlinear equations of the form D: Ay = σ.(x), where A is an m×n matrix of rational constants and y = (Y1,...,yn), σ(x) = (σ1(x),..., σm (x)) are column vectors are considered. Each σi(x) is of the form ri(x) or @@@@ri(x)@@@@, where ri(x) is a rational function of x with rational coefficients. It is shown that the problem of determining for a given system D whether there exists a nonnegative integral solution (y1,...,yn,X) satisfying D is decidable. In fact, the problem is NP-complete when restricted to systems D in which the maximum degree of the polynomials defining the σi(x)'s is bounded by some fixed polynomial in the length of the representation of D. Some recent results connecting Diophantine equations and counter machines are briefly mentioned.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133557072","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}
Tarjan has given an algorithm for deciding isomorphism of two groups of order n (given as multiplication tables) which runs in O(n(log2n+O(1)) steps where n is the order of the groups. Tarjan uses the fact that a group of n is generated by log n elements. In this paper, we show that Tarjan's technique generalizes to isomorphism of quasigroups, latin squares, Steiner systems, and many graphs generated from these combinatorial objects.
{"title":"On the nlog n isomorphism technique (A Preliminary Report)","authors":"G. Miller","doi":"10.1145/800133.804331","DOIUrl":"https://doi.org/10.1145/800133.804331","url":null,"abstract":"Tarjan has given an algorithm for deciding isomorphism of two groups of order n (given as multiplication tables) which runs in O(n(log2n+O(1)) steps where n is the order of the groups. Tarjan uses the fact that a group of n is generated by log n elements. In this paper, we show that Tarjan's technique generalizes to isomorphism of quasigroups, latin squares, Steiner systems, and many graphs generated from these combinatorial objects.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124685703","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}
In this paper, we show that two widely different notions of program equivalence coincide for the language of recursive definitions with simplification rules. The first one is the now classical equivalence for fixed-point semantics. The other one is purely operational in nature and is much closer to a programmer's intuition of program equivalence.
{"title":"Operational and semantic equivalence between recursive programs.","authors":"J. Raoult, J. Vuillemin","doi":"10.1145/800133.804334","DOIUrl":"https://doi.org/10.1145/800133.804334","url":null,"abstract":"In this paper, we show that two widely different notions of program equivalence coincide for the language of recursive definitions with simplification rules. The first one is the now classical equivalence for fixed-point semantics. The other one is purely operational in nature and is much closer to a programmer's intuition of program equivalence.","PeriodicalId":313820,"journal":{"name":"Proceedings of the tenth annual ACM symposium on Theory of computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1978-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121353790","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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