We consider the convex hull $P_{varphi}(G)$ of all satisfying assignments of a given MSO formula $varphi$ on a given graph $G$. We show that there exists an extended formulation of the polytope $P_{varphi}(G)$ that can be described by $f(|varphi|,tau)cdot n$ inequalities, where $n$ is the number of vertices in $G$, $tau$ is the treewidth of $G$ and $f$ is a computable function depending only on $varphi$ and $tau.$ In other words, we prove that the extension complexity of $P_{varphi}(G)$ is linear in the size of the graph $G$, with a constant depending on the treewidth of $G$ and the formula $varphi$. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs. As a corollary of our main result, we obtain an analogous result % for the weaker MSO$_1$ logic on the wider class of graphs of bounded cliquewidth. Furthermore, we study our main geometric tool which we term the glued product of polytopes. While the glued product of polytopes has been known since the '90s, we are the first to show that it preserves decomposability and boundedness of treewidth of the constraint matrix. This implies that our extension of $P_varphi(G)$ is decomposable and has a constraint matrix of bounded treewidth; so far only few classes of polytopes are known to be decomposable. These properties make our extension useful in the construction of algorithms.
我们考虑给定图$G$上给定MSO公式$varphi$的所有满足赋值的凸包$P_{varphi}(G)$。我们证明了多面体$P_{varphi}(G)$的一个扩展公式可以用$f(|varphi|,tau)cdot n$不等式来描述，其中$n$是$G$中的顶点数，$tau$是$G$的树宽，$f$是一个仅依赖于$varphi$和$tau.$的可计算函数。我们证明了$P_{varphi}(G)$的扩展复杂度在图$G$的大小上是线性的，并且取决于$G$的树宽和公式$varphi$。这提供了一个非常一般但又非常简单的元定理，关于与广泛的问题和图相关的多面体的可拓复杂性。作为我们主要结果的一个推论，我们得到了一个类似的结果 % for the weaker MSO$_1$ logic on the wider class of graphs of bounded cliquewidth. Furthermore, we study our main geometric tool which we term the glued product of polytopes. While the glued product of polytopes has been known since the '90s, we are the first to show that it preserves decomposability and boundedness of treewidth of the constraint matrix. This implies that our extension of $P_varphi(G)$ is decomposable and has a constraint matrix of bounded treewidth; so far only few classes of polytopes are known to be decomposable. These properties make our extension useful in the construction of algorithms.
{"title":"Extension Complexity, MSO Logic, and Treewidth","authors":"P. Kolman, Martin Koutecký, Hans Raj Tiwary","doi":"10.23638/DMTCS-22-4-8","DOIUrl":"https://doi.org/10.23638/DMTCS-22-4-8","url":null,"abstract":"We consider the convex hull $P_{varphi}(G)$ of all satisfying assignments of a given MSO formula $varphi$ on a given graph $G$. We show that there exists an extended formulation of the polytope $P_{varphi}(G)$ that can be described by $f(|varphi|,tau)cdot n$ inequalities, where $n$ is the number of vertices in $G$, $tau$ is the treewidth of $G$ and $f$ is a computable function depending only on $varphi$ and $tau.$ \u0000In other words, we prove that the extension complexity of $P_{varphi}(G)$ is linear in the size of the graph $G$, with a constant depending on the treewidth of $G$ and the formula $varphi$. This provides a very general yet very simple meta-theorem about the extension complexity of polytopes related to a wide class of problems and graphs. As a corollary of our main result, we obtain an analogous result % for the weaker MSO$_1$ logic on the wider class of graphs of bounded cliquewidth. \u0000Furthermore, we study our main geometric tool which we term the glued product of polytopes. While the glued product of polytopes has been known since the '90s, we are the first to show that it preserves decomposability and boundedness of treewidth of the constraint matrix. This implies that our extension of $P_varphi(G)$ is decomposable and has a constraint matrix of bounded treewidth; so far only few classes of polytopes are known to be decomposable. These properties make our extension useful in the construction of algorithms.","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127716382","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 : 2015-04-14DOI: 10.4230/LIPIcs.SWAT.2016.13
Andreas Björklund
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k-epsilon)^pw(G)poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between k-colorable graphs having at most s proper k-colorings and non-k-colorable graphs. We also show how to obtain a k-coloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them. We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is k-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a non-zero value if the graph has few k-colorings. Yet every non-k-colorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases.
Lokshtanov, Marx, and Saurabh SODA 2011证明，对于任意常数epsilon>0，除非强指数时间假设(SETH)为假，否则不存在(k-epsilon)^pw(G)多(n)时间算法来决定路径宽度为pw的n顶点图G是否允许k色的适当顶点着色。然而，我们在这里证明，当k> 1 /2 r + 1，其中是图G中的最大度，有一个更好的算法，至少当有很少的着色时。我们提出了一种蒙特卡罗算法，该算法给出了一个图G以及路径宽度为pw(G)的G的路径分解，其运行时间为(lfloor Delta/ 2rfloor + 1)^pw(G)poly(n)s，该算法区分了k个可色图，最多有s个适当的k色图和非k个可色图。我们还展示了如何在相同的渐近运行时间内获得k着色。我们的算法避免了违反SETH的一个，因为高度顶点仍然花费太多，而且前面提到的硬度构造使用了很多顶点。我们利用了著名的Alon- Tarsi定理的一个新变体，它比原始形式具有算法优势。最初的定理表明，一个图在每个顶点都有一个外度小于k的方向，只有当图是k色时，才有不同数量的奇偶欧拉子图，但是没有已知的有效方法来找到这样一个方向。我们的新形式表明，如果我们在随机均匀选择的每个顶点上计算满足模度约束的偶数和奇数子图的另一个差值，如果图有很少的k色，我们有一个公平的机会得到一个非零值。然而，每个非k色图给出的差值都为零，因此一组随机的约束条件很有可能用于分离这两种情况。
{"title":"Coloring Graphs Having Few Colorings Over Path Decompositions","authors":"Andreas Björklund","doi":"10.4230/LIPIcs.SWAT.2016.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.SWAT.2016.13","url":null,"abstract":"Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k-epsilon)^pw(G)poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between k-colorable graphs having at most s proper k-colorings and non-k-colorable graphs. We also show how to obtain a k-coloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them. \u0000 \u0000We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is k-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a non-zero value if the graph has few k-colorings. Yet every non-k-colorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases.","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114902199","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 : 2014-07-02DOI: 10.1007/978-3-319-08404-6_1
P. Afshani, Nodari Sitchinava
{"title":"I/O-Efficient Range Minima Queries","authors":"P. Afshani, Nodari Sitchinava","doi":"10.1007/978-3-319-08404-6_1","DOIUrl":"https://doi.org/10.1007/978-3-319-08404-6_1","url":null,"abstract":"","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117213884","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 : 2014-07-02DOI: 10.1007/978-3-319-08404-6_23
Boris Klemz, T. Mchedlidze, M. Nöllenburg
{"title":"Minimum Tree Supports for Hypergraphs and Low-Concurrency Euler Diagrams","authors":"Boris Klemz, T. Mchedlidze, M. Nöllenburg","doi":"10.1007/978-3-319-08404-6_23","DOIUrl":"https://doi.org/10.1007/978-3-319-08404-6_23","url":null,"abstract":"","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126547588","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 : 2014-07-02DOI: 10.1007/978-3-319-08404-6_20
R. Grossi, Søren Vind
{"title":"Colored Range Searching in Linear Space","authors":"R. Grossi, Søren Vind","doi":"10.1007/978-3-319-08404-6_20","DOIUrl":"https://doi.org/10.1007/978-3-319-08404-6_20","url":null,"abstract":"","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"158 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114057927","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 : 2014-07-02DOI: 10.1007/978-3-319-08404-6_18
Toshihiro Fujito
{"title":"On Matchings and b-Edge Dominating Sets: A 2-Approximation Algorithm for the 3-Edge Dominating Set Problem","authors":"Toshihiro Fujito","doi":"10.1007/978-3-319-08404-6_18","DOIUrl":"https://doi.org/10.1007/978-3-319-08404-6_18","url":null,"abstract":"","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128702974","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 : 2014-07-02DOI: 10.1007/978-3-319-08404-6_3
D. Belazzougui, G. Brodal, J. Nielsen
{"title":"Expected Linear Time Sorting for Word Size Ω(log2 n loglogn)","authors":"D. Belazzougui, G. Brodal, J. Nielsen","doi":"10.1007/978-3-319-08404-6_3","DOIUrl":"https://doi.org/10.1007/978-3-319-08404-6_3","url":null,"abstract":"","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121199966","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 : 2014-07-02DOI: 10.1007/978-3-319-08404-6_31
Anup Joshi, N. Narayanaswamy
{"title":"Approximation Algorithms for Hitting Triangle-Free Sets of Line Segments","authors":"Anup Joshi, N. Narayanaswamy","doi":"10.1007/978-3-319-08404-6_31","DOIUrl":"https://doi.org/10.1007/978-3-319-08404-6_31","url":null,"abstract":"","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124126027","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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