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...ng the chromatic index problem on specific input graphs using a simple mathematical model. 2.1 The chromatic index problem Let G be a simple graph. A proper edge coloring (we suppose all colorings are proper throughout the article) of G is such that no two adjacent edges are assigned the same color. The chromatic index χ′ is the minimum number of colors such that there exists an edge coloring of G. Vizing’s theorem [18, 5] states that the chromatic index is either ∆ or ∆ + 1, where ∆ is the maximum degree of a vertex in G. The problem of determining whether χ′ equals ∆ or ∆ + 1 is NP-complete [7], even if ∆ = 3. 2.2 Input graphs We consider only graphs with ∆ = 3. The Petersen graph [14, 6], shown in Figure 1a, is a graph that has ∆ = 3 and χ′ = 4. We build two graphs similar to the Petersen graph, P1 and P2 (see Figures 1b and 1c), in such a way that only P1 retains the property that χ′ = 4. Lemma 1. The chromatic index of P1 is 4, and the chromatic index of P2 is 3. Proof. Consider the inner C5 (cycle graph on 5 vertices) of the Petersen graph as shown in Figure 1a. Without loss of generality, color two adjacent edges of this C5 with colors 1 and 2. Its only 3-edge-coloring is then ...

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...(b) Graph P1 (c) Graph P2 Figure 1: The Petersen graph and two variations, P1 and P2. The first two graphs have χ′ = 4, while P2 has χ′ = 3. 2 Instances We build IP instances encoding the chromatic index problem on specific input graphs using a simple mathematical model. 2.1 The chromatic index problem Let G be a simple graph. A proper edge coloring (we suppose all colorings are proper throughout the article) of G is such that no two adjacent edges are assigned the same color. The chromatic index χ′ is the minimum number of colors such that there exists an edge coloring of G. Vizing’s theorem [18, 5] states that the chromatic index is either ∆ or ∆ + 1, where ∆ is the maximum degree of a vertex in G. The problem of determining whether χ′ equals ∆ or ∆ + 1 is NP-complete [7], even if ∆ = 3. 2.2 Input graphs We consider only graphs with ∆ = 3. The Petersen graph [14, 6], shown in Figure 1a, is a graph that has ∆ = 3 and χ′ = 4. We build two graphs similar to the Petersen graph, P1 and P2 (see Figures 1b and 1c), in such a way that only P1 retains the property that χ′ = 4. Lemma 1. The chromatic index of P1 is 4, and the chromatic index of P2 is 3. Proof. Consider the inner C5 (cycle graph o...

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...ing variables by exhibiting a family of instances for which an optimal solution is both trivial to find and provably optimal by a fixed-size branch-and-bound tree, but for which state-of-the-art Mixed Integer Programming solvers need an increasing amount of resources. The instances encode the edge-coloring problem on a family of graphs containing a small subgraph requiring four colors, while the rest of the graph requires only three. 1 Introduction Mixed Integer Programming (MIP) solvers depend on branching rules to implicitly search the solution space. Numerous experimental results (see e.g. [2]) provide a good notion of their performances. However, little literature has been dedicated to theoretical results on MIP branching. By contrast, branching in satisfiability (SAT) solvers have been studied in a theoretical setting. Liberatore [10] has proven that choosing a branching candidate that minimizes the tree size is NP-hard. Ouyang [13] provides a family of instances of increasing sizes for which a fixed-size search tree exists. Unfortunately, the findings of Ouyang [13] do not translate in the context of MIP solving, since the given SAT instances – once written in an appropriate for...

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...ese instances are clearly harder to solve than the ones with χ′ = 4, i.e. closing the dual gap is easier than closing the primal gap. Indeed, for these instances, there is a trivial linear-size B&B tree, since there exists a 3-coloring, which objective value matches the dual bound at the root, but it is not clear if a fixed-size tree exists. So in this respect, MIP solvers are not completely fooled. 3.3 Branching rule analysis We study the behavior of the default branching rule of the open-source code SCIP, because we know exactly how it performs. It is called reliability pseudocost branching [4, 2, 3], and mostly relies on strong branching and pseudocost branching. Strong branching assigns high scores to variables producing high dual bounds (when minimizing) in the children of the current node. Pseudocost branching is a history-based rule that tries to predict the score that strong branching would compute. It is an inexpensive alternative to strong branching, but it needs a history of previous branchings to be reliable. In reliability 5 CPLEX GUROBI SCIP Size (k) s n t s n t s n t 1 10 0 0 10 0 0 10 1 0 2 10 0 0 10 0 0 10 1 0 4 10 1 0 10 0 0 10 2 1 8 10 0 0 10 0 0 10 12 4 16 10 144 2 10 17...

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...- strong branching inference Size (k) s n t s n t s n t 1 0 +17 - 0 0 - 0 +708 - 2 0 -21 0 0 +16 0 0 +658 -100 4 0 -56 -25 0 -2 0 0 +559 -75 8 0 -70 -10 0 +18 +10 0 +257 -70 16 0 -46 +17 0 -24 -9 0 +41 -70 32 0 +102 +69 0 +29 -2 0 +51 -63 64 +1 -34 +17 +1 +48 +17 +1 +201 -16 128 0 -76 -13 -2 -46 -40 0 +137 +29 256 +1 -52 +60 -2 +16 -2 -3 +295 +75 512 +1 -59 +57 0 +15 -14 -3 +47 -38 1024 0 - - 0 - - 0 - - Table 4: Parameter tuning with SCIP, with the same notations as Table 3. 9 CPLEX, we have also run the tests using mip strategy probe (set to 3. Probing is a preprocessing technique, see e.g. [15]). The results of these three distinct parameter changes are reported in Table 3. Based on the analysis, we have tried more specific parameters in SCIP. The first test removes all limits (e.g. LP iterations) to the resources allocated to conflict analysis, and enables it in all implemented cases. The second test reduces the extent of strong branching (namely, the maximum number of branching candidates investigated and the total LP iterations are reduced by a factor of 5 and 2 respectively). The third test uses inference branching [2], a SAT solving rule that does not take the objective functio...

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....1 The chromatic index problem Let G be a simple graph. A proper edge coloring (we suppose all colorings are proper throughout the article) of G is such that no two adjacent edges are assigned the same color. The chromatic index χ′ is the minimum number of colors such that there exists an edge coloring of G. Vizing’s theorem [18, 5] states that the chromatic index is either ∆ or ∆ + 1, where ∆ is the maximum degree of a vertex in G. The problem of determining whether χ′ equals ∆ or ∆ + 1 is NP-complete [7], even if ∆ = 3. 2.2 Input graphs We consider only graphs with ∆ = 3. The Petersen graph [14, 6], shown in Figure 1a, is a graph that has ∆ = 3 and χ′ = 4. We build two graphs similar to the Petersen graph, P1 and P2 (see Figures 1b and 1c), in such a way that only P1 retains the property that χ′ = 4. Lemma 1. The chromatic index of P1 is 4, and the chromatic index of P2 is 3. Proof. Consider the inner C5 (cycle graph on 5 vertices) of the Petersen graph as shown in Figure 1a. Without loss of generality, color two adjacent edges of this C5 with colors 1 and 2. Its only 3-edge-coloring is then 1, 2, 1, 2, 3, and there is a unique 3-edge-coloring of the edges between the inner C5 and the o...

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...(b) Graph P1 (c) Graph P2 Figure 1: The Petersen graph and two variations, P1 and P2. The first two graphs have χ′ = 4, while P2 has χ′ = 3. 2 Instances We build IP instances encoding the chromatic index problem on specific input graphs using a simple mathematical model. 2.1 The chromatic index problem Let G be a simple graph. A proper edge coloring (we suppose all colorings are proper throughout the article) of G is such that no two adjacent edges are assigned the same color. The chromatic index χ′ is the minimum number of colors such that there exists an edge coloring of G. Vizing’s theorem [18, 5] states that the chromatic index is either ∆ or ∆ + 1, where ∆ is the maximum degree of a vertex in G. The problem of determining whether χ′ equals ∆ or ∆ + 1 is NP-complete [7], even if ∆ = 3. 2.2 Input graphs We consider only graphs with ∆ = 3. The Petersen graph [14, 6], shown in Figure 1a, is a graph that has ∆ = 3 and χ′ = 4. We build two graphs similar to the Petersen graph, P1 and P2 (see Figures 1b and 1c), in such a way that only P1 retains the property that χ′ = 4. Lemma 1. The chromatic index of P1 is 4, and the chromatic index of P2 is 3. Proof. Consider the inner C5 (cycle graph o...

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...se strong branching tests a particular candidate variable: either both child LPs have value 3, or at least one of them cannot lead to a strictly better feasible solution (since the objective function is integral and a solution of value 4 is known). Such a child is considered infeasible, even if its LP has objective value in )3, 4], and pseudocosts are then not updated. Pseudocosts are thus systematically 0 for all variables throughout all runs, and therefore provide no information on branching candidates. The first tie-breaking strategy of reliability pseudocost branching is conflict analysis [1]. Conflict analysis extracts information from infeasible nodes by considering all variable bound changes (typically, branching decisions) leading to that node, and by finding a small subset of these changes such that the resulting subproblem is still infeasible. The conflict score of a variable increases when it is involed in a newly created conflict, and it decays otherwise. At the root node, heuristics may produce some conflicts, but conflict analysis will still mostly be uninitialized, leading to a branching decision that – in our experiments – systematically varies depending on the row per...

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...ese instances are clearly harder to solve than the ones with χ′ = 4, i.e. closing the dual gap is easier than closing the primal gap. Indeed, for these instances, there is a trivial linear-size B&B tree, since there exists a 3-coloring, which objective value matches the dual bound at the root, but it is not clear if a fixed-size tree exists. So in this respect, MIP solvers are not completely fooled. 3.3 Branching rule analysis We study the behavior of the default branching rule of the open-source code SCIP, because we know exactly how it performs. It is called reliability pseudocost branching [4, 2, 3], and mostly relies on strong branching and pseudocost branching. Strong branching assigns high scores to variables producing high dual bounds (when minimizing) in the children of the current node. Pseudocost branching is a history-based rule that tries to predict the score that strong branching would compute. It is an inexpensive alternative to strong branching, but it needs a history of previous branchings to be reliable. In reliability 5 CPLEX GUROBI SCIP Size (k) s n t s n t s n t 1 10 0 0 10 0 0 10 1 0 2 10 0 0 10 0 0 10 1 0 4 10 1 0 10 0 0 10 2 1 8 10 0 0 10 0 0 10 12 4 16 10 144 2 10 17...

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... x i e ≤ ci ∀v ∈ V, ∀i ∈ C ci ∈ {0, 1} ∀i ∈ C xie ∈ {0, 1} ∀e ∈ E, ∀i ∈ C Variables ci encode whether color i is used or not. Variables xie encode whether color i is assigned to edge e. The objective function minimizes the number of colors used. The second set of constraints ensures each edge is assigned a color. The third set of constraints enforces a proper coloring. This formulation has (∆ + 1)(|E|+ 1) variables, |E|+ (∆ + 1)|V |constraints and (∆ + 1)(3|E|+ |V |) non-zero coefficients. Note that more elaborated MIP techniques have been designed to solve the edge-coloring problem (see e.g. [12, 8, 9]). However, this simple model is sufficient for our purposes. 3 3 Fooling MIP solvers Let Ik be the IP instances resulting from applying the model in Section 2.3 to the graphs Gk given in Section 2.2. In this section we study the behavior of MIP solvers on the instances Ik as k grows. In these instances, finding the optimal solution is trivial. Indeed, every MIP solver we have tested finds the optimal solution at the root of the B&B tree. This is not surprising, as finding a 4-edgecoloring to the graphs Gk can be done in linear time [17]. Thus we assume that the optimal solution is found or av...

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... amount of resources. The instances encode the edge-coloring problem on a family of graphs containing a small subgraph requiring four colors, while the rest of the graph requires only three. 1 Introduction Mixed Integer Programming (MIP) solvers depend on branching rules to implicitly search the solution space. Numerous experimental results (see e.g. [2]) provide a good notion of their performances. However, little literature has been dedicated to theoretical results on MIP branching. By contrast, branching in satisfiability (SAT) solvers have been studied in a theoretical setting. Liberatore [10] has proven that choosing a branching candidate that minimizes the tree size is NP-hard. Ouyang [13] provides a family of instances of increasing sizes for which a fixed-size search tree exists. Unfortunately, the findings of Ouyang [13] do not translate in the context of MIP solving, since the given SAT instances – once written in an appropriate format – are trivially solved by current MIP solvers. Our study is analogous to Ouyang’s, but in a MIP setting. We design a family of increasingly large IP instances encoding the edge-coloring problem. These instances have trivial feasible solutions. ...

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...igned to solve the edge-coloring problem (see e.g. [12, 8, 9]). However, this simple model is sufficient for our purposes. 3 3 Fooling MIP solvers Let Ik be the IP instances resulting from applying the model in Section 2.3 to the graphs Gk given in Section 2.2. In this section we study the behavior of MIP solvers on the instances Ik as k grows. In these instances, finding the optimal solution is trivial. Indeed, every MIP solver we have tested finds the optimal solution at the root of the B&B tree. This is not surprising, as finding a 4-edgecoloring to the graphs Gk can be done in linear time [17]. Thus we assume that the optimal solution is found or available at the root node. Furthermore, for any k ≥ 1, the continuous relaxation of Ik has value 3 [16, p. 63], which means the absolute gap is 1 at the root. The B&B task is thus to close that gap. Note that the objective function is integral, thus a node with an LP optimal value greater than 3 can be fathomed. We will first prove that there exists for all k ≥ 1 a fixed-size B&B tree that proves optimality for Ik. We will then produce experimental results using SCIP, CPLEX and GUROBI, showing that each of them needs an increasing amount ...

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...g a small subgraph requiring four colors, while the rest of the graph requires only three. 1 Introduction Mixed Integer Programming (MIP) solvers depend on branching rules to implicitly search the solution space. Numerous experimental results (see e.g. [2]) provide a good notion of their performances. However, little literature has been dedicated to theoretical results on MIP branching. By contrast, branching in satisfiability (SAT) solvers have been studied in a theoretical setting. Liberatore [10] has proven that choosing a branching candidate that minimizes the tree size is NP-hard. Ouyang [13] provides a family of instances of increasing sizes for which a fixed-size search tree exists. Unfortunately, the findings of Ouyang [13] do not translate in the context of MIP solving, since the given SAT instances – once written in an appropriate format – are trivially solved by current MIP solvers. Our study is analogous to Ouyang’s, but in a MIP setting. We design a family of increasingly large IP instances encoding the edge-coloring problem. These instances have trivial feasible solutions. Moreover, we prove that there exists for each instance (of arbitrary size) a fixed-size branch-and-b...

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...implied by Ik, the LP optimum at each leaf of T ′ is greater or equal than the LP optimum at its counterpart leaf in T . 3.2 Experimental results We present numerical results for three solvers: CPLEX 12.6.0.0, GUROBI 5.6.0 and SCIP 3.1.0 (with the CPLEX LP solver). All experiments have been conducted on the same machine, with a single thread per solver. Instance Ik has 72k variables, 64k constraints and 264k non-zero coefficients, up to small constants. For each instance Ik, ten equivalent programs have been generated by random row permutations (to reduce the impact of performance variability [11]). Table 1 reports the number of instances solved, together with the number of nodes and the time in seconds used by each solver. 4 CPLEX GUROBI SCIP Size (k) s n t s n t s n t 1 10 11 0 10 21 0 10 12 0 2 10 15 0 10 23 0 10 19 1 4 10 38 0 10 30 1 10 41 4 8 10 59 0 10 50 3 10 79 10 16 9 302 3 10 84 15 10 263 23 32 7 213 11 10 175 47 10 419 48 64 9 50 26 10 1921 424 9 1328 178 128 8 276 79 7 1470 1098 10 6542 808 256 6 1366 564 7 699 4182 8 6225 2041 512 2 3265 1700 7 198 3586 6 6125 6347 1024 2 1509 5501 3 112 16943 0 - - Table 1: Number of instances solved (s), and, for the instances solved, t...

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... x i e ≤ ci ∀v ∈ V, ∀i ∈ C ci ∈ {0, 1} ∀i ∈ C xie ∈ {0, 1} ∀e ∈ E, ∀i ∈ C Variables ci encode whether color i is used or not. Variables xie encode whether color i is assigned to edge e. The objective function minimizes the number of colors used. The second set of constraints ensures each edge is assigned a color. The third set of constraints enforces a proper coloring. This formulation has (∆ + 1)(|E|+ 1) variables, |E|+ (∆ + 1)|V |constraints and (∆ + 1)(3|E|+ |V |) non-zero coefficients. Note that more elaborated MIP techniques have been designed to solve the edge-coloring problem (see e.g. [12, 8, 9]). However, this simple model is sufficient for our purposes. 3 3 Fooling MIP solvers Let Ik be the IP instances resulting from applying the model in Section 2.3 to the graphs Gk given in Section 2.2. In this section we study the behavior of MIP solvers on the instances Ik as k grows. In these instances, finding the optimal solution is trivial. Indeed, every MIP solver we have tested finds the optimal solution at the root of the B&B tree. This is not surprising, as finding a 4-edgecoloring to the graphs Gk can be done in linear time [17]. Thus we assume that the optimal solution is found or av...

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... x i e ≤ ci ∀v ∈ V, ∀i ∈ C ci ∈ {0, 1} ∀i ∈ C xie ∈ {0, 1} ∀e ∈ E, ∀i ∈ C Variables ci encode whether color i is used or not. Variables xie encode whether color i is assigned to edge e. The objective function minimizes the number of colors used. The second set of constraints ensures each edge is assigned a color. The third set of constraints enforces a proper coloring. This formulation has (∆ + 1)(|E|+ 1) variables, |E|+ (∆ + 1)|V |constraints and (∆ + 1)(3|E|+ |V |) non-zero coefficients. Note that more elaborated MIP techniques have been designed to solve the edge-coloring problem (see e.g. [12, 8, 9]). However, this simple model is sufficient for our purposes. 3 3 Fooling MIP solvers Let Ik be the IP instances resulting from applying the model in Section 2.3 to the graphs Gk given in Section 2.2. In this section we study the behavior of MIP solvers on the instances Ik as k grows. In these instances, finding the optimal solution is trivial. Indeed, every MIP solver we have tested finds the optimal solution at the root of the B&B tree. This is not surprising, as finding a 4-edgecoloring to the graphs Gk can be done in linear time [17]. Thus we assume that the optimal solution is found or av...